*466*
*78*
*3MB*

*English*
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*Year 2016*

- Author / Uploaded
- Carla C. Morris
- Robert M. Stark

*Table of contents : CoverTitle PageCopyrightContentsChapter 1 Linear Equations and FunctionsChapter 2 The DerivativeChapter 3 Using the DerivativeChapter 4 Exponential and Logarithmic FunctionsChapter 5 Techniques of DifferentiationChapter 6 Integral CalculusChapter 7 Integration TechniquesChapter 8 Functions of Several VariablesChapter 9 Series and SummationsChapter 10 Applications to ProbabilityEULA*

SOLUTIONS MANUAL TO ACCOMPANY FUNDAMENTALS OF CALCULUS

SOLUTIONS MANUAL TO ACCOMPANY FUNDAMENTALS OF CALCULUS CARLA C. MORRIS University of Delaware

ROBERT M. STARK University of Delaware

Copyright © 2016 by John Wiley & Sons, Inc. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Morris, Carla C. Fundamentals of Calculus / Carla C. Morris, Robert M. Stark. pages cm Includes bibliographical references and index. ISBN 978-1-119-01526-0 (cloth) 1. Calculus–Textbooks. I. Stark, Robert M., 1930- II. Title. QA303.2.M67 2015 515–dc23 2014042182 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

CONTENTS

1 LINEAR EQUATIONS AND FUNCTIONS

1

2 THE DERIVATIVE

16

3 USING THE DERIVATIVE

26

4 EXPONENTIAL AND LOGARITHMIC FUNCTIONS

39

5 TECHNIQUES OF DIFFERENTIATION

46

6 INTEGRAL CALCULUS

56

7 INTEGRATION TECHNIQUES

66

8 FUNCTIONS OF SEVERAL VARIABLES

79

9 SERIES AND SUMMATIONS

94

10 APPLICATIONS TO PROBABILITY

110

v

CHAPTER 1

LINEAR EQUATIONS AND FUNCTIONS

EXERCISES 1.1 1. .3x + 1 = 4x − 5 1 = x–5

conditional equation

x=6 3. .5(x + 1) + 2(x − 1) = 7x + 6 5x + 5 + 2x − 2 = 7x + 6 7x + 3 = 7x + 6

contradiction

5. .4(x + 3) = 2(2x + 5) 4x + 12 = 4x + 10

contradiction

7. .5x − 3 = 17 5x = 20 x=4

Solutions Manual to Accompany Fundamentals of Calculus, First Edition. Carla C. Morris and Robert M. Stark. © 2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

1

2

LINEAR EQUATIONS AND FUNCTIONS

9. .

2x = 4x − 10

2x − 4x = −10 −2x = −10 x=5 11. .4x − 5 = 6x − 7 −5 + 7 = 6x − 4x 2 = 2x 1=x 13. .0.6x = 30 30 = 50 0.60 ( ) 4 1 2 15. . = x− 3 5 3 ( ) {( ) } 2 4 1 15 = 15 x− 3 5 3 10 = 12x − 5 x=

15 = 12x 5 =x 4 17. .5(x − 4) = 2x + 3(x − 7) 5x − 20 = 2x + 3x − 21 5x − 20 = 5x − 2 19. .3s − 4 = 2s + 6 s−4=6 s = 10 21. .7t + 2 = 4t + 11 7t − 4t = 11 − 2 3t = 9 t=3

No solution

EXERCISES 1.1

23. .4(x + 1) + 2(x − 3) = 7(x − 1) 4x + 4 + 2x − 6 = 7x − 7 6x − 2 = 7x − 7 6x − 7x = −7 + 2 −x = −5 x=5 25. . x + 8 = 2 2x − 5 (x + 8) = 2(2x − 5) x + 8 = 4x − 10 8 + 10 = 4x − x 18 = 3x 6=x 27. .8 − {4[x − (3x − 4) − x] + 4} = 3(x + 2) 8 − {4[x − 3x + 4 − x] + 4} = 3x + 6 8 − {4[−3x + 4] + 4} = 3x + 6 8 − {−12x + 16 + 4} = 3x + 6 8 − {−12x + 20) = 3x + 6 8 + 12x − 20 = 3x + 6 12x − 12 = 3x + 6 9x = 18 x=2 29. .6x − 3y = 9 for x 6x = 3y + 9 x=

3y + 9 1 3 = y+ 6 2 2

31. .3x + 5y = 15 5y = 15 − 3x

3

4

LINEAR EQUATIONS AND FUNCTIONS

(15 − 3x) 5 ( ) 3 x y=3− 5

y=

33. . V = LWH V =W LH 35. . Z = (x − 𝜇) 𝜎 Z𝜎 = x − 𝜇 x = Z𝜎 + 𝜇 37. Let x = monthly installment ($). Since Sally paid $300, she owes $1300 − $300 = $1000. Therefore, 5x = 1000 or x = $200 monthly installment. 39. The consumption function is C(x) = mx + b. The slope is the “marginal propensity to consume.” Therefore, C(x) = 0.75x + b. The disposable income x = 2 for a consumption y = 11 yields 11 = (0.75)(2) + b, so b = 9.5 and consumption is C(x) = 0.75x + 9.5. 41. .a) d = 4.5(2) = 9 miles b) 18 = 4.5t and t = 18∕4.5 = 4 seconds 43. The tax is 6.2%, or 0.062 as a decimal form, so T = 0.062x, where 0 ≤ x ≤ 87,000. 45. .a) BSA = 1321 + (0.3433)(20,000) = 8187 cm2 b) 1. 330 = 1321 + (0.3433)(Wt) 9 = (0.3433)(Wt) 9 ∕ 0.3433 = 26.2 kg = Wt.

EXERCISES 1.2 1. Setting y = 0 determines the x-intercept and setting x = 0 determines the y-intercept. a) 5x − 3y = 15 x-intercept 3, y-intercept −5 b) y = 4x − 5 x-intercept 5/4, y-intercept −5 c) 2x + 3y = 24 x-intercept 12, y-intercept 8

5

EXERCISES 1.2

d) 9x − y = 18 e) x = 4 f) y = −2

x-intercept 2, y-intercept −18 x-intercept 4, no y-intercept(vertical line) no x-intercept (horizontal line), y-intercept −2 y2 − y 1 3. The slope is m = x2 − x1 4−6 −2 1 a) (3, 6) and (−1, 4) m= = = −1 − 3 −4 2 11 − 6 5 b) (1, 6) and (2, 11) m= = =5 2−1 1 7−3 4 2 c) (6, 3) and (12, 7) m= = = 12 − 6 6 3 7−3 4 d) (2, 3) and (2, 7) m= = undefined 2−2 0 6−6 0 e) (2, 6) and (5, 6) m= = =0 5−2 3 1∕3 1 − 2∕3 1 = = f) (5/3, 2/3) and (10/3, 1) m = 10 5∕3 ∕3 − 5∕3 5 5. a) . . x-intercept 5/2 and y-intercept b) x-intercept 4 and no y-intercept y −5 4

y 6

3 2

4

y = 2x – 5

1

2 0

0 –1 0 –1

x

–2 –1 0 –2

1

2

4

3

x=4

5

x 1

2

3

4

5

–2 –3

–4 –6

c) x-intercept 5 and y-intercept 3

d) x-intercept 7 and y-intercept 2

y

y

6

8 6

3

4

3x + 5y = 15

2

0 –5

0 –3

5

x 10

2x + 7y = 14

0 –14

–7

–2 –4 –6

0

7

14

x 21

6

LINEAR EQUATIONS AND FUNCTIONS

7. .a) For y = (5∕3)x + 2 and 5x − 3y = 10; the slope of the first line is 5/3. Solving for y in the second equation yields y = (5∕3)x − (10∕3). This slope is also 5/3. The lines are parallel (same slope, different intercepts). b) 6x + 2y = 4 and y = (1∕3)x + 1. The slope of the second line is apparent from the slope intercept form as 1/3. Solve for y in the first equation as y = −3x + 2. The slope is −3. The slopes are negative reciprocals, so the lines are perpendicular. c) 2x − 3y = 6 and 4x − 6y = 15. Solving for y in each equation, yields y = (2∕3)x − 2 and y = (2∕3)x − (5∕2). These lines have the same slope (and different intercepts) so they are parallel. d) y = 5x − 4 and 3x − y = 4. The slope of the first line is 5. In the second equation, (y = 3x − 4), the slope is 3. These slopes are neither the same nor negative reciprocals. The lines are neither parallel nor perpendicular. e) y = 5 is a horizontal line while x = 3 is a vertical line. The two lines are perpendicular. 9. Generally, lines have a single x-intercept. The exception y = 0 (the x-axis) with an infinite number of x-intercepts. Any horizontal line (except y = 0) has no x-intercepts. Generally, lines do not have more than one y-intercept. The exception x = 0 (the y-axis) with an infinite number of y-intercepts. Any vertical line (except x = 0) has no y-intercepts. 11. The ordered pairs with time and machine values are (0, 75,000) and 21,000 − 75,000 −54,000 = = −6000. (9, 21,000). The slope m = 9−0 9 The y-intercept is the initial cost, $75,000. Therefore, to model the straight-line depreciation V(t) = −6000t + 75, 000 where V(t) is the machine’s value ($) at time t. 13. Ordered pairs (gallons of gasoline, miles traveled) are (7, 245) and 420 − 245 175 = = 35. Let x = gallons of (12, 420). The slope is 12 − 7 5 gasoline and y = miles traveled. Then either ordered pair, with the point slope formula, yields y − 245 = 35(x − 7) or y = 35x. 15. Total cost is fixed plus variable costs. The fixed cost is monthly rent of $1100. The variable cost is 5x, where x is monthly production.

EXERCISES 1.3

7

The total cost is C(x) = 1100 + 5x. 17. .a) Here, fixed cost is ($50/day) and variable cost is ($0.28/mile). So, C(x) = 50 + 0.28x b) If one has $92, the equation for the travel distance is 92 = 50 + 0.28x Solving, 42 = 0.28x 42 =x 0.28 150 = x The car can be rented and driven 150 miles for $92. 19. Since R is a function of C, the ordered pairs (C, R) are (70, 84) and 48 − 84 36 6 (40, 48). The slope is = = . Either ordered pair deter40 − 70 30 5 mines the equation as the slope is known. Therefore, R − 84 = (6 ∕ 5)(C − 70) or, R = (6 ∕ 5)C.

EXERCISES 1.3 1. 3. 5. 7. 9. 11. 13. 15.

Here, the GCF is 8, so 8x − 24 = 8(x − 3) Here, the GCF is 5x, so 5x3 − 10x2 + 15x = 5x(x2 − 2x + 3) Here, the GCF is 5a3 bc3 , so 5a3 b2 c4 + 10a3 bc3 = 5a3 bc3 (bc + 2) Here, the GCF is 5x2 y3 z5 , so 20x3 y5 z6 + 15x4 y3 z7 + 20x2 y4 z5 = 5x2 y3 z5 (4xy2 z + 3x2 z2 + 4y) This is a difference of squares, so x2 − 25 = (x − 5)(x + 5) There is a GCF of 3 to yield 3(x2 + 9). A sum of squares is not factorable. There is a GCF of 2 to yield 2(x3 − 8). Next, using the difference of cubes formula the expression factors as 2(x − 2)(x2 + 2x + 4). There is a GCF of 7(a + b) to yield 7(a + b)(x2 − 4). Next, using the difference of squares formula yields 7(a + b)(x + 2)(x − 2).

8

LINEAR EQUATIONS AND FUNCTIONS

17. The last term is +4 and since the middle term is positive, one seeks two positive factors of 4 that add to 5. The expression factors as x2 + 5x + 4 = (x + 4)(x + 1). 19. The last term is positive, and since the middle term is positive, one seeks two positive factors of 1 that add to 3. This is not possible. Therefore, x2 + 3x + 1 is not factorable. 21. Here, the last term is negative so seek one positive factor and one negative factor of 16 that add to give −6. Therefore, x2 − 6x − 16 = (x − 8)(x + 2). 23. First, the GCF is 2 so 2x2 + 12x + 16 = 2(x2 + 6x + 8). Next, two positive factors of 8 that add to 6 are needed. The expression is completely factored as 2x2 + 12x + 16 = 2(x + 4)(x + 2). 25. Seek two positive factors of 20 that add to 9. The expression factors as a2 b2 + 9ab + 20 = (ab + 4)(ab + 5). 27. First, the GCF is 2 so 2x2 y2 + 28xy + 90 = 2(x2 y2 + 14xy + 45). Next, seek two positive factors of 45 that add to 14. The expression factors as 2(xy + 9)(xy + 5). 29. Seek two positive factors of 5 that add to 7. Since this is not possible, the expression x2 + 7x + 5 is prime. 31. This is a quadratic in x2 . Seek two negative factors of 4 that add to 5. Therefore, x4 − 5x2 + 4 = (x2 − 4)(x2 − 1). The factors are both differences of squares, so factoring yields x4 − 5x2 + 4 = (x − 2)(x + 2)(x − 1)(x + 1). 33. First, group the expression by powers to yield (x2 − a2 ) + (5x − 5a). Then, factor each pair to yield (x − a)(x + a) + 5(x − a). Next, a GCF of (x − a) is factored from the expression to yield (x − a)[(x + a) + 5]. 35. First, the GCF is 2, so 4ab − 8ax + 6by − 12xy = 2[2ab − 4ax + 3by − 6xy]. Group by pairs as 2[(2ab − 4ax) + (3by − 6xy)]. Factoring each pair yields 2[2a(b − 2x) + 3y(b − 2x)] = 2[(b − 2x)(2a + 3y)]. 37. Using a = 1, the quadratic formula yields √ b = 9, and c = 8 in √ 2 −(9) ± (9) − 4(1)(8) −9 ± 49 −9 ± 7 = = . So x = −8 or x= 2(1) 2 2 x = −1.

EXERCISES 1.4

9

39. Using a = 1,√b = 17, and c = 72 in the √ quadratic formula yields −(17) ± (17)2 − 4(1)(72) −17 ± 1 −17 ± 1 x= = = . 2(1) 2 2 So, x = −8 or x = −9. 41. Using a = 1, the quadratic formula yields √ b = 4, and c = 7 in √ −(4) ± (4)2 − 4(1)(7) −4 ± −12 x= = . There are no real 2(1) 2 solutions. 43. First, rewrite as x2 − 9x + 18 = 0. Next, using a = 1, b = −9, and c = 18 in the quadratic formula yields √ √ 2 −(−9) ± (−9) − 4(1)(18) 9 ± 9 9 ± 3 x= = = . So, x = 6 or 2(1) 2 2 x = 3. 45. Using a = 2, √ b = −3, and c = 1 in the √ quadratic formula yields 2 −(−3) ± (−3) − 4(2)(1) 3 ± 1 x= = . So, x = 1 or x = 1 ∕ 2. 2(2) 4

EXERCISES 1.4 1. .

(3, 7) 3

3. .

7

(5, ∞) 5

5. .

(−2, 1) −2

7. 9. 11. 13. 21.

(4, ∞) (−3, 7) [1, 8) [5, 8) .a) f (0) = 3 b) f (1) = 15

1

15. All real numbers 17. [5∕2, ∞) 19. (−∞, −3) ∪ (−3, 1) ∪ (1, ∞) c) f (x + 3) = 7(x + 3)3 + 5(x + 3) + 3

23. .a) f (−1) = 10 c) f (x + h) = (x + h)5 + 11 b) f (a2 ) = a10 + 11 25. It is not a function. It fails the vertical line test. 27. It is not a function. It fails the vertical line test.

10

LINEAR EQUATIONS AND FUNCTIONS

29. f (x) = x2 − 4 f (x) 4 3 2 1 x

0 –4

–2

0

2

4

–1 –2 –3

f (x) = x2 – 4

–4 –5

3

31. .f (x) = x − 8 f (x) 8

4

x

0 –3

–2

–1

0

1

2

3

–4 f (x) = x3 − 8 -8

–12

–16

33. It is the piecewise graph. f (x) 6 4 2 x

0 –3

–2

–1

0 –2 –4 –6 –8

1

2

3

4

5

6

EXERCISES 1.5

35. .a) b) c) d) 37. .a)

(3x5 + 7x3 + 8) − (4x5 − 2x3 + 2x) = −x5 + 9x3 − 2x + 8 (4x5 − 2x3 + 2x) − (3x5 + 7x3 + 8) = x5 − 9x3 + 2x − 8 (4x5 − 2x3 + 2x)(3x5 + 7x3 + 8) 3(4x5 − 2x3 + 2x)5 + 7(4x5 − 2x3 + 2x)3 + 8

2x5 + h b) (x + h)2 + 4

c) (2a5 )(a2 + 4) d) 2(x + 1)5 [(x + 2)2 + 4]

EXERCISES 1.5 (√ )3 1. .13∕7 = 7 1 = 1 √ 3. .(25)3∕2 = ( 25)3 = (5)3 = 125 5. .(64)5∕6 =

(√ 6

)5 64

= (2)5 = 32

( )−2 ( )2 3 9 7. . 2 = = 3 2 4 √ 3 9. .(0.008)1∕3 = 0.008 = 0.20 ( )3 3 11. .15 = 15 = 33 = 27 5 53 13. .x3 x5 = x8 15. .(2xy)3 = 23 x3 y3 = 8x3 y3 3 5 8 17. .x x = x = x12 x−4 x−4 4 5 19. . x y = x4−2 y5−(−2) = x2 y7 x2 y−2 ( )2 (2)2 (x3 )2 4x6 21. . 2x3 = = y2 (y2 )2 y4 √ √ √ 3 3 3 23. . x5 x4 = x9 = x3 √ 4 25. .(81x4 y8 )1∕4 = 81x4 y8 = 3xy2

11

12

LINEAR EQUATIONS AND FUNCTIONS

4 5 3∕2 3∕2 6 15∕2 √ 27. .(16x√y ) = (16) x y = ( 16)3 x6 y7 = 64x6 y7 1∕2 y y 5 7 2∕3 2∕3 10∕3 14∕3 29. .(8x√y ) = (8) x y = 4x3 y4 1∕3 y2∕3 3 2 x xy

EXERCISES 1.6 1. f (x) = 6x + 11 f (x + h) − f (x) [6(x + h) + 11] − [6x + 11] 6h = = =6 h h h f (x) − f (a) [6x + 11] − [6a + 11] 6x − 6a 6(x − a) b) = = = =6 x−a x−a x−a x−a 3. f (x) = 7x − 4 f (x + h) − f (x) [7(x + h) − 4] − [7x − 4] 7h = = =7 a) h h h f (x) − f (a) [7x − 4] − [7a − 4] 7x − 7a 7(x − a) b) = = = =7 x−a x−a x−a x−a 2 5. f (x) = x − 7x + 4 2 2 a) .f (x + h) − f (x) = [(x + h) − 7(x + h) + 4] − [x − 7x + 4] h h 2 h(2x + h − 7) 2xh + h − 7h = = 2x + h − 7 h h 2 2 b) .f (x) − f (a) = [x − 7x + 4] − [a − 7a + 4] x−a x−a (x2 − a2 ) − (7x − 7a) = x−a (x − a)(x + a) − 7(x − a) = =x+a−7 x−a 7. f (x) = x2 + 6x − 8 2 2 a) .f (x + h) − f (x) = [(x + h) + 6(x + h) − 8] − [x + 6x − 8] h h h(2x + h + 6) 2xh + h2 + 6h = = 2x + h + 6 h h 2 2 b) .f (x) − f (a) = [x + 6x − 8] − [a + 6a − 8] x−a x−a (x2 − a2 ) + (6x − 6a) = x−a (x − a)(x + a) + 6(x − a) = =x+a+6 x−a a)

EXERCISES 1.6

9. f (x) = 5x2 − 2x − 3 2 2 a) .f (x + h) − f (x) = [5(x + h) − 2(x + h) − 3] − [5x − 2x − 3] h h 10xh + 5h2 − 2h h(10x + 5h − 2) = = 10x + 5h − 2 h h 2 2 b) .f (x) − f (a) = [5x − 2x − 3] − [5a − 2a − 3] x−a x−a (5x2 − 5a2 ) − (2x − 2a) = x−a 5(x − a)(x + a) − 2(x − a) = = 5x + 5a − 2 x−a 11. f (x) = x3 − 4x + 5 3 3 a) .f (x + h) − f (x) = [(x + h) − 4(x + h) + 5] − [x − 4x + 5] h h 3x2 h + 3xh2 + h3 − 4h h(3x2 + 3xh + h2 − 4) = h h = 3x2 + 3xh + h2 − 4 3 3 b) .f (x) − f (a) = [x − 4x + 5] − [a − 4a + 5] x−a x−a (x3 − a3 ) − (4x − 4a) = x−a (x − a)(x2 + ax + a2 ) − 4(x − a) = x−a = x2 + ax + a2 − 4 13. f (x) = 2x3 − 7x + 3 3 3 a) .f (x + h) − f (x) = [2(x + h) − 7(x + h) + 3] − [2x − 7x + 3] h h 6x2 h + 6xh2 + 2h3 − 7h h(6x2 + 6xh + 2h2 − 7) = h h = 6x2 + 6xh + 2h2 − 7 3 3 b) .f (x) − f (a) = [2x − 7x + 3] − [2a − 7a + 3] x−a x−a (2x3 − 2a3 ) − (7x − 7a) = x−a 2(x − a)(x2 + ax + a2 ) − 7(x − a) = x−a = 2(x2 + ax + a2 ) − 7

13

14

LINEAR EQUATIONS AND FUNCTIONS

15. f (x) =

3 x3

3 3 a) . − 3 3 f (x + h) − f (x) 3x3 − 3(x + h)3 (x + h) x = = h h x3 h(x + h)3 −9x2 h − 9xh2 − 3h3 = x3 h(x + h)3 −3h(3x2 + 3xh + h2 ) −3(3x2 + 3xh + h2 ) = = x3 h(x + h)3 x3 (x + h)3 3 3 3a3 − 3x3 b) . − 3 3 f (x) − f (a) 3(a − x)(a2 + ax + x2 ) a = a3 x 3 = x = x−a x−a x−a (x − a)a3 x3 2 2 −3(a + ax + x ) = a3 x 3

SUPPLEMENTARY EXERCISES CHAPTER 1 1. .9(x − 3) + 2x = 3(x + 1) − 2 9x − 27 + 2x = 3x + 3 − 2 11x − 27 = 3x + 1 8x = 28 x=

7 2

x−𝜇 3. Z = so Z𝜎 = x − 𝜇 and 𝜇 = x − Z𝜎. 𝜎 5. 3x + 5y = 15 y 9 6 3

3x + 5y = 15

0 –15

–10

–5

x 0

–3 –6 –9

5

10

15

EXERCISES 1.6

15

1 − 7 −6 = = 2. Next, use the point slope 2 − 5 −3 form with either point to yield y − 7 = 2(x − 5). 4 Rewriting in slope intercept form 4x − 3y = 12 as y = x − 4. 3 4 The slope is . A line parallel has the same slope, so the line of 3 4 interest is y − 5 = (x − 2). 3 The GCF of 2x3 − 18x2 − 20x is 2x, so the initial factoring yields 2x(x2 − 9x − 10). Complete factoring yields 2x(x − 10)(x + 1). Group as (2ax − 2ay) + (bx − by) before factoring the GCF from each pair of terms. Then 2a(x − y) + b(x − y) = (x − y)(2a + b) is the completely factored expression. Multiplying by 3 to eliminate the fraction yields 2(x − 1) < 3(x − 2). Next, 2x − 2 < 3x − 6 yields 4 < x so the interval notation solution is (4, ∞). To determine the domain, factor the denominator. This yields 2x + 5 . The domain is all real numbers except x = 0, x = −8, x(x + 8)(x + 1) or x = −1. .a) (f − g)(x) = −x3 + 4x2 + 3x + 5

7. First, find the slope m = 9.

11. 13.

15.

17.

19.

b) (f ⋅ g)(x) = (x2 + 3x + 1)(x3 − 3x2 − 4) 3 2 2 3 2 c) (g(x)) ( (f ∘5 g)(x) ) (= f−2 ) = (x 3 − 3x − 4) + 3(x − 3x − 4) + 1 −2x −1 3x −6x 21. = 4 ⋅ = 12x7 3x2 4x5 2x )−3 ( 3 2 )−3 ( 4 )3 ( 2x y 2x3 3z 27z12 23. = = = 3y−2 z4 3z4 2x3 y2 8x9 y6 3 25. f (x) = x + 3x + 1 3 3 a) .f (x + h) − f (x) = [(x + h) + 3(x + h) + 1] − [x + 3x + 1] h h 2 + 3xh + h2 + 3) 2 2 3 h(3x 3x h + 3xh + h + 3h = h h = 3x2 + 3xh + h2 + 3 3 3 b) .f (x) − f (a) = [x + 3x + 1] − [a + 3a + 1] x−a x−a (x3 − a3 ) + (3x − 3a) = x−a (x − a)(x2 + ax + a2 ) + 3(x − a) = = x2 + ax + a2 + 3 x−a

CHAPTER 2

THE DERIVATIVE

EXERCISES 2.1 1. 3. 5. 7.

9.

11. 13. 15. 17. 19.

The slope of the tangent is the derivative, so y′ = 0. The derivative is 2x. When x = 5, the derivative (slope) is 10. The derivative is 2x. When x = 1∕2, the derivative (slope) is 1. At x = 5∕2, the derivative or slope is 2(5∕2) = 5. The point is (5∕2, (5∕2)2 ) or (5/2, 25/4). Using point slope yields y − 25∕4 = 5(x − 5∕2). At x = −7∕2, the derivative (slope) is 2(−7∕2) = −7. The point is (−7∕2, (7∕2)2 ) or (−7∕2, 49∕4). Using point slope yields y − 49∕4 = −7(x + 7∕2). One seeks the point where the derivative 2x = 7∕4. Therefore, 2x = 7∕4 and x = 7∕8. The point is (7/8, 49/64). One seeks the point where the derivative 2x = 4∕5. Therefore, 2x = 4∕5 and x = 2∕5. The point is (2/5, 4/25). The line has a slope of 2/3, so one seeks where the derivative is also 2/3. Therefore, 2x = 2∕3 so x = 1∕3. The point is (1/3, 1/9). Evaluating 3x2 at x = 2 yields a slope of 12. Evaluating 3x2 at x = 1∕2 yields a slope of 3/4.

Solutions Manual to Accompany Fundamentals of Calculus, First Edition. Carla C. Morris and Robert M. Stark. © 2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

16

EXERCISES 2.2

17

21. Evaluating 3x2 at x = 4 yields a slope of 48. The point is (4, (4)3 ) or (4, 64). The equation of the tangent line is y − 64 = 48(x − 4). 23. Evaluating 3x2 at x = 3∕4 yields a slope of 27/16. The point is (3∕4, (3∕4)3 ) or (3/4, 27/64). The equation of the tangent line is y − 27∕64 = 27∕16(x − 3∕4). 25. One seeks where 3x2 = 48 or x2 = 16. The two solutions are x = 4 and x = −4. The points are (4, 64) and (−4, −64). 27. Solving for y, the slope yields y = (4∕3)x − 2 and a line par2 allel also has slope (4/3. Therefore, ) ( set 3x) = 4∕3. This yields 2 2 8 2 8 x = ± and the points , and − , − . 3 3 27 3 27 EXERCISES 2.2 1. .a) lim f (x) = 10

b) lim f (x) = 4

c) lim f (x) does not x→3 exist

3. .a) lim f (x) = −3

b) lim f (x) = −3

c) lim f (x) = −3

5. .a) lim f (x) = 0

b) lim f (x) = 0

c) lim f (x) = 0

x→3−

x→−2−

x→0−

x→3+

x→−2+

x→0+

x→−2

x→0

7. .lim 4 = 4 x→−2

9. .lim (3x + 2) = 3(5) + 2 = 17 x→5

11. .lim (9x + 5) = 9(4) + 5 = 41 x→4

13. .lim (5x2 + 9x + 3) = 5(1)2 + 9(1) + 3 = 17 x→1

15. .lim (2x7 + 3x4 + 9x + 3) = 2(0)7 + 3(0)4 + 9(0) + 3 = 3 x→0

17. .lim (2x3 + 5x2 − 4) = 2(−1)3 + 5(−1)2 − 4 = −1 x→−1

3(2) 6 3x = = is undefined, so the limit does not exist. x→2 x − 2 2−2 0 x 2 − 9 32 − 9 0 = = . This is an indeterminate form. Try factoring 21. lim x→3 x − 3 3−3 0 (x − 3)(x + 3) = lim(x + 3) = 3 + 3 = 6. as lim x→3 x→3 x−3 19. lim

18

THE DERIVATIVE

x 3 − 8 23 − 8 0 = = . This is an indeterminate form. x→2 x − 2 2−2 0 Try factoring as (x − 2)(x2 + 2x + 4) = lim(x2 + 2x + 4) = 12 lim x→2 x→2 x−2 −4 −4 25. lim 3 = 3 so the limit approaches 0. x→∞ x ∞

23. lim

27. lim (2x3 + 5x + 1) approaches infinity. The limit does not exist. x→∞

29. .lim f (x) = 1 = −1 x→−1 −1 31. .lim f (x) = 2(1) + 5 = 7 x→1

33. .lim f (x) = (4)2 − 3(4) − 4 = 0 x→4

35. .lim f (x) = 10 x→10

EXERCISES 2.3 1. If f (x) = 4x + 11 [4(x + h) + 11] − [4x + 11] 4h = lim = 4 = f ′ (x). h→0 h→0 h h [4x + 11] − [4a + 11] 4x − 4a b) lim = lim x→a x→a x − a x−a 4(x − a) = lim = 4 = f ′ (a). x→a x − a Therefore, f ′ (x) = 4. a) lim

3. If f (x) = x2 + 5x + 1 [(x + h)2 + 5(x + h) + 1] − [x2 + 5x + 1] a) .lim h→0 h h(2x + h + 5) 2xh + h2 + 5h = lim = lim = 2x + 5 = f ′ (x) h→0 h→0 h h (x2 − a2 ) + (5x − 5a) [x2 + 5x + 1] − [a2 + 5a + 1] b) l.im = lim x→a x→a x−a x−a (x − a)(x + a) + 5(x − a) ′ = 2a + 5 = f (a). = lim x→a x−a Therefore, f ′ (x) = 2x + 5.

EXERCISES 2.3

5. If f (x) = x2 − 6x + 1 2 2 a) .lim [(x + h) − 6(x + h) + 1] − [x − 6x + 1] h→0 h h(2x + h − 6) 2xh + h2 − 6h = lim = 2x − 6 = f ′ (x) = lim h→0 h→0 h h 2 2 2 2 b) lim . [x − 6x + 1] − [a − 6a + 1] = lim (x − a ) − (6x − 6a) x→a x→a x−a x−a (x − a)(x + a) − 6(x − a) = lim = 2a − 6 = f ′ (a). x→a x−a Therefore, f ′ (x) = 2x − 6. 7. If f (x) = x3 + 5 3 3 2 2 3 a) .lim [(x + h) + 5] − [x + 5] = lim 3x h + 3xh + h h→0 h→0 h h 2 2 h(3x + 3xh + h ) 2 ′ = lim = 3x = f (x) h→0 h 3 3 3 3 b) lim . [x + 5] − [a + 5] = lim (x − a ) x→a x→a x − a x−a 2 2 (x − a)(x + ax + a ) = lim = 3a2 = f ′ (a) x→a x−a Therefore, f ′ (x) = 3x2 . 9. f. ′ (x) = 0 11. f. ′ (x) = 0 13. f. ′ (x) = 7 15. f. ′ (x) = 14 17. f. ′ (x) = 4x + 7 19. f. ′ (x) = 30x2 − 18x + 3 21. . d (7x5 − 4x4 + 3x2 + 40) = 35x4 − 16x3 + 6x dx ( ) 23. . d 9x + 4 = 9 − 4 dx x x2 ) (√ 6 2 5 8 25. . d x5 + 4 + 8 = x−1∕6 − 5 dx 6 x x

19

20

THE DERIVATIVE

y′ (1) = 3(1)2 + 4(1) + 3 = 10 f (2) = (2)2 + 4(2) + 2 = 14 and f ′ (2) = 2(2) + 4 = 8. f (−1) = (−1)3 + 5(−1)2 + 2 = 6 and f ′ (−1) = 3(−1)2 + 10(1) = −7. y′ (2) = 6(2) + 5 = 17 and y(2) = 24. The derivative is the slope of the tangent, so the equation is y − 24 = 17(x − 2). 35. y′ (0) = 35(0)4 + 8 = 8 and y(0) = 7(0)5 + 8(0) + 25 = 25. The derivative is the slope of the tangent, so y − 25 = 8(x − 0). 27. 29. 31. 33.

EXERCISES 2.4 1. 3. 5. 7. 9. 11. 13.

15.

17.

19.

21.

There is a break in the graph, so it is not differentiable at x = −1. There is a cusp at x = 2, so the function is not differentiable there. It is differentiable at x = 1∕2. It is not continuous at x = −1 since there is a break in the graph there. It is continuous at x = 2 since there is no break there. It is continuous at x = 1∕2 since there is no break there. .a) It is a polynomial and differentiable everywhere, including at x = 0. b) It is a polynomial and continuous everywhere, including at x = 0. .a) The piecewise function has a cusp at x = 0, so it is not differentiable there. b) The segments join at x = 0, so the function is continuous there. .a) The piecewise function has a cusp at x = 0, so it is not differentiable there. b) The segments join at x = 0, so the function is continuous there. .a) It is a polynomial and differentiable everywhere, including at x = 1. b) It is a polynomial and continuous everywhere, including at x = 1. .a) The piecewise function has a cusp at x = 1, so it is not differentiable there. b) The segments join at x = 1, so the function is continuous there.

EXERCISES 2.5

21

23. .a) It is a polynomial and differentiable everywhere, including at x = 2. b) It is a polynomial and continuous everywhere, including at x = 2. 25. .a) The piecewise function has a cusp at x = 2, so it is not differentiable there. b) The segments join at x = 2, so the function is continuous there. 27. For the function to be continuous for all x it must have a limit at exceptional values of x. The limit here is (x + 6)(x − 6) lim = lim (x − 6) = −12. Any function that results in x→−6 x→−6 (x + 6) −12 when x = −6 is acceptable. The two simplest possibilities are a constant or the reduced function, x − 6. Therefore, ⎧ x2 − 36 ⎪ f (x) = ⎨ x + 6 ⎪ x−6 ⎩

x ≠ −6 x = −6

29. For the function to be continuous for all x it must first have a limit as x approaches 5. The limit is (x)(x − 5)(x + 5) lim = lim(x)(x + 5) = 50. Any function value of 50 x→5 x→5 (x − 5) when x is 5 is acceptable. Therefore, ⎧ x3 − 25x ⎪ f (x) = ⎨ x − 5 ⎪x2 + 5x ⎩

x≠5 x=5

EXERCISES 2.5 1. .f ′ (x) = 4(4x2 + 1)3 (8x) = 32x(4x2 + 1)3 3. .y′ = 7(5x2 + 3)6 (10x) = 70x(5x2 + 3)6 5. .f ′ (x) = 28(3x2 + 1)3 (6x) = 168x(3x2 + 1)3 7. .f ′ (x) = 5(9x10 + 6x5 − x)4 (90x9 + 30x4 − 1) 9. .y′ = 10(12x7 + 3x4 − 2x + 5)9 (84x6 + 12x3 − 2)

22

THE DERIVATIVE

( ) −20 1 27x2 + x−1∕2 √ 2 (9x3 + x + 3)6 ) ( .f ′ (x) = 6(7x8∕5 + 5x + 1)5 56 x3∕5 + 5 5 ( ) √ 1 .f ′ (x) = 5( x + 1)4 x−1∕2 2 ( )5 ( ) √ 2 3 y′ = 6 4x5 + x2 + 1 20x4 + x−1∕3 3 140 ′ 2 y = 21x + 16x − (4x − 3)8 One seeks y′ (1). It is 2[5(1) + 1](5) = 60 The point is (2, 27). The slope is m = y′ (2) = (3∕2)[4(2) + 1]1∕2 (4) = 18. The equation of the tangent line is y − 27 = 18(x − 2). −15 The point is (1, 5). The slope is y′ (1) = [3(1)2 − 2] 3 [(1) − 2(1) + 2]4 or −15. The equation of the tangent line is y − 5 = −15(x − 1).

11. .f ′ (x) = 13. 15. 17. 19. 21. 23.

25.

EXERCISES 2.6 1. .f ′ (z) = 4z3 + 6z 3. .ds = 12r3 + 3r2 + 4r dr 5. .f ′ (p) = 24p7 + 30p5 + 6p2 + 4 7. .f ′ (t) = 10(5t2 + 3t + 1)9 (10t + 3) ( ) √ 9. .S′ (p) = 12(3p10 + p + 5)11 30p9 + 1 p−1∕2 2 11. .d (3t7∕5 − 5t − 4) = 21 t2∕5 − 5 dt 5 13. . d (5p4 − 3p2∕3 ) = 20p3 − 2p−1∕3 dp 15. .d (2a7 t5 − 9bt3 + t2 + 3) = 10a7 t4 − 27bt2 + 2t dt d (2a7 t5 − 9bt3 + t2 + 3) = −9t3 db

EXERCISES 2.7

23

17. f ′ (x) = 27x2 + 4 and f ′′ (x) = 54x 3 3 19. y′ = 6x2 + 3 + x−1∕4 and y′′ = 12x − x−5∕4 4 16 21. v′ = 6t2 + 18 and v′′ = 12t 23. The derivative is 3x2 − 7. Evaluating at x = 1 yields 3(1)2 − 7 = −4 25. The second derivative is 30t4 . Evaluating at t = −1 yields 30. 27. The second derivative is 15(2x + 7)1∕2 . Evaluating at x = 1 yields 45. 29. y′ = 5x4 + 9x2 + 9, y′′ = 20x3 + 18x, y′′′ = 60x2 + 18, and yiv = 120x. EXERCISES 2.7 1. .Δf (x) = [6(x + 1) + 4] − [6x + 4] = 6x + 6 + 4 − 6x − 4 = 6 3. .Δf (x) = [4(x + 1)2 ] − [4x2 ] = 4x2 + 8x + 4 − 4x2 = 8x + 4 5. .Δf (x) = [5(x + 1)2 + 2(x + 1) + 3] − [5x2 + 2x + 3] = 5x2 + 10x + 5 + 2x + 2 + 3] − 5x2 + 2x + 3] = 10x + 7 7. .Δf (x) = [(x + 1)3 + 3(x + 1) + 1] − [x3 + 3x + 1] = [x3 + 3x2 + 3x + 1 + 3x + 3 + 1] − [x3 + 3x + 1] = 3x2 + 3x + 4 9. Δf (x) = [9(x + 1) − 1] − [9x − 1] = 9x + 9 − 1 − 9x + 1 = 9 since Δf (x) is a constant Δ2 f (x) = 0. Alternatively, Δ2 f (x) = f (x + 2) − 2f (x + 1) + f (x) = [9(x + 2) − 1] − 2[9(x + 1) − 1] + [9x − 1] = 0. 11. .Δ2 f (x) = [2(x + 2)2 + 5] − 2[2(x + 1)2 + 5] + [2x2 + 5] = 2x2 + 8x + 8 + 5 − 4x2 − 8x − 4 − 10 + 2x2 + 5 = 4 13. .Δ2 f (x) = [5(x + 2)3 + 2] − 2[5(x + 1)3 + 2] + [5x3 + 2] = 5x3 + 30x2 + 60x + 40 + 2 − 10x3 − 30x2 − 30x − 10 − 4 + 5x3 + 2 = 30x + 30

24

THE DERIVATIVE

SUPPLEMENTARY EXERCISES CHAPTER 2 1. Evaluating the derivative at x = 1∕4 yields m = 2(1∕4) = 1∕2. 3. When 2x = 3∕4. x = 3∕8, and the point is (3/8, 9/64). 5. .lim 3x + 4 = 3(2) + 4 = 8 x→2 x 2 7. This is an indeterminate form. Therefore, (x − 4)(x + 6) x2 + 2x − 24 lim = lim = 4 + 6 = 10 x→4 x→4 x−4 x−4 9. If f (x) = 3x2 + 5x + 1 2 2 a) .lim [3(x + h) + 5(x + h) + 1] − [3x + 5x + 1] h→0 h h(6x + 3h + 5) 6xh + 3h2 + 5h = lim = 6x + 5 = f ′ (x) = lim h→0 h→0 h h 2 2 b) lim . [3x + 5x + 1] − [3a + 5a + 1] x→a x−a 2 2 (3x − 3a ) + (5x − 5a) = lim x→a x−a 3(x − a)(x + a) + 5(x − a) = lim = 6a + 5 = f ′ (a). x→a x−a Therefore, f ′ (x) = 6x + 5.

11. f (1) = 5(1)3 + 2(1)2 + 3 = 10 and f ′ (1) = 15(1)2 + 4(1) = 19. 13. Evaluating y′ = 35x6 − 6x at x = 1 yields a slope of 29. The point is (1, 2) so the equation of the tangent line is y − 2 = 29(x − 1). 15. First f (1) must exist. Here, f (1) = 2(1) − 3 = −1. Next the limit at (2x − 3)(x − 1) x = 1 must exist. So, lim = 2(1) − 3 = −1. x→1 (x − 1) Since f (1) = lim f (x), the function is continuous. x→1 d 3 2 17. . (9x + 4x + 3x + 1)25 = 25(9x3 + 4x2 + 3x + 1)24 (27x2 + 8x + 3) dx ( √ )3 ( ) 4 −1∕3 19. .f ′ (x) = 4 2 3 x2 + 3x + 1 +3 x 3

EXERCISES 2.7

25

5 21. y′ = (4x + 1)3∕2 (4) = 10(4x + 1)3∕2 at x = 6 the derivative is 1250. 2 Therefore, m = 1250 and the point (6, 3125). The tangent line is y − 3125 = 1250(x − 6). d 23. (5a3 p4 + 3ap2 + 2bp + c) = 20a3 p3 + 6ap + 2b) and dp d (5a3 p4 + 3ap2 + 2bp + c) = 15a2 p4 + 3p2 da 25. The first derivative is 36x8 − 21x6 + 7 and the second derivative is 288x7 − 126x5 . Evaluating, y′′ (1) = 288(1)7 − 126(1)5 = 162

CHAPTER 3

USING THE DERIVATIVE c and d are increasing for all x. e has slope that is always increasing as x increases. b, d, g, and h have an inflection point. The graph is increasing on (2, ∞) and decreasing on (−∞, 2). It has a local and absolute minimum at (2, −1). The graph is concave up on (−∞, ∞) with no inflection points. The y-intercept is (0, 3) and x-intercepts (1, 0) and (3, 0). There are no undefined points and no asymptotes. 9. The graph is decreasing on (−∞, −1) ∪ (0, 1) and increasing on (−1, 0) ∪ (1, ∞). There is a local maximum at (0, 0) and local and absolute minimums at (±1, −1). The graph is concave down on (−3∕4, 3∕4) and concave up on (−∞, −3∕4) ∪ (3∕4, ∞). There are inflection points at (±3 ∕ 4, −1 ∕ 2). The y-intercept is (0, 0) and x-intercepts at (0, 0), (±1.4, 0). There are no asymptotes and no undefined points. 11. There are endpoint extrema at A and G. Local extrema are at B, D, and F. 1. 3. 5. 7.

Solutions Manual to Accompany Fundamentals of Calculus, First Edition. Carla C. Morris and Robert M. Stark. © 2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

26

EXERCISES 3.2

27

13. There are inflection points at C and E. 15. There is an absolute maximum at D and absolute minimum at F.

EXERCISES 3.2 1. A positive first derivative means that the function (graph) is increasing. Graphs c and d depict a positive first derivative for all x. 3. A positive second derivative means that the function (graph) is concave up. Graph e depicts a positive second derivative for all x. 5. One possibility is depicted in the graph as follows.

–2

–1

y 6 5 4 3 2 1 0 –1 0 –2 –3 –4 –5

x 1

2

3

4

5

6

7. One possibility is depicted in the following graph. 4

y

3 2 1 x

0 –2

–1

0

1

2

3

4

5

–1 –2

9. The table is filled in as follows.

A B C D

f + − − 0

f′ − 0 − +

f ′′ + + 0 +

6

28

USING THE DERIVATIVE

11. f ′ (5) is positive since the graph is shown above the axis here. A positive first derivative indicates an increasing function. 13. A local maximum occurs when the derivative changes sign from positive to negative. This must occur at one of the x-intercepts of the derivative (see graph). The x-intercept is at x = 1. 15. The ordered pair is (2, −2∕3) and the slope is f ′ (2) = −6. The tangent line is y + (2∕3) = −6(x − 2). EXERCISES 3.3 1. Setting the derivative to zero yields 2x − 4 = 0 or x = 2 as a possible extremum. The derivative is negative on (−∞, 2) and positive on (2, ∞) indicating a local minimum at x = 2. The minimum is f(2) = −9. Alternatively, the second derivative being positive at x = 2 also indicates a minimum there. 3. Setting the derivative to zero yields 3x2 − 6x + 3 = 3(x − 1)2 = 0 or x = 1 as the only possible extremum. The derivative is positive on (−∞, 1) and remains positive on (1, ∞). Since the first derivative does not change in sign, there is no extrema at x = 1. The function has no local extrema. 5. Using the first and second derivatives, f ′ (x) = 2x + 8 and x = −4 is the only possible extremum. The first derivative indicates that the function decreases on (−∞, −4) and increases on (−4, ∞). The second derivative is f ′′ (x) = 2 so, the graph is concave up everywhere and there is no inflection. Next, the y-intercept is 7 and x-intercepts are −1 and −7. At least five points are graphed as a check. The graph is: 11 f (x) = x2 + 8x + 7

y

7 3

–9

–8

–7

–6

–5

–4

–3

–2

–1

–1

x 0

1

–5 (−4, −9)

–9 –13

′ 2 7. The first derivative = 3x 4 indicates ( f (x) √ ) −( ) that the function is √ 2 3 −2 3 increasing on −∞, ∪ , ∞ and decreasing on 3 3 ( √ ( √ √ ) √ ) −2 3 2 3 2 3 −16 3 , . There is a local minimum at , 3 3 3 9

EXERCISES 3.3

29

√ √ ) −2 3 16 3 . The second derivative and a local maximum at , 3 9 f ′′ (x) = 6x indicates that the function is concave down on (−∞, 0) and concave up on (0, ∞) with an inflection point at (0, 0). The function has x-intercepts at −2, 0, and 2, (from factoring) and the graph is (

f (x) 20 15 10 f(x) = x3 − 4x –4

–3

–2

5 0

–1

–5

0

1

2

3

4

x

–10 –15 –20

9. The first derivative f ′ (x) = 12x2 − 12x + 3 indicates that the function is always increasing, so there are no local extrema. The second derivative f ′′ (x) = 24x − 12 indicates that the function is concave ( ) ( ) 1 1 down on −∞, and concave up on , ∞ with an inflection 2 ) 2 ( 1 9 , . The function has a y-intercept of 4 and the graph is: point at 2 2 f (x) 20 15 10 4x3 − 6x2 + 3x + 4 5 x

0 –2

0

–1

1

2

3

–5 –10 –15

11. The first derivative f ′ (x) = −3x2 + 3x indicates that the function is increasing on (0, 1) and decreasing on (−∞, 0) ∪( (1, ∞). ) There is a 7 local minimum at (0, 3) and a local maximum at 1, . The second 2

30

USING THE DERIVATIVE

derivative,(f ′′ (x))= −6x + 3, indicates that ( the )function is concave 1 1 , ∞ and concave up on −∞, with an inflection down on 2 ( 2 ) 1 13 , . The function has a y-intercept of 3 and the graph is point at 2 4 f (x) 10 8 6 4 2 x

0 –2

–1

–2

1

0

–4

2

3

4

−x3 + 3/2x2 + 3

–6 –8 –10 2 13. The first derivative f ′ (x) = −x√ − 4x + 1√indicates that the function is increasing on √ (−2 − 5, −2 + 5) and decreasing on √ (−∞, −2 − 5) ∪ (−2 + 5, ∞). Using two place decimal approximations, there is a local minimum at (−4.23, −8.79) and a local maximum at (0.23, 6.13). The second derivative, f ′′ (x) = −2x − 4, indicates that the function is concave down (on (−2, ∞) ) and concave 4 up on (−∞, −2) with an inflection point at −2, − . The function 3 has a y-intercept of 6 and the graph is:

f (x) 20 15 10 (–1/3)x3 – 2x2 + x + 6

5 0

–8

–6

–4

–2

–5

x 0

2

4

–10 –15

f ′ (x)

15. The first derivative = 4x3 − 4x indicates that the function is decreasing on (−∞, −1) ∪ (0, 1) and increasing on (−1, 0) ∪ (1, ∞). There are local minimums at (±1, −6) and a local maximum at (0, −5). The second derivative f ′′ (x) = 12x2 − 4

31

EXERCISES 3.3

( √ √ ) 3 − 3 and indicates that the function is concave down on , 3 3 ( ) ( ) √ √ − 3 3 concave up on −∞, ∪ , ∞ with inflection points 3 3 ( √ ) ± 3 −50 at , . The function has a y-intercept at the origin, 3 9 √ √ x-intercepts at x = ± 1 + 6. The graph is: f (x) 4 2 –2 .5

–2

–1 .5

–1

0

–0 .5

0

0.5

1

1.5

2

2.5

x

–2 x4 – 2x2 – 5 –4 –6 –8

1 17. The first derivative f ′ (x) = 1 − 2 indicates that the function is x increasing on (−∞, −1) ∪ (1, ∞) and decreasing on (−1, 0) ∪ (0, 1). The function is undefined at x = 0. There is a local minimum at (1, 2) 2 and a local maximum at (−1, −2). The second derivative f ′′ (x) = 3 x indicates that the function is concave down on (−∞, 0) and concave up on (0, ∞) with no inflection points. There are asymptotes at x = 0 and at y = x. The function has no y-intercept, and the graph is: f (x) 5 4 3 2

y=x

1

x

0 –6

–4

–2

–1 0 –2 –3

x + 1/x

–4 –5

2

4

6

32

USING THE DERIVATIVE

EXERCISES 3.4 1. Setting the derivative to zero yields f ′ (x) = 2x − 6 = 0. Therefore, at x = 3, there is a possible extremum. The derivative is negative when x < 3 and positive when x > 3 indicating a local minimum at x = 3. (Alternatively, the positive second derivative indicates a minimum at the critical value x = 3 from the first derivative.) 3. .f ′ (x) = −4x3 + 6x −2x + + – – 2x2 – 3 + + – – + + f ′(x) = −2x(2x2 – 3) – – 6 – 6 0 2

5.

7.

9.

11.

2

√ ± 6 The critical point table indicates local maxima at x = . The max2 imum value is 9/4. Also, negative signed second derivatives at the critical values (first derivative) signal a maximum. f ′ (x) = 16 − 2x. The derivative is positive when x < 8 and negative when x > 8 indicating a local maximum at x = 8. The maximum value is 64. Alternatively, the negative second derivative indicates a maximum at the critical value x = 8. x + y = 8 and y = 8 − x so the product is P(x) = x(8 − x) = 8x − x2 . P′ (x) = 8 − 2x indicates a possible extremum at x = 4. The derivative is positive when x < 4 and negative when x > 4 indicating a maximum at x = 4. The maximum of the product is 16. Alternatively, the negative second derivative indicates a maximum at the critical value x = 4. x + y = 20 and y = 20 − x so the sum of the squares, S(x) = x2 + (20 − x)2 = 2x2 − 40x + 400. S′ (x) = 4x − 40, indicates a possible extremum at x = 10. The derivative is negative when x < 10 and positive when x > 10 indicating a minimum at x = 10. The minimum sum of squares is 200. Alternatively, the positive second derivative indicates a minimum at the critical value x = 10. First, for a number to exceed its square, it is between 0 and 1. One seeks to maximize f (x) = x − x2 . The first derivative f ′ (x) = 1 − 2x indicates a possible extremum at x = 1 ∕ 2. The derivative is positive when x < 1 ∕ 2 and negative when x > 1 ∕ 2. Therefore, 1/2 exceeds its square by the largest amount. Alternatively, the negative second derivative also indicates a maximum at the critical value x = 1 ∕ 2.

EXERCISES 3.4

33

13. Here, one seeks to maximize the area xy subject to 2x + 2y = 60. Solving for y in the constraint yields y = 30 − x. Therefore, one seeks to maximize A(x) = x(30 − x) = 30x − x2 . The first derivative, A′ (x) = 30 − 2x, indicates a possible extremum when x = 15. The first derivative is positive when x < 15 and negative when x > 15 indicating a maximum at x = 15. Therefore, x = y = 15. The maximum area is a square of 225 ft2 . Alternatively, the negative second derivative also indicates a maximum at the critical value x = 15. 15. Here, one seeks to maximize xy ∕ 2 subject to the constraint x + y = 10. The objective written in terms of a single variable is −1 2 1 x + 5x. The first derivative A′ (x) = −x + 5 A(x) = (x)(10 − x) = 2 2 indicates a possible extremum at x = 5. The derivative is positive when x < 5 and negative when x > 5, so there is a maximum at x = 5. Alternatively, the second derivative being negative at the critical value x = 5 also indicates a maximum there. If x = 5 then y is also 5, so the largest area for the right triangle is 25/2 square centimeters. 17. First, rewrite the constraint as y = 8 − x. The objective function to be minimized is S(x) = x2 + (8 − x)2 = 2x2 − 16x + 64. S′ (x) = 4x − 16 indicates a possible extremum at x = 4. When x < 4, the first derivative is negative and when x > 4, the first derivative is positive. There is a minimum at x = 4. Alternatively, the second derivative being positive at the critical value x = 4 also indicates a minimum there. The minimum value for the sum of squares is 32. 19. The objective is to maximize the area xy subject to the constraint 6 12x + 10y = 120. Solving for y yields y = 12 − x. Rewriting 5 ( ) 6 6 the objective as A(x) = x 12 − x = 12x − x2 and taking the 5 5 12 derivative yields A′ (x) = 12 − x. There is a possible extremum 5 at x = 5. The derivative is positive when 0 < x < 5 and negative when x > 5 indicating a local maximum at x = 5. Alternatively, the second derivative being negative at the critical value x = 5 indicates a maximum there. The maximum area is 30 ft2 . 21. To maximize xy subject to the constraint 15x + 20y = 2400, solve for( y and substitute into the objective function to yield ) 3 3 A(x) = x 120 − x = 120x − x2 . Taking the first derivative yields 4 4 3 A′ (x) = 120 − x indicating a possible extremum at x = 80. The 2 derivative is positive when 0 < x < 80 and negative when x > 80 indicating a maximum at x = 80 ft and y = 60 ft. Alternatively, the

34

USING THE DERIVATIVE

second derivative being negative at the critical value x = 80 indicates a maximum there. 23. To maximize x2 h subject to the constraint 2x2 + 4xh = 60, solve for y and substitute to yield an objective function ( ) 15 1 1 V(x) = x2 − x = 15x − x3 and a first derivative of x 2 2 3 V ′ (x) = 15 − x2 . The derivative indicates a possible extremum 2 √ (no negative lengths). The first at x = 10 √ √ derivative is positive negative when x > 10 indicating a local on 0 < x < 10 and √ maximum = h = 10 ft. The box is a cube with dimensions √ at x√ √ 10 × 10 × 10. Alternatively, the second derivative being negative at the critical value indicates a maximum there. 25. Here, the surface area is 2𝜋rh + 2(2r)2 = 2𝜋rh + 8r2 and the volume V is V = 𝜋r2 h. Using the volume and solving for h yields h = 2 . 𝜋r ( ) V 2 = 2V + 8r 2 . The objective is to minimize S(r) = 2𝜋r + 8r r 𝜋r2 −2V ′ The first derivative is S (r) = 2 + 16r indicates a possible r V 1∕3 V 1∕3 4V 1∕3 extremum at r = . For a minimum, r = and h = . 2 2 1∕3 𝜋 4 4V h = . This = The optimal height to diameter ratio is 2r 𝜋(V 1∕3 ) 𝜋 relationship holds regardless of the volume, V, and indicates that the can is slightly taller than in Example 3.4.8. 27. Let c = cost per square foot of the sides and 2c = cost per square ( )2 d 𝜋d2 foot for the top and bottom. The area of the bottom is 𝜋 = , 2 4 where d is the diameter of the circular end. The volume of the cylinder 𝜋d2 4V is V = . The cost as a function (h). Solving for h yields h = 2 4 𝜋d ( 2) ) ( 𝜋d 4cV 4V of d is C(d) = 2 (c) = 𝜋d2 c + (2c) + 𝜋d . The 4 d 𝜋d2 4cV first derivative is C′ (d) = 2𝜋cd − 2 indicating a possible extremum d √ √ 2V 3 2V . The minimum occurs when d = 3 and when d = 𝜋 𝜋 √ 16V h , so the best height to diameter ratio = 2. h= 3 𝜋 d

EXERCISES 3.5

35

EXERCISES 3.5 1. C . ′ (x) = 6x2 + 9 3. C′ (x) = 7x6 − 15x2 + 20 so C′ (1) = 7(1)6 − 15(1)2 + 20 = $12 5. MC = C′ (x) = 3x2 − 18x + 15 = 3(x − 1)(x − 5). Setting MC = 0 yields x = 1 or 5. 7. MC = C′ (x) = 3x2 − 24x + 60. Next, the second derivative is needed to find a possible minimum. MC′ (x) = 6x − 24. Setting this derivative to zero yields x = 4, a local minimum. The minimum marginal cost is MC(4) = C′ (4) = 3(4)2 − 24(4) + 60 = 12. 9. .R′ (x) = −2x + 30 11. Revenue is price multiplied by demand. ( ) 1 −x2 So, R(x) = xp(x) = x − + 12 = + 12x. 4x 4 −x + 12. Therefore, R′ (x) = 2 13. The restrictions for x and p are 0 ≤ x ≤ 40 and 0 ≤ p ≤ 240, respectively. Next, solving for p in terms of x yields p = 240 − 6x. Using this information, the revenue function is formed as R(x) = (x)(240 − 6x) = 240x − 6x2 . Using Profit = revenue − cost yields Pr(x) = (240x − 6x2 ) − (x3 + 9x2 − 360x + 2000) = −x3 − 15x2 + 600x − 2000. The first derivative yields Pr′ (x) − 3x2 − 30x + 600. Setting it to zero yields x = 10 as a possible extremum (−20 is rejected). Other possibilities are x = 0 and x = 240 as endpoint extrema. Evaluating the profit at the potential candidates yields the maximum profit of $1500 when 10 units are produced at $180 each. SUPPLEMENTARY EXERCISES CHAPTER 3 1. c and e are increasing for all x. 3. a and c are concave up for all x. −500 5. Here, y′ = since the denominator is always positive y′ is (x + 1)2 always negative indicating that the function is decreasing for x ≥ 0. 7. The first derivative is f ′ (x) = 4x − 1 and x = 1∕4 is the only possible extremum. The first derivative indicates that the function decreases on (−∞, 1∕4) and increases on (1∕4, ∞) so the minimum value is −49 ∕ 8 when x = 1∕4. The second derivative f ′′ (x) = 4 indicates that the

36

USING THE DERIVATIVE

graph is concave up everywhere and has no points of inflection. Next, there is a y-intercept of −6 and x-intercepts of 2 and −3 ∕ 2. The graph is: y 6 4 2 0 –3

–2

–1

–2

x 0

1

2

3

y = 2x2 – x – 6

–4 –6 –8

3 − 18x indicates that the function 9. The first derivative (f ′ (x) = 4x√ ) ( √ ) −3 2 3 2 is decreasing on −∞, ∪ 0, and increasing 2 2 ) ( √ ) ( √ 3 2 3 2 ,0 ∪ , ∞ . There are local minimums at on − 2 2 ( √ ) ±3 2 −81 , and a local maximum at (0, 0). The second deriva2 4

tive f ′′ (x) = 12x2 − 18 indicates that the function is concave down on ( √ √ ) ( ) √ ) (√ 6 6 − 6 − 6 , and concave up on −∞, ∪ , ∞ with 2 2 2 2 ) ( √ ± 6 −45 . The function has a y-intercept at , inflection points at 2 4 the origin, x-intercepts at x = −3, 0, and 3. The graph is: y 50 40 30 20

y = x4 − 9x2

10 x

0 –4

–2

–10 –20 –30

0

2

4

EXERCISES 3.5

37

11. Here, f ′ (x) = 12x3 − 12x2 and f ′′ (x) = 36x2 − 24x = 12x(3x − 2). 12x 3x – 2 f ′′(x) = 12x(3x – 2)

– – +

+ – – 0

+ + + 2/3

The critical point table shows that there are inflection ( points ) at 2 2 −16 x = and at x = 0. The inflection points are (0, 0) and , . 3 3 27 13. To maximize the area A(x) = xy subject to the cost constraint 20x + 30y = 480, solve for y and substitute to yield ( ) 2 2 A(x) = x 16 − x = 16x − x2 . 3 3 4 The first derivative is A′ (x) = 16 − x. When x < 12, the first 3 derivative is positive and when x > 12, it is negative indicating a maximum occurs at x = 12 yards. The maximum area is 96 square yards when x = 12 yards and y = 8 yards. Alternatively, the second derivative being negative also indicates a maximum at the critical value x = 12. (Note as a quick check, the same sum is spent in each direction. In this case, 12(20) = 8(30) = 240 dollars in each of the two directions x and y.) 15. .a) In this case, one seeks to minimize 2x2 + 4xh subject to x2 h = 64. Solving for y and substituting yields an objective function ( ) 64 256 and a first derivative of S(x) = 2x2 + 4x 2 = 2x2 + x x 256 S′ (x) = 4x − 2 . The derivative indicates a possible extremum x at x = 4. The first derivative is negative on 0 < x < 4 and positive when x > 4 indicating a local minimum at x = h = 4 in. Alternatively, the second derivative being positive at the critical value x = 4 also indicates a minimum there. The box is a 4-inch cube. b) Here, since the box is to be open, minimize x2 + 4xh subject to x2 h = 64. Solving for y and substituting yields an objective ( ) 64 256 function S(x) = x2 + 4x 2 = x2 + and a first derivative of x x 256 S′ (x) = 2x − 2 . The derivative indicates a possible extremum x √ √ 3 3 at x = 4 2. The first derivative is negative on 0 < x < 4 2 and

38

USING THE DERIVATIVE

√ √ 3 3 positive when x > 4 2 indicating a local minimum at x = 4 2 in. √ 3 and h = 2 2 in. Alternatively, the second derivative being positive at the critical value also indicates a minimum there. The box has a square base whose side is twice the height. c21 c22 + subject to 17. Here, one seeks to minimize V(x) = x1 x2 B − c1 x1 c1 x1 + c2 x2 = B. Using the constraint one determines x2 = c2 so one rewrites the objective as c21 c22 c21 c32 V(x) = + = + . Differentiating yields B − c1 x1 x1 x1 B − c1 x1 c2 −c21 c32 c1 ′ . Setting the derivative to zero yields V (x) = 2 + (B − c1 x1 )2 x1 the following (c31 − c32 )x12 − 2c21 Bx1 + c1 B2 = 0. Using the quadratic √ c21 B ± c2 B c1 c2 formula yields a possible optimum when x1 = c31 − c32 19. Here, R′ (x) = 4000 − 4x indicates an extremum at x = 100. At this point, the maximum revenue is $2,000,000. 21. The restrictions for x and p are 0 ≤ x ≤ 40 and 0 ≤ p ≤ 240. Next, solving for p in terms of x yields p = 240 − 6x. Using this information, the revenue function is formed as R(x) = (x)(240 − 6x) = 240x − 6x2 . Using Profit = revenue − cost yields Pr(x) = (240x − 6x2 ) − (x3 − 21x2 − 360x + 8350) = −x3 + 15x2 + 600x − 8350. Now, the first derivative yields Pr′ (x) = −3x2 + 30x + 600. Setting the derivative to zero yields 20 as a possible extremum (−10 is rejected as a possibility). Other possibilities are x = 0 and x = 240 as endpoint extrema. Evaluating the profit at the potential extrema yields the maximum profit of $1650 when 20 units are produced at $120 each.

CHAPTER 4

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

EXERCISES 4.1 1. .a) The expression must have powers of 2 or 3. In this case, 8 is a power of 2, so, (8)3x = (23 )3x = 29x . b) Rewrite as (27)2x = (33 )2x = 36x . c) Rewrite as (16)5x = (24 )5x = 220x . ( )−4x ( )−4x 1 1 3. .a) = 3 = (2−3 )−4x = (2)12x 8 2 ( )6x ( )6x 1 1 b) = 2 = (3−2 )6x = (3)−12x 9 3 ( )−2x ( )−2x 1 1 c) = 3 = (3−3 )−2x = (3)6x 27 3 105x 5. .a) 5x = (2)5x 5 322x b) = (2)2x 162x ( )3x 43x 1 c) = = (3−1 )3x = (3)−3x 3 123x Solutions Manual to Accompany Fundamentals of Calculus, First Edition. Carla C. Morris and Robert M. Stark. © 2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

39

40

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

7. .7x3 x5 y−2 7x8 7x6 = = x 2 y4 x 2 y4 y2 y6 ( ) 5 3 9. . x ÷ x = (x3 y2 ) ⋅ y = x 2 y7 x y−2 y5 5x+3 4x+1 25x+3 (22 )x+1 25x+3 22x+2 11. .2 = = = 24x+3 8(23x−1 ) 23 (23x−1 ) 23x+2

13. Since the bases are the same, exponents are equal. Therefore, 3x = 15 yields x = 5. 15. Equate bases as 27−x = 25 to yield 7 − x = 5 and x = 2. 17. Factoring yields 5x [(1 + x) + (3–2x)] = 0 5x [(4–x)] = 0 x Since 5 cannot be zero, 4 − x = 0 and x = 4. 19. Factoring, 7x [(x2 + 4x) + (x + 6)] = 0. Simplifying, 7x [x2 + 5x + 6] = 0. Therefore, [x2 + 5x + 6] = (x + 3)(x + 2) = 0. The solutions are x = −2 or x = −3. 21. .23+h = 2h (23 ) 23. .7x+5 − 72x = 72x (7−x+5 − 1) 25. This is a difference of cubes (7h )3 − 8 = (7h –2)[72h + 2(7h ) + 4].

EXERCISES 4.2 1. Rewrite as 3y = x to determine ordered pairs and graph as: 4

y

3

y = 1og3 x

2 1 0

x 0

–1 –2 –3 –4

3

6

9

12

15

18

21

EXERCISES 4.3

41

3. log10 1,000,000 = x so 10x = 106 , which yields x = 6. 5. log2 64 = x so 2x = 26 , which yields x = 6. 1 = x so 2x = 2−5 , which yields x = −5. 7. log2 32 9. Since ln ex = x, then ln e3 = 3. 11. Since ln ex = x, then ln e7.65 = 7.65. 13. ln(ln e) = ln(1) = 0. 15. log9 27 = x so 9x = 27, which yields 32x = 33 . Therefore, 2x = 3 or x = 3∕2. 17. log4 32 = x so 4x = 25 . This is rewritten as 22x = 25 , so 2x = 5 and x = 5∕2. 19. log2 128 = x so 2x = 128 = 27 . Therefore, x = 7. 21. Rewriting logx 27 = 3 in exponential form yields x3 = 27 and x = 3. 23. Rewriting log3 (5x + 2) = 3 in exponential form yields 33 = 5x + 2. Solving for x yields x = 5. 25. Here, 5x = 35 or x = 7. 27. Since ln(ln 4x) = 0, ln 4x = 1. Therefore, 4x = e and x = e∕4. 2 6 29. .log (x + 1) (x − 3) = log (x + 1)2 + log (x–3)6 –log (3x + 5)3 4 4 4 4 (3x + 5)3 = 2log4 (x + 1) + 6log4 (x–3)–3log4 (3x + 5). 31. First, the coefficients become exponents to yield ln x2 − ln (y + 1)3 + ln (z + 1)4 . x2 (z + 1)4 This results in ln . (y + 1)3 33. Using logarithms, 102x−1 = 105 is written as log10 102x−1 = log10 105, 1 + log10 105 which yields 2x − 1 = log10 105. Therefore, x = . 2 x−1 35. First, rewrite as e = 4∕3. Taking the natural logarithm of both sides of the equation yields ln ex−1 = ln(4∕3) or x − 1 = ln 4∕3. Therefore, x = 1 + ln(4∕3). 37. The difference between 4.1 and 6.8 quakes is 2.7 on the Richter scale. Therefore, one seeks 102.7 = 501.2. The “Halloween” Quake was roughly 500 times stronger than the Northern Illinois Quake. EXERCISES 4.3 1. Multiplying exponents yields e3x , rewriting with a negative exponent yields e−5x .

42

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

3. Add exponents to yield e7x , (e10x )1∕2 = e5x . 5. Since e2 is a constant then f ′ (x) = 0. 7. .f ′ (x) = 2e2x 9. .f ′ (x) = 5(6e6x ) = 30e6x 11. .f ′ (x) = 5(25e25x ) = 125e25x 9

13. .f ′ (x) = 6e8x [72x8 ] = 432x8 e8x

9

3

15. .f ′ (x) = e2x +5x+1 [6x2 + 5] 7

4

17. .f ′ (x) = 6e8x +9x [56x6 + 36x3 ] 19. Evaluating y and its derivative at x = 0 gives the point (0, 2) and y′ (0) = 5e5(0) + 3 = 8 = m. The tangent line is y − 2 = 8(x − 0) or y = 8x + 2. EXERCISES 4.4 1. .f ′ (x) = 25 = 1 25x x 3. .f ′ (x) = 3x2 + 10x + 3 = 3x2 + 10x + 1 3x x 5. .f ′ (x) = 4 4x + 7 7. .f ′ (x) =

15x2 + 12x + 9 5x3 + 6x2 + 9x + 2

4 3 9. .f ′ (x) = 45x + 24x + 8 5 4 9x + 6x + 8x

+ 1)−1∕2 (5) 5 11. .f ′ (x) = (1∕2)(5x = √ 2(5x + 1) 5x + 1 ( ) 13. .f ′ (x) = 8(17x2 + 15x + ln 2x)7 34x + 15 + 1 x 3x 15. .f ′ (x) = 3e e3x + 2

17. The point is (e, 1) and the derivative at x = 1 is 1/e, so the tangent line is y − 1 = (1∕e)(x − e).

EXERCISES 4.6

43

19. Marginal cost is the derivative of the cost function, so 300 MC = C′ (x) = . 3x + 1 21. Rewriting as 5ln(3x + 5) + 2ln(4x3 − 7x + 1) yields ) ( ( ) 3 12x2 − 7 ′ f (x) = 5 . +2 3x + 5 4x3 − 7x + 1 EXERCISES 4.5 1. Using t = 5 to represent 2015 yields 50,000e0.3(5) ≈ 58,092. 3. There are seven doublings, so 500(2)7 = 64,000. 5. The formula for doubling is 250(1.06)t = 500. Solving for t yields the following (1.06)t = 2 t ln(1.06) = ln 2 t=

ln 2 = 11.896 ≈ 12 days ln 1.06

7. The half-life is P0 e−𝜆(28) = 0.50P0 . Therefore, 𝜆 = 0.02476. If there are 500 g today, in 200 years there will be 500e−0.02476(200) ≈ 3.54g. 9. The half-life is P0 e−𝜆(35) = 0.50P0 . Therefore, 𝜆 = 0.0198. One seeks 1 the time to decay to 1/5 of the original. So, P0 e−0.0198t = P0 . 5 Solving for t yields t ≈ 81.3 days. 11. Using the model in the text, one has 2.5e−(0.00012097(10,000) = 0.7457g. EXERCISES 4.6 1. The compound interest formula yields ( ) 0.08 4⋅4 50,000 1 + = 50,000(1.02)16 = $68,639.29. 4 3. The continuous compound interest formula yields 500,000e(0.07)(6) = 500,000e0.42 = $760,980.78. 5. Comparing, 250,000(1 + 0.03)5 = 250,000(1.03)5 = $289,818.52 and 225,000e(0.05)(4) = 225,000e0.20 = $274,815.62. Therefore, $250,000 is worth more.

44

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

7. Solve 25,000e0.04t = $100,000 so e0.04t = 4 and ln 4 t= ≈ 34.66 years. 0.04 ln(0.3) 9. S0 e−0.2t = 0.3S0 and therefore, t = ≈ 6.02 weeks. −0.2 f ′ (t) 6t + 2 11. . When t = 3, the percentage rate of change is = 2 f (t) 3t + 2t + 1 58.82%. f ′ (t) 0.4e0.4t = 0.4t = 0.4. When t = 5, the percentage rate of change is 13. f (t) e 40%. This rate is independent of t. −1 ′ f (t) (t + 3)2 −1 15. = = . When t = 2, the percentage rate of change 1 f (t) t+3 t+3 is −20%. −p(−500e−0.25p (0.25) 17. For E(p) = > 1, 0.25p > 1 and p > 4. 500e−0.25p

SUPPLEMENTARY EXERCISES CHAPTER 4 1. a) c)

( )3x 163x 16 = = 23x 8 83x

b)

( )5x−1 505x−1 50 = = 55x−1 10 105x−1

(52 )2x+1 53x+2 252x+1 53x+2 54x+2 53x+2 = = = 54x+4 125x (53 )x 53x

3. .a) Rewrite as 5−2x = 54 so x = −2. b) Rewrite as 23x−1 = 25 so 3x − 1 = 5 and x = 2. 5. This is a difference of squares and factors as (3x − 5)(3x + 5). 7. .f ′ (x) = 4(9e9x ) = 36e9x 3

9. .f ′ (x) = 12x2 (e4x ) 11. f ′ (x) = 6e6x and at x = 0, one has m = f ′ (0) = 6. Since f (0) = 2, one seeks a line through (0, 2) with slope 6 so y − 2 = 6(x − 0) or y = 6x + 2. 13. Solve by setting 3x = 729. Then 3x = 36 and x = 6. Therefore, log3 729 = 6.

EXERCISES 4.6

15. Solve by setting 9x = 27. Then 32x = 33 so 2x = 3 and x = 3∕2. Therefore, log9 27 = 3∕2. 17. ln e7.23 = 7.23 since ln ex = x. 19. Using rules for logarithms, ln (2x + 3)4 (x + 1)2 (4x + 5)7 = ln (2x + 3)4 + ln (x + 1)2 + ln (4x + 5)7 = 4ln(2x + 3) + 2ln(x + 1) + 7ln(4x + 5). 21. The earthquake is 108.9−6.3 = 101.4 ≈ 25.1 times as strong. [ ] 15x2 + 4x + 8 23. .f ′ (x) = 5x3 + 2x2 + 8x + 7 25. 650 = 250e𝜆(5) so 𝜆 = 0.1911. When t = 10, one has 250e(0.1911)(10) ≈ 1690. 27. Solving P0 e𝜆(1) = 3P0 yields 𝜆 = ln 3, so 400e(ln 3)(5) = 97,200. An alternative method uses 400(3)5 = 97,200. ′ −t 29. .f (t) = −0.02e + 0.03 f (t) 0.3 + 0.03t + 0.02e−t When t = 0, the percentage rate of change is 3.125%. When t = 2, the percentage rate of change is 7.53%.

45

CHAPTER 5

TECHNIQUES OF DIFFERENTIATION

EXERCISES 5.1 1. f ′ (x) = (2x2 + 3x + 8)[5] + (5x − 1)[4x + 3] 3. f ′ (x) = (x2 + 5x + 3)4 [3(3x2 − 2x + 6)2 (6x − 2)] + (3x2 − 2x + 6)3 [4(x2 + 5x + 3)3 (2x + 5)] ] [ 2x + ln(x2 + 5)[5] 5. f ′ (x) = 5x 2 x +5 1 10 7. f ′ (x) = x3 [5e5x ] + e5x [3x2 ] + + x−2∕7 x 7 9. First, since the numerator is a constant, it is easier to determine the derivative by rewriting as 3(x5 + 7x − 3)−8 . Therefore, −24[5x4 + 7] f ′ (x) = −24(x5 + 7x − 3)−9 [5x4 + 7] or 5 (x + 7x − 3)9 (5x2 + 7x + 3)[3] − (3x + 2)[10x + 7] 11. f ′ (x) = (5x2 + 7x + 3)2 2 (5x + 7x + 3)[12x2 + 2] − (4x3 + 2x + 1)[10x + 7] 13. f ′ (x) = (5x2 + 7x + 3)2

Solutions Manual to Accompany Fundamentals of Calculus, First Edition. Carla C. Morris and Robert M. Stark. © 2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

46

EXERCISES 5.1

15. f ′ (x) =

47

(7x3 + 5x + 3)4 [11(5x6 + 3x − 4)10 (30x5 + 3)] (7x3 + 5x + 3)8 −

(5x6 + 3x − 4)11 [4(7x3 + 5x + 3)3 (21x2 + 5)]

(7x3 + 5x + 3)8 17. Here, the power rule is applied first, and since the numerator is a constant, the expression in parenthesis is viewed as 30(x7 − 5x4 + 3)−3 to yield: )99 [

( 30

′

f (x) = 100 ( )3 x7 − 5x4 + 3

)] ( −90 7x6 − 20x3 (x7 − 5x4 + 3)4

[ ] 1 − ln 7x[2] x 19. f ′ (x) = (2x − 5)2 ] [ 1 21. f ′ (x) = 25(x3 + 7x + ln 8x)24 3x2 + 7 + x 23. Here, the quotient rule is applied and the product rule for the derivative of the numerator: (2x − 5)

f ′ (x) =

(2x + 7)3 [x3 (5e5x ) + e5x (3x2 ))] − (x3 e5x )[3(2x + 7)2 (2)] (2x + 7)6

25. Here, the power rule is applied first and then the quotient rule is applied within the parenthesis to yield: ( √ )3 3x − 7 f (x) = 4 4 x + 5x + 3 [ ] ⎡ (x4 + 5x + 3) 1 (3x − 7)−1∕2 (3) − √3x − 7[4x3 + 5] ⎤ ⎢ ⎥ 2 ⎢ ⎥ 4 2 (x + 5x + 3) ⎢ ⎥ ⎣ ⎦ [ (√ )9 ( )] 3 −2∕5 5 +2 x 27. f ′ (x) = (3x + 1)4 10 x3 + 2x 5 (√ )10 5 + x3 + 2x [4(3x + 1)3 (3)] ] [ ( √ 8 2 x3 √ )7 ( 2 1 −1∕2 ) 3 ′ 3 x 3 18x − x 29. f (x) = (6x − x) [3x e ] + e 8 6x − x 2 ′

48

TECHNIQUES OF DIFFERENTIATION

31. f ′ (x) =

(2x + 1)5 [3e3x ] − e3x [5(2x + 1)4 (2)] (2x + 1)10

33. f ′ (x) =

(ex + 1)[4e4x ] − e4x [ex ] (ex + 1)2

35. First, f ′ (x) = (5x − 4)(3) + (3x + 7)(5). Next, evaluating at x = 1 yields a slope of the tangent as f ′ (1) = 1(3) + 10(5) = 53 = m. The equation is y − 10 = 53(x − 1). 37. Here, f ′ (x) = (x − 1)4 [2] + (2x + 3)[4(x − 1)3 (1)]. The slope is the derivative at x = 2 so, f ′ (2) = (1)(2) + 7(4) = 30. The equation is of the tangent line is y − 7 = 30(x − 2). 39. Here, f ′ (x) = (4x − 3)3∕2 [1] + (x + 1)[3∕2(4x − 3)1∕2 (4)]. The slope at x is f ′ (3) = (27)(1) + 4(3∕2)(3)(4) = 99. The tangent line is y − 108 = 99(x − 3). 41. For a horizontal tangent, its slope is zero. Setting the first derivative to (2x − 1)⌊3x2 ⌋ − x3 [2] zero yields 0 = . The fraction is zero when its (2x − 1)2 numerator is zero, so 4x3 − 3x2 = 0. This indicates a horizontal ( ) 3 27 tangent at (0, 0) and , . 4 32 EXERCISES 5.2 1. Substituting g(x) into f (x) yields f (g(x)) = (3x − 1)2 + 2(3x − 1) + 4. 3. Substituting g(x) into f (x) yields f (g(x)) = (9x2 + 5x + 7)4 + 2(9x2 + 5x + 7)3 + 5(9x2 + 5x + 7). 5. Letting f (x) = x4 and g(x) = 5x + 1 yields f (g(x)) = h(x) = (5x + 1)4 . 2 7. Letting f (x) = x7 + 3 and g(x) = 2x + 1 yields x 2 . f (g(x)) = h(x) = (2x + 1)7 + (2x + 1)3 9. The function is a composite of x19 and x2 + 5x + 3. Using the chain rule, f ′ (x) = 19(x2 + 5x + 3)18 (2x + 5). 11. The function is a composite of x5 and 5x7 − 3x4 + 3. Using the chain rule, f ′ (x) = 5(5x7 –3x4 + 3)4 (35x6 –12x3 ). 13. The function is a composite of x1 ∕ 5 and 2x4 + 3x2 + 3. 1 Using the chain rule, f ′ (x) = (2x4 + 3x2 + 3)−4∕5 (8x3 + 6x). 5 dy 5 3∕2 du 15. Here, = u and = 8. Multiplying, du 2 dx

EXERCISES 5.3

49

dy du 5 3∕2 ⋅ = u (8) = 20u3∕2 . However, the answer is in terms of x, du dx 2 dy not u. Substitution yields = 20(8x − 3)3∕2 . dx dy = u[5(2u + 3)4 (2)] + (2u + 3)5 [1] = 3(2u + 3)4 (4u + 1) and 17. Here, du du = 6x2 + 7. Multiplying, yields dx dy du ⋅ = 3(2u + 3)4 (4u + 1)(6x2 + 7). However, the answer is in du dx terms of x, not u. Substitution yields dy = 3(4x3 + 14x + 5)4 (8x3 + 28x + 5)(6x2 + 7). dx dy (u + 1)[2] − (2u − 1)[1] 3 du 19. Here, = and = = 16x. du dx (u + 1)2 (u + 1)2 dy du 48x 3 (16x) = . However, Multiplying yields ⋅ = du dx (u + 1)2 (u + 1)2 dy 48x the answer is in terms of x, not u. Substitution yields . = 2 dx (8x + 4)2 EXERCISES 5.3 dy −8x −4x dy = 0 so = = dx dx 18y 9y 2 dy dy 3 − 6x 6x2 + 28y3 = 3 so = dx dx 28y3 dy dy dy 2(x − 1) 2x − 2 + 3y2 = 3 so = 2 dx dx 3(1 − y ) dx ) ( −y5 ) dy ( −2 12 dy 2 − 5 = 14 so = 14 + 3 dx 12 x3 y dx x ] [ dy dy 18 − 2xy2 9 − xy2 x2 2y + y2 [2x] = 18 so = = dx dx 2x2 y x2 y ] [ dy 12x − 2 − 3x2 y5 dy x3 5y4 + y5 [3x2 ] + 2 = 12x so = dx dx 5x3 y4 [ ] [ ] dy dy 4x 3y2 + 4y3 − x2 − 2xy + 3x2 = 9 and rearranging terms dx dx dy yields (12xy2 − x2 ) = −4y3 + 2xy − 3x2 + 9. Therefore, dx dy −4y3 + 2xy − 3x2 + 9 . = dx 12xy2 − x2

1. 8x + 18y 3. 5. 7. 9. 11. 13.

50

TECHNIQUES OF DIFFERENTIATION

dy implicitly and then evaluate this derivative dx [ ] dy dy −y when x = −3 and y = 5. Therefore, x + y[1] = 0 and = . dx dx x dy −5 5 At (−3, 5), = = . dx −3 3 dy To solve, determine implicitly and then evaluate this derivative dx dy dy when x = −1 and y = 1. Therefore, 4y3 + 2 − 9x2 = 2 so dx dx dy 11 dy 2 + 9x2 = = . . At (−1, 1), dx 4y3 + 2 dx 6 dy To solve, determine implicitly and then evaluate this derivative dx [ ] [ ] dy dy 2 + y[1] + 3y = 0 and when x = 5 and y = 2. Therefore, x dx dx dy −2 dy −y . At (5, 2), = = . 2 dx x + 3y dx 17 [ ] dy −y dy −y4 3 Here, x 4y = + y4 = 0 and = . At (3,2), 3 dx dx 4xy 4x dy −2 −1 −1 = = so the equation of the tangent is y − 2 = (x − 3). dx 12 6 6 ) ( dy dy −x3 dx dx 4x3 + 4y3 = 0 and = 3 dt dt dt dt y dy dy dx dx dx 2x + 4x + 4y = 7 + 2y or dt dt dt dt dt dy dx (2x + 4y − 7) = (2y − 4x) and therefore, dt dt ( ) dy 2x + 4y − 7 dx = . dt 2y − 4x dt 4 The volume of a sphere is V = 𝜋r3 . Half of the diameter yields a 3 radius of 0.5 ft. Since air is being pumped into the balloon, its volume dV dr dV changes as = 3. Differentiating, = 4𝜋r2 . Solving, when dt dt dt 3 dr = ft∕min. r = 0.5 ft yields dt 𝜋 −100 dI . When R = 5, = Given I = 100∕R, and differentiating, dR R2 dI = −4. dR

15. To solve, determine

17.

19.

21.

23. 25.

27.

29.

EXERCISES 5.4

51

31. Using implicit differentiation ] [ [ ] 5y3∕4 3 −1∕4 dy 1 15x1∕4 dy 1∕4 20x y + 20y3∕4 x−3∕4 = 0 so = − 4 dx 4 y1∕4 dx x3∕4 dy −y dy −81 and = . Substituting x = 16 and y = 81 yields = . dx 3x dx 48

EXERCISES 5.4 1. Δf (x) = (x + 1)3 − x3 = x3 + 3x2 + 3x + 1 − x3 = 3x2 + 3x + 1 Δ2 f (x) = f (x + 1) − f (x) = 3(x + 1)2 + 3(x + 1) + 1 − (3x2 + 3x + 1) = 6x + 6. An alternate method is to use Δ2 f (x) = f (x + 2) − 2f (x + 1) + f (x) = (x + 2)3 − 2(x + 1)3 + x3 = 6x + 6. 3. First, one adds the functions as f (x) + g(x) = x2 + 3x + 5 + 3x . Next, differencing [(x + 1)2 + 3(x + 1) + 5] + 3x+1 − (x2 + 3x + 5 + 3x ) = x2 + 2x + 1 + 3x + 3 + 5 + 3x+1 − x2 − 3x − 5 − 3x = 2x + 4 + 3x+1 − 3x = 2x + 4 + 2(3x ) (agreeing with Example 5.4.4) First, subtract the functions as f (x) − g(x) = x2 + 3x + 5 − 3x . Next, differencing [(x + 1)2 + 3(x + 1) + 5] − 3x+1 − (x2 + 3x + 5 − 3x ) = x2 + 2x + 1 + 3x + 3 + 5 − 3x+1 − x2 − 3x − 5 + 3x = 2x + 4 − 3x+1 + 3x = 2x + 4 − 2(3x ) (agreeing with Example 5.4.4) First, multiply the functions as f (x)g(x) = x2 (3x ) + 3x(3x ) + 5(3x ) Next, differencing (x + 1)2 (3x+1 ) + 3(x + 1)(3x+1 ) + 5(3x+1 ) − [x2 (3x ) + 3x(3x ) + 5(3x )] = (x2 + 2x + 1 + 3x + 3 + 5)(3x+1 ) − [(x2 + 3x + 5)(3x )] = (x2 + 5x + 9)(3)(3x ) − [(x2 + 3x + 5)(3x )] = (3x2 + 15x + 27 − x2 − 3x − 5)(3x ) = (2x2 + 12x + 22)(3x ) = 2(x2 + 6x + 11)(3x ) (agreeing with Example 5.4.4) f (x) x2 + 3x + 5 = First, divide the functions as g(x) 3x

52

TECHNIQUES OF DIFFERENTIATION

Differencing as

(x + 1)2 + 3(x + 1) + 5 x2 + 3x + 5 − 3x 3x+1

(x + 1)2 + 3(x + 1) + 5 3(x2 + 3x + 5) − 3x+1 3x+1 x2 + 2x + 1 + 3x + 3 + 5 − 3x2 − 9x − 15 = 3x+1

=

−2x2 − 4x − 6 −2(x2 + 2x + 3) = (agreeing with Example 5.4.4) 3x+1 3x+1 n! = n(n − 1) · · · (n − r + 1) 5. First, n Pr = (n − r)! =

Δn Pr = (n + 1)(n)(n − 1) · · · ((n + 1) − r + 1) − n(n − 1) · · · (n − r + 1) Δn Pr = (n)(n − 1) · · · ((n + 1) − r + 1)[(n + 1) − (n − r + 1] = (n)(n − 1) · · · ((n + 1) − (r − 1))[r] Δn Pr = (r)[n Pr−1 ] 7. At a maximum, f (x∗ ) is larger than any other value of the function, so f (x∗ ) ≥ f (x∗ + 1) and f (x∗ ) ≥ f (x∗ − 1). If f (x∗ ) ≥ f (x∗ + 1), subtract f (x∗ ) from each side of the inequality 0 ≥ f (x∗ + 1) − f (x∗ ) = f (x∗ ). Therefore, one condition for a maximum is that Δf (x∗ ) ≤ 0. Another condition is f (x∗ ) ≥ f (x∗ − 1) and by subtracting f (x∗ − 1) from each side of the inequality f (x∗ ) − f (x∗ − 1) ≥ 0. Since f (x∗ ) − f (x∗ − 1) = Δf (x∗ − 1), another condition for a maximum is Δf (x∗ − 1) ≥ 0. Combining both conditions yields the desired result Δf (x∗ ) ≤ 0 ≤ Δf (x∗ − 1) 9. When a = 1 and b = 3 then 3 ∑ Δf (x)g(x) = Δ[f (1)g(1)] + Δ[f (2)g(2)] + [f (3)g(3)] = [f (2)g(2) x=1

− f (1)g(1)] + [f (3)g(3) − f (2)g(2)] + [f (4)g(4) − f (3)g(3)] = f (4)g(4) − f (1)g(1) = f (3 + 1)g(3 + 1) − f (1)g(1) = f (b + 1)g(b + 1) − f (a)g(a) n ∑ 11. a) One seeks x2 . To evaluate the summation by antidifferences, x=1

seek an antidifference F(x) such that ΔF(x) = x2 . Since x2 is a polynomial, it suggests as a trial, F(x) = ax3 + bx2 + cx + d. The coefficients a, b, c, and d are determined so that ΔF(x) = x2 . That is, ΔF(x) = [a(x + 1)3 + b(x + 1)2 + c(x + 1) + d] − [ax3 + bx2 + cx + d] = (3a)x2 + (3a + 2b)x + (a + b + c)

EXERCISES 5.4

53

For ΔF(x) to equal x2 requires 3a = 1 and (3a + 2b) = 0, and (a + b + c) = 0. 1 −1 1 Therefore, a = , b = , c= . 3 2 6 n n ( ) ∑ ∑ 1 1 1 2 Now, one can write the following x = Δ x3 − x2 + x , 3 2 6 x=1 x=1 which yields [

] [ ] 1 1 1 1 1 1 − + (n + 1)3 − (n + 1)2 + (n + 1) − 3 2 6 3 2 6 1 1 1 = n3 + n2 + n. 3 2 6

( x ) a b) Using the result ax = Δ leads to a−1 n n ( ) ∑ ∑ ax an+1 a1 an+1 − a ax = Δ = − = and, after a−1 a−1 a−1 a−1 x=1 x=1 a(an − 1) . simplifying, a−1 n ∑ x3 . To evaluate the summation by antidifferences, c) One seeks x=1

seek an antidifference F(x) such that ΔF(x) = x3 . Since x3 is a polynomial, it suggests as a trial. F(x) = ax4 + bx3 + cx2 + dx + e. The coefficients a, b, c, d, and e are determined so that ΔF(x) = x3 . That is ΔF(x) = [a(x + 1)4 + b(x + 1)3 + c(x + 1)2 + d(x + 1) + e] − [ax4 + bx3 + cx2 + dx + e] = (4a)x3 + (6a + 3b)x2 + (4a + 3b + 2c)x + (a + b + c + d) For ΔF(x) to equal x3 requires 4a = 1, (6a + 3b) = 0, (4a + 3b + 2c) = 0, and (a + b + c + d) = 0. Therefore, 1 a= , 4

b=

−1 , 2

1 c= , 4

d = 0.

54

TECHNIQUES OF DIFFERENTIATION

Now, one can write

n ∑ x=1

[

x3 =

n ( ) ∑ 1 1 1 Δ x4 − x3 + x2 , which yields 4 2 4 x=1

] 1 1 1 (n + 1)4 − (n + 1)3 + (n + 1)2 4 2 4 [ ] 1 1 1 1 4 1 3 1 2 − − + = n + n + n. 4 2 4 4 2 4

SUPPLEMENTARY EXERCISES CHAPTER 5 1. f ′ (x) = (5x3 − 2x + 1)4 [2] + (2x + 3)[4(5x3 − 2x + 1)3 (15x2 − 2)] 3. f ′ (x) = (2x4 − 3x2 + e3x )2 [3(2x + 1)2 (2)] + (2x + 1)3 [2(2x4 − 3x2 + e3x )(8x3 − 6x + 3e3x )] ] ( ) [ 5x + 1 9 (3x − 5) [5] − (5x + 1)[3] ′ 5. f (x) = 10 3x − 5 (3x − 5)2 [ ( )] 1 (25x + 3)5 8(2x + 3 + ln 4x)7 2 + x 7. f ′ (x) = (25x + 3)10 −

(2x + 3 + ln 4x)8 ⌊5(25x + 3)4 (25)⌋ (25x + 3)10

(4x2 − 11x − 9)6

9. f ′ (x) = −

(x3 + 5x + 1)3 ln 3x⌊6(4x2 − 11x − 9)5 (8x − 11)⌋ (4x2 − 11x − 9)12

[

11.

f ′ (x)

[( ] )3 ( 1 ) x3 + 5x + 1 + (ln 3x)3(x3 + 5x + 1)2 (3x2 + 5) x (4x2 − 11x − 9)12

= (3x +

1)4

] 15x2 − 18x + 2 + ln(5x3 − 9x2 + 2x + 1) 5x3 − 9x2 + 2x + 1

[4(3x + 1)3 (3)] 13. When x = 2, by substitution y = 11. The derivative f ′ (x) = (3x − 5)7 [5] + (5x + 1)[7(3x − 5)6 (3)] evaluated when x = 2 yields 236. Therefore, one seeks a line through (2, 11) with slope 236. The tangent line is y − 11 = 236(x − 2). 15. When x = 0, by substitution y = 1. The derivative f ′ (x) = (x + 1)5 [3e3x ] + (e3x )[5(x + 1)4 ] evaluated when x = 0

EXERCISES 5.4

55

yields 8. Therefore, one seeks a line through (0, 1) with slope 8. The tangent is y − 1 = 8(x − 0) or y = 8x + 1. dy du 17. = 14u6 and = 15x4 − 18x + 5 so du dx dy du ⋅ = 14u6 (15x4 − 18x + 5). Writing, in terms of x, yields du dx dy = 14(3x5 − 9x2 + 5x + 10)6 (15x4 − 18x + 5). dx 19. a) f (g(x)) = f (x7 + 2x5 + x2 + 1) = (x7 + 2x5 + x2 + 1)10 b) The derivative of x10 is 10x9 so the chain rule begins with 10(x7 + 2x5 + x2 + 1)9 and if multiplied by (7x6 + 10x4 + 2x) to yield 10(x7 + 2x5 + x2 + 1)9 (7x6 + 10x4 + 2x) c) The derivative of f (g(x)) is 10(x7 + 2x5 + x2 + 1)9 (7x6 + 10x4 + 2x), which agrees with part b. ] [ ] [ dy 4 2 dy 3 3 21. 3x 3y + y [12x ] = 10y + 12x and rearranging terms, dx dx dy dy 12x − 12x3 y3 (9x4 y2 − 10y) = 12x − 12x3 y3 . Therefore, . = dx dx 9x4 y2 − 10y ( ) ( ) dp dv 23. Here, p +v = 0 and rearranging yields ( ) ( ) dt ( )dt dp p ( dv ) dp dv = −p or that =− . Given v = 75, v dt dt dt v dt dp dv p = 30 and = 5, substituting yields = −2. The pressure is dt dt 2 decreasing at 2lbs∕in. min. 25. Δf (x) = f (x + 1) − f (x) = (x + 1)4 − x4 =x4 + 4x3 + 6x2 + 4x + 1 − x4 = 4x3 + 6x2 + 4x + 1 2 Δ f (x) = Δ(Δf (x)) = 4(x + 1)3 + 6(x + 1)2 + 4(x + 1) + 1 − [4x3 + 6x2 + 4x + 1] = 12x2 + 24x + 14

CHAPTER 6

INTEGRAL CALCULUS

EXERCISES 6.1 1. . 7dx = 7x + C ∫ 5x2 +C 3. . 5xdx = ∫ 2 3x−4 +C 5. . 3x−5 dx = − ∫ 4 7. . 2tdt = t2 + C ∫ 2 x1∕2 dx = x3∕2 + C 3 3 x2∕3 dx = x5∕3 + C 11. First, rewrite as x2∕3 dx to yield ∫ ∫ 5 1 −3 1 1 −3 x dx to yield x dx = − x−2 + C 13. First, rewrite as ∫ 2 ∫ 2 4 15. Integrating term by term yields ( 3∕5 ) 5 x + x−2∕3 dx = x8∕5 + 3x1∕3 + C ∫ 8 9. First, rewrite as

∫

x1∕2 dx to yield

∫

Solutions Manual to Accompany Fundamentals of Calculus, First Edition. Carla C. Morris and Robert M. Stark. © 2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

56

EXERCISES 6.2

57

7t 17. . e7t dt = e + C ∫ 7

19. Integrating term by term yields ) ( 3 4x + 3x2 + 2x + 9 dx = x4 + x3 + x2 + 9x + C ∫ e4x 21. Integrating term by term yields (e4x + 1)dx = +x+C ∫ 4 ( ) 2 23. Integrating term by term yields 4− dx = 4x − 2 ln |x| + C ∫ x ) ( √ ( −1∕2 ) 3 − 2t1∕2 dt. Integration yields 25. Rewrite √ − 2 t dt as 3t ∫ ∫ t ) ( −1∕2 4 − 2t1∕2 dt = 6t1∕2 − t3∕2 + C 3t ∫ 3 27. First, integration yields f (x) = x3 + C. Next, the initial condition indicates that 6 = (−1)3 + C and that C = 7, so f (x) = x3 + 7 is the desired function. 29. First, integration yields f (x) = x3 − x2 + 4x + C. Next, the initial condition indicates that 10 = (2)3 − (2)2 + 4(2) + C and that C = −2, so f (x) = x3 − x2 + 4x − 2 is the desired function. 2 2 5 31. Differentiating a) yields [2xex ] = 5xex and b) yields 2 2 2 5x[2xex ] + ex [5], therefore, part a) indicates that 2 5 2 5xex dx = ex + C ∫ 2

EXERCISES 6.2 1. The subinterval width, Δx, is

b−a , where a and b are endpoints. n

4−0 = 1. Divide the interval 0 to 4 into four subintervals: 4 [0, 1], [1, 2], [2, 3], and [3, 4]. The four left endpoints are 0, 1, 2, and 3. b−a , where a and b are endpoints. Here, 3. The subinterval width, Δx, is n 3−0 3 = . Divide the interval 0 to 3 into four subintervals: Δx = 4 4 [0, 0.75], [0.75, 1.5], [1.5, 2.25], and [2.25, 3]. The four left endpoints are 0, 0.75, 1.5, and 2.25. Here, Δx =

58

INTEGRAL CALCULUS

5. The subinterval width, Δx, is

7.

9.

11.

13.

b−a , where a and b are endpoints. Here, n

15 − 1 = 2. Divide the interval 1 to 15 into seven subintervals: Δx = 7 [1, 3], [3, 5], [5, 7], [7, 9], [9, 11], [11, 13], and [13, 15]. The seven right endpoints are 3, 5, 7, 9, 11, 13, and 15. b−a The subinterval width, Δx, is , where a and b are endpoints. Here, n 27 − 3 = 4. Divide the interval 3 to 27 into six subintervals: Δx = 6 [3, 7], [7, 11], [11, 15], [15, 19], [19, 23], and [23, 27]. The six right endpoints are 7, 11, 15, 19, 23, and 27. b−a The subinterval width, Δx, is , where a and b are endpoints. Here, n 4−0 1 = . Divide the interval 0 to 4 into eight subintervals: Δx = 8 2 [0, 0.5], [0.5, 1.0], [10, 1.5], [1.5, 2.0], [2.0, 2.5], [2.5, 3.0], [3.0, 3.5], and [3.5, 4.0]. The eight midpoints are 0.25, 0.75, 1.25, 1.75, 2.25, 2.75, 3.25, and 3.75. b−a The subinterval width, Δx, is , where a and b are endpoints. Here, n 21 − 5 = 4. Divide the interval 5 to 21 into four subintervals: Δx = 4 [5, 9], [9, 13], [13, 17], and [17, 21]. The four midpoints are 7, 11, 15, and 19. b−a The subinterval width, Δx, is , where a and b are endpoints. Here, n 91 − 1 = 2. Divide the interval 1 to 9 into four subintervals: Δx = 4 [1, 3], [3, 5], [5, 7], and [7, 9]. The four right endpoints are 3, 5, 7, 4 b − a∑ f (xi ) to yield and 9. Next, use n i=1 2[f (3) + f (5) + f (7) + f (9)] = 2[10 + 26 + 50 + 82] = 336.

15. The subinterval width, Δx, is

b−a , where a and b are endpoints. Here, n

20 − 0 = 4. Divide the interval 0 to 20 into five subintervals: 5 [0, 4], [4, 8], [8, 12], [12, 16], and [16, 20]. The five right endpoints are 4, 8, 12, 16, and 20. Next, use

Δx =

EXERCISES 6.2

59

b − a∑ f (xi ) to yield 4[f (4) + f (8) + f (12) + f (16) + f (20)] n i=1 5

= 4[64 + 512 + 1728 + 4096 + 8000] = 57, 600. 17. The subinterval width, Δx, is

b−a , where a and b are endpoints. n

23 − 3 = 5. Divide the interval 3 to 23 into four Here, Δx = 4 subintervals: [3, 8], [8, 13], [13, 18], and [18, 23]. The four left 4 b − a∑ f (xi ) to yield endpoints are 3, 8, 13, and 18. Next, use n i=1 5[f (3) + f (8) + f (13) + f (18)] = 5[19 + 44 + 69 + 94] = 1130. b−a 19. The subinterval width, Δx, is where a and b are endpoints. n , 9−1 = 2. Divide the interval 1 to 9 into four subintervals: Here, Δx = 4 [1, 3], [3, 5], [5, 7], and [7, 9]. The four left endpoints are 1, 3, 5, and 7. 4 b − a∑ Next, use f (xi ) to yield n i=1 352 2[f (1) + f (3) + f (5) + f (7)] = ≈ 3.35. 105 b−a 21. The subinterval width, Δx, is where a and b are endpoints. n , 9−1 Here, Δx = = 2. Divide the interval 1 to 9 into four subintervals: 4 [1, 3], [3, 5], [5, 7], and [7, 9]. The four midpoints are 2, 4, 6, and 8. 4 b − a∑ f (xi ) to yield 2[f (2) + f (4) + f (6) + f (8)] = 248. Next, use n i=1 b−a 23. The subinterval width, Δx, is where a and b are endpoints. Here, n , 3−1 1 = . Divide the interval 1 to 3 into four subintervals: Δx = 4 2 [1, 1.5], [1.5, 2], [2, 2.5], and [2.5, 3]. The four midpoints are 1.25, 1.75, 2.25, and 2.75. Next, use 4 b − a∑ 1 f (xi ) to yield [f (1.25) + f (1.75) + f (2.25) + f (2.75)] n i=1 2 1 = [e5 + e7 + e9 + e11 ] ≈ 34, 611.14. 2

60

INTEGRAL CALCULUS

25. The subinterval width, Δx, is

b−a , where a and b are endpoints. n

7−1 = 1. Divide the interval 1 to 7 into six subin6 tervals [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], and [6, 7]. The six right 6 b − a∑ f (xi ) to yield endpoints are 2, 3, 4, 5, 6, and 7. Next, use n i=1 1[f (2) + f (3) + f (4) + f (5) + f (6) + f (7)]. Using the graph yields 1[6 + 10 + 7 + 4 + 7 + 10] = 44. b−a , where a and b are endpoints. Here, 27. The subinterval width, Δx, is n 10 − 2 = 1. Divide the interval 2 to 10 into eight subintervals: Δx = 8 [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], and [9, 10]. The eight 8 b − a∑ f (xi ) to left endpoints are 2, 3, 4, 5, 6, 7, 8, and 9. Next, use n i=1 Here, Δx =

yield 1[f (2) + f (3) + f (4) + f (5) + f (6) + f (7) + f (8) + f (9)] and using the graph yields 1[6 + 10 + 7 + 4 + 7 + 10 + 14 + 12] = 70. b−a , where a and b are endpoints. 29. The subinterval width, Δx, is n 9−1 = 2. Divide the interval 1 to 9 into four subintervals: Here, Δx = 4 [1, 3], [3, 5], [5, 7], and [7, 9]. The four midpoints are 2, 4, 6, and 8. 4 b − a∑ Next, use f (xi ) to yield 2[f (2) + f (4) + f (6) + f (8)] and using n i=1 the graph yields 2[6 + 7 + 7 + 14] = 68. 31. .a) Using a rectangle and a right triangle the area, geometrically, is (1 × 4) + (1∕2)(4 × 16) = 36 square units. b−a is unity and the four right endpoints b) Using a Riemann sum, n are 1, 2, 3, and 4. The approximate area is 1[f (1) + f (2) + f (3) + f (4)] = 1[5 + 9 + 13 + 17] = 44. This overestimates the actual area since 4x + 1 is increasing on the interval [0, 4]. 4

c)

∫0

(4x + 1)dx = (2x2 + x)|40 = 36

EXERCISES 6.3

EXERCISES 6.3 9

1. . dx = x|94 = 9 − 4 = 5 ∫4 4

3. . 3 dx = 3x|41 = 12 − 3 = 9 ∫1 5 ( )5 5. . (2x + 3)dx = x2 + 3x || = 40 − 10 = 30 |2 ∫2 2

7. . (4x3 + 2x + 5)dx = x4 + x2 + 5x|21 ∫1 = (16 + 4 + 10) − (1 + 1 + 5) = 23 ( 3p )|15 15 e 1 | 3p (e )dp = | = (e45 − 1) 9. .∫ | 3 3 0 |0 6

1 11. . dt = ln t|63 = ln 6 − ln 3 = ln 2 ∫3 t ( ) ( )|3 4 4 4 8 dt = − || = − + 4 = t |1 3 3 t2 ( )|4 4 5x2 | 3 4 (8x + 5x + 2)dx = 2x + + 2x | .∫ | 2 −1 |−1 ( ) 5 1115 = (512 + 40 + 8) − 2 + − 2 = 2 2 9√ )|9 ( 2 3∕2 | 2 52 . t dt = = 18 − = t | ∫1 3 3 3 |1 8√ ( )8 3 3 5∕3 || 96 3 93 . x x2 dx = | = 5 −5= 5 ∫1 5 |1 ( 4 )8 8( ) √ x 3 4∕3 || 3 3 2 3 . x + 3x + x dx = +x + x | | ∫1 4 4 |1 ( ) 1 3 = (1024 + 512 + 12) − +1+ = 1546 4 4 ) ( )3 ( 3( ) 27 x3 || 39 x2 . dx = 4x − − (0 − 0) = 4− | = 12 − ∫0 4 12 ||0 12 4 3

13. . ∫1 15.

17. 19. 21.

23.

61

62

INTEGRAL CALCULUS

25. In the graph (below) the rectangular area between x = 2 and x = 7 with a height of 6 is 6(5) = 30. y

y=6

7 6 5 4 3 2 1 0

x 0

1

2

3

4

5

6

7

8

Using integration 7

∫2

6 dx = (6x)|72 = 42 − 12 = 30.

27. In the graph (below) the area between x = 1 and x = 3 is 10 9 8 7 6 5 4 3 2 1 0

y y = 3x

0

0.5

1

1.5

2

2.5

3

x 3.5

(2 × 3) + (1∕2)(2 × 6) = 12. Using integration 3

∫1

(

3x2 3x dx = 2

)|3 27 3 | − = 12. | = | 2 2 |1

EXERCISES 6.4 1. Setting 8 − x2 = −2x + 5 yields x2 − 2x − 3 = 0. This factors as (x − 3)(x + 1) = 0. The limits of integration are −1 and 3. The area bounded by the curves is 3 [( ) ] 8 − x2 − (−2x + 5) dx. Integration yields ∫−1 ( 3 )|3 3[ ] −x 32 | 2 2 + x + 3x | = . −x + 2x + 3) dx = | ∫−1 3 3 |−1

EXERCISES 6.4

3. Setting x2 =

√

x yields x2 −

63

√ x = 0. Solve this equation for the

limits of integration 0 and 1. The area bounded by the curves is 1 √ ( x − x2 )dx. Integration yields ∫0 )1 ( 3∕2 1 (√ ) 1 x3 || 2x x − x2 dx = − | = . ∫0 3 3 ||0 3 5. Setting x2 + 3 = 4x + 3 yields x2 − 4x = 0. This factors as (x)(x − 4) = 0. The limits of integration are 0 and 4. 4[ ] The area bounded by the curves is (4x + 3) − (x2 + 3) dx. ∫ )4 ( 0 4( ) −x3 || 32 Integration yields 4x − x2 dx = 2x2 − | = . ∫0 3 ||0 3 7. Setting x2 − 3x + 1 = −x + 4 yields x2 − 2x − 3 = 0. This factors as (x − 3)(x + 1) = 0. The limits of integration are −1 and 3. The area bounded by the 3[ ] curves is (−x + 4) − (x2 − 3x + 1) dx. ∫−1 Integration yields ( 3 )|3 3[ ] −x 32 | 2 2 −x + 2x + 3) dx = + x + 3x | = . | ∫−1 3 3 |−1 9. Setting x2 − 4x + 3 = −x2 + x + 3 yields 2x2 − 5x = 0. This factors as (x)(2x − 5) = 0. The limits of integration are 0 and 5/2. The area bounded by the curves is 5∕2 [( ) ( )] −x2 + x + 3 − x2 − 4x + 3 dx. ∫0 Integration yields ( ) 5∕2 5∕2 [ ] −2x3 5x2 || 125 2 −2x + 5x dx = + . | = ∫0 3 2 ||0 24 11. Since the curve y = x2 is above the x-axis on the interval (−2, 3), one ( 3 )|3 3 x | 35 2 integrates (x )dx = | = . | ∫−2 3 |−2 3 13. Setting x3 − x2 + 1 = 2x + 1 yields x3 − x2 − 2x = 0. This factors as (x)(x − 2)(x + 1) = 0. The limits of integration are −1, 0, and 2.

64

INTEGRAL CALCULUS

The area bounded by the curves is 0 [( 2[ ) ] ] 3 2 x − x + 1 − (2x + 1) dx + (2x + 1) − (x3 − x2 + 1) dx. ∫0 ∫−1 Integration yields 0

2

[(x3 − x2 − 2x)]dx + [(−x3 + x2 + 2x)]dx ∫−1 ∫0 ( 4 )|0 )|2 ( 4 −x 37 x3 x3 x | | − − x2 | + + + x2 | = = | | 4 3 4 3 |−1 |0 12 15. The average value is 4 )|4 1 1( 2 2x + 3x | = 13. (4x + 3) dx = |1 4 − 1 ∫1 3 17. The average value is 6( ) 1 1 1 1 1 dx = (ln x)|62 = (ln 6 − ln 2) = ln 3. 6 − 2 ∫2 x 4 4 4 8( ) 1 3 3 19. The average value is x1∕3 dx = x4∕3 |80 = . 8 − 0 ∫0 32 2 21. The average value is ( )|20 20( ) 1 x3 x2 1 1 2 | x + x + 100 dx = + + 100x | ≈205.67. | 20 − 6 ∫6 2 14 6 2 |6 23. One seeks ( )5 5[( 5( ) ] ) 250 x3 || 2 2 −x + 34 − (9) dx = 25 − x dx = 25x − . | = ∫0 ∫0 3 ||0 3

SUPPLEMENTARY EXERCISES CHAPTER 6 4t3 dt = t4 + C. The derivative of t4 + C is 4t3 , ∫ which is in agreement with the integrand. e3x + 3x2 + 5x + C. 3. Integration yields (e3x + 6x + 5)dx = ∫ 3 21 − 3 = 3. 5. The width of the subintervals is Δx = 6 The right endpoints are 6, 9, 12, 15, 18, and 21. 11 − 3 = 2.The left endpoints 7. The width of the subintervals is Δx = 4 are 3, 5, 7, and 9. The fourth Riemann Sum is 2[f (3) + f (5) + f (7) + f (9)] = 2[9 + 25 + 49 + 81] = 328. 1. Integration yields

EXERCISES 6.4

65

9. Integration yields

( )4 2e3x || 2 (5 + 2e )dx = 5x + | = 15 + (e12 − e3 ) ∫1 3 ||1 3 4 ( )|4 2 5∕2 | 62 11. Integration yields x3∕2 dx = = x | ∫1 5 5 |1 3 3 13. Setting x + 1 = x + 1 yields x − x = 0. This factors as (x)(x − 1)(x + 1) = 0. The limits of integration are −1, 0, and 1. The area bounded by the curves is 0 1 4

3x

∫−1

[(x3 + 1) − (x + 1)]dx + 0

∫0

[(x + 1) − (x3 + 1)]dx. 1

Integration yields [(x3 − x)]dx + [(−x3 + x)]dx ∫−1 ∫0 ( 4 )0 )1 ( 4 −x 1 x2 || x2 || x − + = | + | = . 4 2 ||−1 4 2 ||0 2 15. Setting x2 − 7 = −x2 + 2x + 5 yields 2x2 − 2x − 12 = 0. This factors as 2(x − 3)(x + 2) = 0 and the limits of integration are −2 and 3. The area bounded by the curves is 3

∫−2

3

[−x2 + 2x + 5) − (x2 − 7)]dx =

Integration yields 3

∫−2

2

(−2x + 2x + 12)dx =

(

∫−2

(−2x2 + 2x + 12)dx

−2x3 + x2 + 12x 3

17. The average value is ( )2 2 1 8 x3 || 1 2 (4 − x )dx = 4x − | = . 2 − 0 ∫0 2 3 ||0 3

)|3 125 | . | = | 3 |−2

CHAPTER 7

INTEGRATION TECHNIQUES

EXERCISES 7.1 1 1. Let u = 5x + 3, du = 5dx, and du = dx. Therefore, (5 ) 1 −3∕4 −3∕4 1 dx = u du . Rewriting as u−3∕4 du (5x + 3) ∫ ∫ 5 5∫ 4 yields u1∕4 + C. The expression is in terms of x so 5 ∫

4 (5x + 3)−3∕4 dx = (5x + 3)1∕4 + C. 5

1 3. Let u = x2 + 5, du = 2xdx, and du = xdx. Therefore, ( )2 2 4 4 1 x(x + 5) dx = u du ∫ ∫ 2 1 1 Rewriting as u4 du yields u5 + C. The expression is in terms ∫ 2 10 of x so

∫

x(x2 + 5)4 dx =

1 (x2 10

+ 5)5 + C.

Solutions Manual to Accompany Fundamentals of Calculus, First Edition. Carla C. Morris and Robert M. Stark. © 2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

66

EXERCISES 7.1

67

5. Let u = x4 + x2 + 11, du = (4x3 + 2x)dx, and 2du = (8x3 + 4x)dx. Therefore,

∫

(8x3 + 4x)(x4 + x2 + 11)9 dx =

Rewriting as 2

∫

∫

u9 2du

u9 du yields 15 u10 + C. The expression is in terms

1 (8x3 + 4x)(x4 + x2 + 11)9 dx = (x4 + x2 + 11)10 + C. ∫ 5 2 7. Let u = x , du = (2x)dx, and 4du = 8xdx. Therefore, of x so

2

∫

8xex dx =

eu (4du)

∫

Rewriting as 4

∫

x so

9.

11.

13.

15.

2

eu du yields 4eu + C. The expression is in terms of 2

8xex dx = 4ex + C.

∫ Let u = x2 + 3x + 5 and du = (2x + 3)dx. Therefore, 2x + 3 1 = du = ln |u| + C. The expression is in terms of ∫ x2 + 3x + 5 ∫ u 2x + 3 = ln|x2 + 3x + 5| + C. x so ∫ x2 + 3x + 5 1 Let u = x3 − 3x2 + 1, du = (3x2 − 6x)dx, and du = (x2 − 2x)dx. 3 ( ) 1 1 x2 − 2x dx = Therefore, du . ∫ u 3 ∫ x3 − 3x2 + 1 1 1 1 Rewriting as du yields ln|u| + C. The expression is in 3∫ u 3 1 x2 − 2x dx = ln|x3 − 3x2 + 1| + C. terms of x so ∫ x3 − 3x2 + 1 3 1 Let u = x5 + 1, du = (5x4 )dx, and du = (x4 )dx. Therefore, 5 ( ) x4 1 1 dx = du ∫ x5 + 1 ∫ u 5 1 1 1 Rewriting as du yields ln|u| + C. The expression is in ∫ 5 u 5 x4 1 terms of x so dx = ln|x5 + 1| + C. ∫ x5 + 1 5 1 Let u = x2 + 2, du = 2xdx, and du = xdx. Therefore, 2 ) ( ) ( x 1 dx = u−1∕2 du . √ ∫ ∫ 2 x2 + 2

68

INTEGRATION TECHNIQUES

1 u−1∕2 du yields u1∕2 + C. The expression is in ∫ 2 ) ( √ x dx = x2 + 2 + C. terms of x so √ ∫ x2 + 2 2 17. Let u = x + 9, du = 2xdx, and 3du = 6xdx. Therefore, √ 6x x2 + 9dx = u1∕2 (3du). Rewriting as 3 u1∕2 du yields ∫ ∫ ∫ 3∕2 2u + C. The expression is in terms of x so √ 6x x2 + 9 dx = 2(x2 + 9)3∕2 + C. ∫ 1 19. Let u = 3x4 + 5x2 + 8, du = (12x3 + 10x)dx, and du = (6x3 + 5x)dx. 2 ( ) 1 Therefore, (6x3 + 5x)(3x4 + 5x2 + 8)10 dx = u10 du ∫ ∫ 2 1 1 11 Rewriting as u10 du yields u + C. The expression is in 2∫ 22 terms of x so 1 (6x3 + 5x)(3x4 + 5x2 + 8)10 dx = (3x4 + 5x2 + 8)11 + C. ∫ 22 Rewriting as

EXERCISES 7.2 1. Using integration by parts, let f (x) = x, f ′ (x) = 1 dx, g′ (x) = e9x dx, and g(x) = ( ∫

9x

xe dx = x

e9x 9

)

( −

∫

e9x 9

) (1dx) =

e9x . Therefore, 9

xe9x e9x − + C. 9 81

3. Using integration by parts, let f (x) = x, f ′ (x) = dx, g′ (x) = e−x dx, and g(x) = −e−x . Therefore, ∫

xe−x dx = −xe−x +

∫

e−x dx = −xe−x − e−x + C.

EXERCISES 7.2

5. Using integration by parts for the first term, let f (x) = x, f ′ (x) = dx, g′ (x) = e7x dx, and g(x) =

∫

(xe7x + 4x + 3)dx =

e7x . Therefore, 7

xe7x dx + (4x + 3)dx ∫ ( ) xe7x e7x = − dx + 2x2 + 3x + C ∫ 7 7 =

∫

xe7x e7x − + 2x2 + 3x + C. 7 49

7. Using integration by parts, let 1 x4 f (x) = ln 5x, f ′ (x) = dx g′ (x) = x3 dx, and g(x) = . Therefore, x 4 4 3 4 x x x x4 (x3 ln 5x)dx = ln 5x − dx = ln 5x − + C. ∫ ∫ 4 4 4 16 9. Using integration by parts, let 1 f (x) = ln 9x, f ′ (x) = dx g′ (x) = 6x5 dx, and g(x) = x6 . Therefore, x x6 (6x5 ln 9x) dx = x6 ln 9x − x5 dx = x6 ln 9x − + C. ∫ ∫ 6 11. Using integration by parts for the first term, let 1 x9 f (x) = ln 3x, f ′ (x) = dx g′ (x) = x8 , and g(x) = . Therefore, x 9 x8 ln 3xdx + (e2x + 6) dx (x8 ln 3x + e2x + 6) dx = ∫ ∫ ( ) 9 8 2x x x e = ln 3x − dx + + 6x + C ∫ 9 9 2 ∫

=

x9 x9 e2x ln 3x − + + 6x + C. 9 81 2

69

70

INTEGRATION TECHNIQUES

13. Using integration by parts f (x) = 5x, f ′ (x) = 5dx, g′ (x) = (x + 2)−1∕2 dx, and g(x) = 2(x + 1)1∕2 . Therefore, ∫

√

5x

dx =

x+2

∫

5x(x + 2)−1∕2 dx

= 10x(x + 1)1∕2 −

10(x + 2)1∕2 dx ∫ 20 = 10x(x + 2)1∕2 − (x + 2)3∕2 + C. 3

15. Using integration by parts f (x) = x, f ′ (x) = dx, g′ (x) = (x + 4)−2 dx, and g(x) = −(x + 4)−1 . Therefore, −x −x x(x + 4)−2 dx = + (x + 4)−1 dx = + ln|x + 4| + C. ∫ x+4 ∫ x+4 3 17. Using integration by parts f (x) = ln x3 , f ′ (x) = dx, g′ (x) = dx, and x 3 3 3dx = x ln x3 − 3x + C. g(x) = x. Therefore, ln x dx = x ln x − ∫ ∫ 19. Here, the first term requires an integration by parts, the second, a u-substitution, and the last, the basic rules for integration. For first term, let f (x) = 3x, f ′ (x) = 3dx, g′ (x) = ex dx, and g(x) = ex . For u-substitution, let u = x2 and 3du = 6xdx. Therefore, [ ] x x2 5x x x 3e dx (3xe + 6xe + e )dx = 3xe − ∫ ∫ [ ] u + e (3du) + e5x dx ∫ ∫ 2

= 3xex − 3ex + 3ex +

e5x + C. 5

EXERCISES 7.3 1. Use u = x3 , so du = (3x2 )dx. The limits of integration are (1)3 and (4)3 or 1 and 64. Therefore, 4

∫1

64

3x2 ex 3 dx =

∫1

eu du = eu ||1 = e64 − e 64

EXERCISES 7.3

71

3. Using integration by parts, f (x) = x, f ′ (x) = 1dx, g′ (x) = (x − 1)−1∕2 , and g(x) = 2(x − 1)1∕2 . Therefore, 5

)

( √

∫2

x x−1

1∕2

5

dx = 2x(x − 1)1∕2 |52 −

∫2

2(x − 1)

dx

|5 4 (x − 1)3∕2 || 3 |2 ] [ 4 20 4 = [10(2) − 4(1)] − (8) − (1) = . 3 3 3 = 2x(x − 1)1∕2 |52 −

5. Use u = x4 , so 2du = 8x3 dx, and the limits of integration are (1)4 = 1 and (2)4 = 16. Therefore, 2

∫1

16

4

8x3 ex dx =

∫1

16 eu (2du) = 2 eu |16 1 = 2e − 2e

7. Use u = x2 + 144, so 4du = 8xdx, and the limits of integration are (0)2 + 144 = 144 and (5)2 + 144 = 169. Therefore, 5

∫0

√ 8x x2 + 144 dx =

8 3∕2 || u | ∫144 3 |144 8 8 3752 . = (13)3 − (12)3 = 3 3 3 169

169

u1∕2 (4du) =

9. Use u = x2 , so du = 2xdx, and (1)2 = 1 and (2)2 = 4 as the limits of integration for the first term. Use basic integration for the other terms. Therefore, 2

∫1

4

2

(2xex + 4x + 3) dx =

∫1

2

eu du +

∫1

(4x + 3)dx

= eu |41 + (2x2 + 3x)|21 = (e4 − e) + (8 + 6) − (2 + 3) = e4 − e + 9

72

INTEGRATION TECHNIQUES

11. Use u = x3 + 2x2 + 1, so 3du = (9x2 + 12x)dx, and the limits of integration are (0)3 + 2(0)2 + 1 = 1 and (1)3 + 2(1)2 + 1 = 4. Therefore, 1

4

(9x2 + 12x)(x3 + 2x2 + 1)3 dx =

∫0

∫1

u3 (3du)

768 3 765 3u4 || = = − = . 4 ||1 4 4 4 4

1 13. Using integration by parts, f (x) = ln x, f ′ (x) = dx, g′ (x) = x4 , and x x5 g(x) = . Therefore, 5 e

e e 4 |e |e x x5 x5 || x5 | | x ln x dx = ln x| − dx = ln x| − | 5 5 |1 ∫1 5 |1 25 |1 ( 5 ) ( 5 ) e e 1 4e5 1 = −0 − − = + . 5 25 25 25 25 4

∫1

15. Using integration by parts, f (x) = 6x, f ′ (x) = 6dx, g′ (x) = (x + 3)−3 , (x + 3)−2 and g(x) = . Therefore, −2 5

∫1

5

6x(x + 3)−3 dx = −3x(x + 3)−2 |51 +

∫1

3(x + 3)−2 dx

5 3 || (x + 3) ||1 ) ( ) ( 3 3 3 21 −15 + − − = . = 64 16 8 4 64

= −3x(x + 3)−2 |51 −

EXERCISES 7.4 A B 2 . Solving, + = x − 1 x + 1 x2 − 1 A(x + 1) + B(x − 1) = 2 When x = 1, 2A = 2 so A = 1 and when x = −1, −2B = 2 so B = −1. Therefore, ( ) ( ) 2 1 1 dx = + dx = ln|x − 1| − ln|x + 1| + C. ∫ x2 − 1 ∫ x−1 x+1

1. Using partial fractions,

EXERCISES 7.4

73

3. Since the numerator is not of a lesser degree than the denominator, a x+1 . long division reduces the integrand to x + 3 + 2 x −1 A B x+1 . Using partial fractions, + = x − 1 x + 1 x2 − 1 Solving, A(x + 1) + B(x − 1) = x + 1 When x = 1, 2A = 2 and A = 1 and when x = −1, −2B = 0 so B = 0 (note that the result is the same as reducing the fraction algebraically). Therefore, ( 3 ) ) ( x + 3x2 − 2 1 dx dx = x + 3 + ∫ ∫ x−1 x2 − 1 =

x2 + 3x + ln|x − 1| + C. 2

A B 1 . + = 2 x−3 x+3 x −9 Solving, A(x + 3) + B(x − 3) = 1 When x = 3, 6A = 1 and A = 1∕6 and when x = −3, −6B = 1 so B = −1∕6. Therefore,

5. Using partial fractions,

dx = 2 ∫ x −9 ∫ =

[ ( ) ( )] 1 1 1 1 − dx 6 x−3 6 x+3

1 1 ln|x − 3| − ln|x + 3| + C. 6 6

B C x2 + 2 A . Solving, + + = 3 x x − 1 x − 2 x − 3x2 + 2x A(x − 1)(x − 2) + B(x)(x − 2) + C(x)(x − 1) = x2 + 2. When x = 0, 2A = 2 and A = 1, when x = 1, −B = 3 so B = −3, and when x = 2, 2C = 6 so C = 3. Therefore, ) ( [ ] 1 3 3 x2 + 2 dx = − + dx ∫ x x−1 x−2 ∫ x3 − 3x2 + 2x

7. Using partial fractions,

= ln|x| − 3 ln|x − 1| + 3 ln|x − 2| + C

74

INTEGRATION TECHNIQUES

B C 6x2 + 7x − 4 A + + = 3 . Solving, x x+2 x−1 x + x2 − 2x A(x + 2)(x − 1) + B(x)(x − 1) + C(x)(x + 2) = 6x2 + 7x − 4. When x = 0, −2A = −4 and A = 2, when x = −2, 6B = 6 so B = 1, and when x = 1, 3C = 9 so C = 3. Therefore, ( 2 ) [ ] 6x + 7x − 4 2 1 3 dx = + + dx ∫ ∫ x x+2 x−1 x3 + x2 − 2x

9. Using partial fractions,

= 2 ln|x| + ln|x + 2| + 3 ln|x − 1| + C.

EXERCISES 7.5 5−2 1 = . The subintervals are, 6 2 [2, 2.5], [2.5, 3], [3, 3.5], [3.5, 4], [4, 4.5], and [4.5, 5]. The six midpoints are 2.25, 2.75, 3.25, 3.75, 4.25, and 4.75. 5−2 1 3. Here, a = 2, b = 5, n = 6, and Δx = = . The endpoints are 6 2 2, 2.5, 3, 3.5, 4, 4.5, and 5. 4−0 5. Here, a = 0, b = 4, n = 4, and Δx = = 1. The four midpoints 4 are 0.5, 1.5, 2.5, and 3.5. Using the midpoint rule, 1. Here, a = 2, b = 5, n = 6, and Δx =

4

(x2 + 5) dx ≈ 1[f (0.5) + f (1.5) + f (2.5) + f (3.5)]

∫0

= 5.25 + 7.25 + 11.25 + 17.25 = 41. 9−1 = 1. The eight midpoints 8 are 1.5, 2.5, 3.5, 4.5, 5.5, 6.5, 7.5, and 8.5. Using the midpoint rule,

7. Here, a = 1, b = 9, n = 8, and Δx =

9

∫1

(x2 + 9x + 8) dx ≈ 1[f (1.5) + f (2.5) + f (3.5) + f (4.5) + f (5.5) + f (6.5) + f (7.5) + f (8.5)] = [23.75 + 36.75 + 51.75 + 68.75 + 87.75 + 108.75 + 131.75 + 156.75] = 666.

EXERCISES 7.5

9. Here, a = 0, b = 3, n = 6, and Δx = rule,

75

3−0 1 = . Using the trapezoidal 6 2

[ ( ) ( ) 1 1 3 f (0) + 2f + 2f (1) + 2f 4 2 2 ] ( ) 5 + f (3) +2f (2) + 2f 2 1 = [4 + 13.25 + 20 + 29.75 + 44 + 64.25 + 46] 4 = 55.31

3

(x3 + 5x + 4)dx ≈

∫0

11. Here, a = 0, b = 4, n = 4, and Δx = 4

(x2 + 5) dx ≈

∫0

=

4−0 = 1. Using Simpson’s rule, 4

1[ f (0) + 4f (0.5) + 2f (1) + 4f (1.5) + 2f (2) 6 ] +4f (2.5) + 2f (3) + 4f (3.5) + f (4) 1 [5 + 4 (5.25) + 2(6) + 4(7.25) + 2(9) 6

]

+4 (11.25) + 2(14) + 4(17.25) + 21 = 41.3

13. Here, a = 1, b = 9, n = 8, and Δx =

9−1 = 1. Using Simpson’s rule, 8

⎡ f (1) + 4f (1.5) + 2f (2) + 4f (2.5) ⎤ ⎢+2f (3) + 4f (3.5) + 2f (4) + 4f (4.5) ⎥ 1 (x2 + 9x + 8)dx ≈ ⎢+2f (5) + 4f (5.5) + 2f (6) + 4f (6.5) ⎥ ∫1 6 ⎢+2f (7) + 4(f 7.5) + 2f (8) + 4f (8.5)⎥ ⎢ ⎥ ⎣+ f (9) ⎦ 9

⎡18 + 95 + 60 + 147 + 88 + 207 ⎤ 1⎢ ⎥ +120 + 275 + 156 + 351 + 196 6 ⎢+435 + 240 + 527 + 288 + 627 + 170⎥ ⎣ ⎦ 4000 = 666.6. = 6 =

76

INTEGRATION TECHNIQUES

15. Here, a = 2, b = 6, n = 4, trapezoidal rule, 6

∫2

and

Δx =

6−2 = 1. 4

Using

1 f (x)dx ≈ [f (2) + 2f (3) + 2f (4) + 2f (5) + f (6)] 2 1 = [4.1 + 2(5.7) + 2(7.4) + 2(4.5) + 7.8] 2 1 = [47.1] = 23.55. 2

EXERCISES 7.6 ∞

1. . ∫5

t

1 1 dx = lim dx t→∞ ∫5 x − 3 x−3 = lim [ln|t − 3| − ln 2] = ∞. (diverges) t→∞

t [ ] 2 −2 2 2 −4 dx = lim 2x dx = lim + = . t→∞ ∫1 t→∞ 3t3 ∫1 x 4 3 3 ] [ 16 4 4 e4t e16 e 4x 4x e dx = lim e dx = lim − = . 5. t→−∞ ∫t t→−∞ ∫−∞ 4 4 4 ∞

3.

∞

t

e

7. .∫1

3x+1

t→∞ ∫1

] e3t+1 e4 = lim − = ∞. (diverges) t→∞ 3 3 [

∞

9.

e3x+1 dx

dx = lim

t

2

2x e−x dx = lim

t→∞ ∫1

∫1 ∞

11. .∫2

2

2

2xe−x dx = lim [−e−t + e−1 ] = e−1 . t→∞

t

1 1 dx = lim dx t→∞ ∫ x ln x 2 x ln x = lim [ln|ln t| − ln|ln 2|] = ∞. (diverges) t→∞

∞

13. .∫2

√

x2 x3 − 4

∞

dx = lim

√

(x2 )

dx x3 − 4 [ ( )1∕2 2 1∕2 ] 2 3 = ∞. (diverges) − (4) t −4 = lim t→∞ 3 3 t→∞ ∫2

the

EXERCISES 7.6

15. .

2

77

2

∫−∞

2 2 dx = lim dx 3 3 t→−∞ ∫ (4 − x) t (4 − x) [ ] 1 1 1 = lim − = . 2 t→−∞ (4 − 2)2 4 (4 − t)

SUPPLEMENTARY EXERCISES CHAPTER 7 8 1. Let u = x3 + 9, du = 3x2 dx, and du = xdx. Therefore, 3 ( ) √ 8 8x2 x3 + 9 dx = u1∕2 du ∫ ∫ 3 16 16 = u3∕2 + C = (x3 + 9)3∕2 + C. 9 9 3. Let u = x4 + x2 + 5, and du = (4x3 + 2x)dx. Therefore, 4x3 + 2x 1 du = ln|u| + C = ln|x4 + x2 + 5| + C. dx = ∫ u + x2 + 5

∫

x4

5. Let f (x) = 5x, f ′ (x) = 5dx, g′ (x) = (5x + 4)−1∕3 dx, and 3 g(x) = (5x + 4)2∕3 . Therefore, 10 ( ) 3 5x 3 (5x + 4)2∕3 dx dx = x(5x + 4)2∕3 − √ 3 ∫ ∫ 2 2 5x + 4 3 9 = x(5x + 4)2∕3 − (5x + 4)5∕3 + C 2 50 7. Let f (x) = 5x + 1, f ′ (x) = 5dx, g′ (x) = e3x dx, and g(x) =

∫

(5x + 1)e3x dx = (5x + 1)

5 3x e3x e3x 5e3x − e dx = (5x + 1) − + C. 3 ∫ 3 3 9

9. Let u = 5x2 and 2du = 20xdx so 2

∫1

20

2

20xe5x dx =

∫5

e3x . 3

2eu du = 2eu |20 = 2e20 − 2e5 . 5

78

INTEGRATION TECHNIQUES

B 2x2 − 25x − 33 A C + = + . x + 1 (x + 1)2 x − 5 (x + 1)2 (x − 5) 2 2 Solving, A(x + 1)(x − 5) + B(x − 5) + C(x + 1) = 2x − 25x − 33. When x = −1, −6B = 6, and B = 1. When x = 5, 36C = −108 so C = −3. Since (A + C)x2 = 2x2 , A = 5. Therefore, [ ] 2x2 − 25x − 33 3 5 1 − dx = + dx ∫ (x + 1)2 (x − 5) ∫ x + 1 (x + 1)2 x − 5

11. Using partial fractions,

= 5 ln|x + 1| − 13. Here, a = 2, b = 6, Δx = midpoint rule yields

1 − 3 ln|x − 5| + C. x+1

6−2 = 1, and f (x) = ex . Using the 4

1[f (2.5) + f (3.5) + f (4.5) + f (5.5)] = e2.5 + e3.5 + e4.5 + e5.5 . Using the trapezoidal rule yields 1 1 1 [f (2) + 2f (3) + 2f (4) + 2f (5) + f (6)] = e2 + e3 + e4 + e5 + e6 . 2 2 2 Using Simpson’s Rule yields 1 [f (2) + 4f (2.5) + 2f (3) + 4f (3.5) + 2f (4) 6 + 4f (4.5) + 2f (5) + 4f (5.5) + f (6)] 1 = [e2 + 4e2.5 + 2e3 + 4e3.5 + 2e4 + 4e4.5 + 2e5 + 4e5.5 + e6 ] 6 3

15.

∫−∞

[

3

e

2x+1

dx = lim

t→−∞ ∫ t

e

2x+1

dx = lim

t→−∞

] e7 e7 e2t+1 − = . 2 2 2

CHAPTER 8

FUNCTIONS OF SEVERAL VARIABLES

EXERCISES 8.1 1. f (2, 5) = 2(2) + 3(5) = 19, f (3, −1) = 2(3) + 3(−1) = 3, and f (4, −3) = 2(4) + 3(−3) = −1. 3. f (1, 0) = 4(1) + 3(0)2 = 4, f (2, −1) = 4(2) + 3(−1)2 = 11, and f (2, 1) = 4(2) + 3(1)2 = 11. 5. f (1, 2) = (1)2 + 3(1) + (2)3 + 2(2) + 5 = 21, f (0, 1) = (0)2 + 3(0) + (1)3 + 2(1) + 5 = 8, and f (−1, −2) = (−1)2 + 3(−1) + (−2)3 + 2(−2) + 5 = −9. 7. f (2, 0) = (2) + 2(0)3 + e0 = 3, f (0, 1) = (0) + 2(1)3 + e1 = 2 + e, and f (3, 0) = (3) + 2(0)3 + e0 = 4. 9. f (0, 1, 2) = (0)2 + 3(1) + (2)3 = 11, f (0.5, 1, 1) = (0.5)2 + 3(1) + (1)3 = 4.25, and f (−1, 0, 2) = (−1)2 + 3(0) + (2)3 = 9. 11. .f (3 + h, 4) − f (3, 4) = [(3 + h)2 + 42 ] − (32 + 42 ) = (9 + 6h + h2 + 16] − (25) = h2 + 6h

Solutions Manual to Accompany Fundamentals of Calculus, First Edition. Carla C. Morris and Robert M. Stark. © 2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

79

80

FUNCTIONS OF SEVERAL VARIABLES

13. .f (2a, 2b) = 7(2a)2∕3 (2b)1∕3 = 7(2)2∕3 (2)1∕3 a2∕3 b1∕3 = 2(7a2∕3 b1∕3 ) = 2f (a, b). 15. The level curves for f (x, y) = x + y and c = 1, 4, and 9 are shown below: y 15 x+y=9

10 5

x+y=4 x

0 –6

–4

–2

–5

0

2 4 x+y=1

6

8

10

12

–10 –15

17. The level curves for f (x, y) = xy and c = 1, 4, and 9 are shown below: 20 15 10 5 0 –4

–3

–2

–1

0

1

–5 –10 –15 –20

EXERCISES 8.2 1. fx = 20x3 − 6 and fy = −6y + 2 3. fx = 25(5x7 + 4y5 + 7y + 3)24 (35x6 ) and fy = 25(5x7 + 4y5 + 7y + 3)24 (20y4 + 7) 3 5

3 5

5. fx = 3x2 y5 (ex y ) + 36x3 and fy = 5x3 y4 (ex y ) + 3 7. fx =

−y5 5y4 and f = y x x2

2

3

4

EXERCISES 8.3

81

9. First, rewrite the function as f (x, y) = (x2 y)1∕3 . The partial derivatives 1 2 are fx = (x2 y)−2∕3 (2xy) = x−1∕3 y1∕3 and 3 3 1 1 fy = (x2 y)−2∕3 (x2 ) = x2∕3 y−2∕3 3 3 11. Think of the function as the product f (x, y) = (x3 ex )(y8 ), fx = (x3 ex + 3x2 ex )y8 , and fy = x3 ex (8y7 ) 13. First, rewrite the function]as x2 (y − x)1∕3 , [ 1 fx = x2 (y − x)−2∕3 (−1) + (y − x)1∕3 [2x] and 3 [ ] 1 fy = (y − x)−2∕3 (1) (x2 ) 3 15. fx = 15x2 , fy = 4y, and fz = 12z3 17. fx = 3x2 y2 + 4ze4xz , fy = 5 + 2x3 y + 3z4 , and fz = 12yz3 + 4xe4xz 19. fx = 6xy3 + 2 and f (4, 1) = 6(4)(1)3 + 2 = 26 fy = 9x2 y2 + 18y5 and f (4, 1) = 9(4)2 (1)2 + 18(1)5 = 162 21. The marginal productivities are fx = 3x−1∕4 y1∕4 and fy = x3∕4 y−3∕4 . EXERCISES 8.3 1. The first partial derivatives are fx = 15x2 − 4x and fy = 4. The second partial derivatives are fxx = 30x − 4, fyy = 0, and fxy = fyx = 0. 3. The first partial derivatives are fx = 10xy + 9y5 and fy = 5x2 + 45xy4 . The second partial derivatives are fxx = 10y, fyy = 180xy3 , and fxy = fyx = 10x + 45y4 . 5. The first partial derivatives are fx = 3x2 y + 9y4 and fy = x3 + 36xy3 . The second partial derivatives are fxx = 6xy, fyy = 108xy2 , and fxy = fyx = 3x2 + 36y3 . −y2 2y 7. The first partial derivatives are fx = 2 and fy = . The second x x 2y2 −2y 2 partial derivatives are fxx = 3 , fyy = , and fxy = fyx = 2 . x x x 3 y4 2 4 x 7 9. The first partial derivatives are fx = 3x y e + 32x + 28x3 y5 and 3 4

fy = 4x3 y3 ex y + 35x4 y4 . The second partial derivatives are 3 4

3 4

3 4

3 4

fxx = 3x2 y4 [3x2 y4 (ex y )] + (ex y )[6xy4 ] + 224x6 + 84x2 y5 , fyy = 4x3 y3 [4x3 y3 (ex y )] + (ex y )[12x3 y2 ] + 140x4 y3 , and 3 4 3 4 fxy = fyx = 3x2 y4 [4x3 y3 (ex y )] + (ex y )[12x2 y3 ] + 140x3 y4 .

82

FUNCTIONS OF SEVERAL VARIABLES

11. Possible extrema occur when the first partial derivatives fx = 2x and fy = −2y are set to zero. The only possible extremum is at (0, 0). 13. Possible extrema occur when the first partial derivatives fx = 2x + 8y and fy = 8x + 2y are set to zero. The only possible extremum is at (0, 0). 15. To determine possible local extrema for a bivariate function, the first partial derivatives are set to zero. Then, second partial derivatives determine D and the nature of the possible extrema. fx = 2x + 2, fy = 2y − 6. Simultaneously setting these to zero yields (−1, 3) as a possible extremum. Using fxx = 2, fyy = 2, and fxy = fyx = 0, D(−1, 3) = (2)(2) − (0)2 > 0. Since D(−1, 3) is positive and fxx > 0, there is a local minimum f (−1, 3) = −10. 17. To determine possible local extrema for a bivariate function, the first partial derivatives are set to zero. Then, second partial derivatives determine D and the nature of the possible extrema. fx = 6x − 4y + 8, fy = −4x + 6y − 17. Simultaneously setting these to zero yields (1, 7/2) as a possible extremum. Using fxx = 6, fyy = 6, and fxy = fyx = −4, D(1, 7∕2) = (6)(6) − (−4)2 > 0. Since D(1, 7/2) is positive and fxx > 0, there is a local minimum f (1, 7∕2) = 17∕4. 19. To determine possible local extrema for a bivariate, the first partial derivatives are set to zero. Then, second partial derivatives determine D and the nature of the possible extrema. fx = 40 − 2x − y, fy = 50 − 2y − x. Simultaneously setting these to zero yields (10, 20) as a possible extremum. Using fxx = −2, fyy = −2, and fxy = fyx = −1, D(10, 20) = (−2)(−2) − (−1)2 > 0. Since D(10, 20) is positive and fxx < 0, there is a local maximum f (10, 20) = 700. 21. To determine possible local extrema for a bivariate function, the first partial derivatives are set to zero. Then, second partial derivatives determine D and the nature of the possible extrema. fx = 6x2 − 6, fy = −4y + 8. Simultaneously setting these to zero yields (1, 2) and (−1, 2) as possible extrema. Using fxx = 12x, fyy = −4, and fxy = fyx = 0, D(1, 2) = (12)(−4) − (0)2 < 0 indicating no extremum at (1, 2). Next, D(−1, 2) = (−12)(−4) − (0)2 > 0 is positive and fxx < 0, there is a local maximum f (−1, 2) = 27.

EXERCISES 8.4

83

23. To determine possible local extrema for a bivariate function, the first partial derivatives are set to zero. Then, the second partial derivatives determine D and the nature of the possible extrema. fx = 3x2 − 2y, fy = −2x + 12y3 . Simultaneously setting these to √ 4 5 . The possible extrema are (0, 0) and zero yields x = 0 or x = 81 (0.5479, 0.4503). Using fxx = 6x, fyy = 36y2 , and fxy = fyx = −2, D(0, 0) = (6)(0) − (−2)2 < 0 indicating no extremum at (0, 0). Next, D(0.5479, 0.4503) = (3.2874)(7.3) − (−2)2 > 0 and fxx > 0 indicating a local minimum at f (0.5479, 0.4503) = −0.2056. 25. First, xyz = 125 and, z = 125 ∕ xy. One seeks to minimize ) ( ) ( 125 250 250 125 + 2x = 2xy + + . The first partial 2xy + 2y xy xy x y 250 250 derivatives are fx = 2y − 2 and fy = 2x − 2 . Setting these to x y zero yields the trivial solution, x = 0, which is not possible. The other possibility is that x = 5. The possible extrema occur when x and y (and also z) = 5. The second partial derivatives are 500 500 fxx = 3 , fyy = 3 , and fxy = fyx = 2. D = (4)(4) − (2)2 > 0 and x y fxx > 0, so there is a minimum when a cube with edge 5 ft with 150 ft2 of insulation. 27. The first partial derivatives are fx = 3 − y − 2x and fy = 4 − x − 4y. ) ( 8 5 , as a possible extremum. The Setting these to zero yields 7 7 second partial derivatives are fxx = −2, fyy = −4, and fxy = fyx = −1. D = (−2)(−4) − (−1)2 > 0 and fxx < 0. The maximum revenue is ( ) 8 5 154 22 f , = = . 7 7 49 7

EXERCISES 8.4 1. First, E = (6 − B − 3A)2 + (9 − B − 4A)2 + (15 − B − 5A)2 . The first partial derivatives are EA = 2(6 − B − 3A)(−3) + 2(9 − B − 4A)(−4) + 2(15 − B − 5A)(−5) = 100A + 24B − 258

84

FUNCTIONS OF SEVERAL VARIABLES

EB = 2(6 − B − 3A)(−1) + 2(9 − B − 4A)(−1) + 2(15 − B − 5A)(−1) = 24A + 6B − 60 Solving the system 100A + 24B = 258 24A + 6B = 60 9 yields A = 9∕2 and B = −8, so the least squares line is ̂ y = x − 8. 2 3. First, E = (6 − B − 4A)2 + (8 − B − 5A)2 + (4 − B − 6A)2 . The first partial derivatives are EA = 2(6 − B − 4A)(−4) + 2(8 − B − 5A)(−5) + 2(4 − B − 6A)(−6) = 154A + 30B − 176 EB = 2(6 − B − 4A)(−1) + 2(8 − B − 5A)(−1) + 2(4 − B − 6A)(−1) = 30A + 6B − 36 Solving the system 154A + 30B = 176 30A + 6B = 36 yields A = −1 and B = 11, so the least squares line is ̂ y = −x + 11. 5. First, E = (5 − B)2 + (11 − B − 4A)2 + (18 − B − 8A)2 . The first partial derivatives are EA = 2(11 − B − 4A)(−4) + 2(18 − B − 8A)(−8) = 160A + 24B − 376 EB = 2(5 − B)(−1) + 2(11 − B − 4A)(−1) + 2(15 − B − 8A)(−1) = 24A + 6B − 68 Solving the system 160A + 24B = 376 24A + 6B = 68 yields A = 13∕8 and B = 29∕6, so the least squares line is 29 13 ̂ y= x+ . 8 6

EXERCISES 8.4

85

7. First, E = (12 − B − A)2 + (11 − B − 2A)2 + (9 − B − 3A)2 + (6 − B − 4A)2 . The first partial derivatives are EA = 2(12 − B − A)(−1) + 2(11 − B − 2A)(−2) + 2(9 − B − 3A)(−3) + 2(6 − B − 4A)(−4) = 60A + 20B − 170 EB = 2(12 − B − A)(−1) + 2(11 − B − 2A)(−1) + 2(9 − B − 3A)(−1) + 2(6 − B − 4A)(−1) = 20A + 8B − 76 Solving the system 60A + 20B = 170 20A + 8B = 76 yields A = −2 and B = 29∕2, so the least squares line is 29 ̂ y = −2x + . 2 9. First, E = (10 − B − 3A)2 + (27 − B − 4A)2 + (52 − B − 5A)2 + (102 − B − 6A)2 . The first partial derivatives are EA = 2(10 − B − 3A)(−3) + 2(27 − B − 4A)(−4) + 2(52 − B − 5A)(−5) + 2(102 − B − 6A)(−6) = 172A + 36B − 2020 EB = 2(10 − B − 3A)(−1) + 2(27 − B − 4A)(−1) + 2(52 − B − 5A)(−1) + 2(102 − B − 6A)(−1) = 36A + 8B − 382 Solving the system 172A + 36B = 2020 36A + 8B = 382 yields A = 301∕10 and B = −877∕10, so the least squares line is 877 301 x− . ̂ y= 10 10

86

FUNCTIONS OF SEVERAL VARIABLES

11. First, E = (9 − B − 2A)2 + (7 − B − 3A)2 + (6 − B − 4A)2 + (10 − B − 5A)2 . The first partial derivatives are EA = 2(9 − B − 2A)(−2) + 2(7 − B − 3A)(−3) + 2(6 − B − 4A)(−4) + 2(10 − B − 5A)(−5) = 108A + 28B − 226 EB = 2(9 − B − 2A)(−1) + 2(7 − B − 3A)(−1) + 2(6 − B − 4A)(−1) + 2(10 − B − 5A)(−1) = 28A + 8B − 64 Solving the system 108A + 28B = 226 28A + 8B = 64 yields A = 1∕5 and B = 73∕10, so the least squares line is 1 73 ̂ y= x+ . 5 10 13. First, E = (5 − B − A)2 + (9 − B − 3A)2 + (11 − B − 4A)2 . The first partial derivatives are EA = 2(5 − B − A)(−1) + 2(9 − B − 3A)(−3) + 2(11 − B − 4A)(−4) = 52A + 16B − 152 EB = 2(5 − B − A)(−1) + 2(9 − B − 3A)(−1) + 2(11 − B − 4A)(−1) = 16A + 6B − 50 Solving the system 52A + 16B = 152 16A + 6B = 50 yields A = 2 and B = 3, so the least squares line is ̂ y = 2x + 3. The least squares error is E = (5 − 2 − 3)2 + (9 − 3 − 3(2))2 + (11 − 4(2) − 3)2 = 0. Since the points are collinear, there is no estimation error. The points (1, 5), (3, 9), and (4, 11) lie on the line y = 2x + 3 and, therefore, it is the least squares line.

EXERCISES 8.5

87

15. First, E = (25 − B − A)2 + (30 − B − 4A)2 + (32 − B − 5A)2 . The first partial derivatives are EA = 2(25 − B − A)(−1) + 2(30 − B − 4A)(−4) + 2(32 − B − 5A)(−5) = 84A + 20B − 610 EB = 2(25 − B − A)(−1) + 2(30 − B − 4A)(−1) + 2(32 − B − 5A)(−1) = 20A + 6B − 174 Solving the system 84A + 20B = 610 20A + 6B = 174 yields A = 45∕26 and B = 302∕13, so the least squares line is 45 302 ̂ y= x+ . 26 13 The price 2 months after time t is $26.69. EXERCISES 8.5 1. To maximize x2 − y2 subject to the constraint 2x + y = 3, one writes the Lagrangian as F(x, y, 𝜆) = x2 − y2 + 𝜆(3 − 2x − y) The partial derivatives with respect to x and y are Fx = 2x − 2𝜆 Fy = −2y − 𝜆 Setting these to zero yields that x = −2y. Using the partial derivative with respect to 𝜆 yields F𝜆 = 3 − 2x − y. Setting this derivative to zero and using the relationship x = −2y yields x = 2 and y = −1, so the maximum of x2 − y2 is (2)2 − (−1)2 = 3.

88

FUNCTIONS OF SEVERAL VARIABLES

3. To maximize xy subject to the constraint x + y = 16, the Lagrangian is F(x, y, 𝜆) = xy + 𝜆(16 − x − y) The partial derivatives with respect to x and y are Fx = y − 𝜆 Fy = x − 𝜆 Setting these first partials to zero yields x = y. The partial derivative with respect to 𝜆 yields F𝜆 = 16 − x − y. Setting this derivative to zero and using the relationship x = y yields x = 8 and y = 8, so the maximum of xy is (8)(8) = 64. 5. For the point on the parabola y2 = 4x that is closest to (1, 0) minimize the squared distance (x − 1)2 + (y − 0)2 subject to y2 = 4x. The Lagrangian is F(x, y, 𝜆) = (x − 1)2 + y2 + 𝜆(4x − y2 ) The partial derivatives with respect to x and y are Fx = 2(x − 1) + 4𝜆 Fy = 2y − 2y𝜆 Focusing on Fy yields that y = 0 or 𝜆 = 1. If the latter is true, then x = −1, which is not possible (F𝜆 = 4x − y2 ). Therefore, y = 0 and then x = 0. The point (0, 0) is the point on the parabola y2 = 4x that is closest to (1, 0). 7. To find two positive numbers that add to 9 that maximize x2 y, the Lagrangian F(x, y, 𝜆) = x2 y + 𝜆(9 − x − y) The partial derivatives with respect to x and y are Fx = 2xy − 𝜆 Fy = x2 − 𝜆

EXERCISES 8.6

89

The partial derivatives indicate that x = 0 or that x = 2y. If x = 0, the product x2 y is a minimum not a maximum. Therefore, using x = 2y and F𝜆 = 9 − x − y, one determines that x = 6 and y = 3 and that the maximum value of x2 y is 108. 9. Here, one seeks to maximize xy subject to the constraint 16x + 10y = 280 (y is the length of the east and west sides). Forming the Lagrangian, one has F(x, y, 𝜆) = xy + 𝜆(280 − 16x − 10y) The partial derivatives with respect to x and y are Fx = y − 16𝜆 Fy = x − 10𝜆 These partial derivatives indicate that y = (8∕5)x. Substituting this relationship into the partial derivative F𝜆 = (280 − 16x − 10y) yields x = 8.75 ft and y = 14 ft. 11. To minimize 2x2 + 4xy subject to the constraint x2 y = 64,000. The Lagrangian is F(x, y, 𝜆) = 2x2 + 4xy + 𝜆(64,000 − x2 y) The partial derivatives with respect to x and y are Fx = 4x + 4y − 2xy𝜆 Fy = 4x − x2 𝜆 The partial derivative with respect to y has x = 0 and discarded or x = 4. Substituting this information into the partial derivative with 4 respect to x yields that y = . Therefore, x = y. Using 𝜆 F𝜆 = (64, 000 − x2 y), x = 40 in. and y = 40 in. forming a cube with the least amount of material.

EXERCISES 8.6 2

1. First, evaluate

∫−1

(5x − 2y)dy to yield

(5xy − y2 )|2−1 = (10x − 4) − (−5x − 1) = 15x − 3.

90

FUNCTIONS OF SEVERAL VARIABLES

Therefore, ] 3[ 2 3 (15x − 3)dx (5x − 2y) dy dx = ∫0 ∫−1 ∫0 ( )|3 117 15x2 | = − 3x | = . | 2 2 |0 3

(6xy2 − x4 )dy to yield ∫−2 (2xy3 − x4 y)|3−2 = (54x − 3x4 ) − (−16x + 2x4 ) = 70x − 5x4 . Therefore, ] 3[ 3( 3 ) 2 4 (70x − 5x4 )dx = (35x2 − x5 )|31 = 38. 6xy − x dy dx = ∫1 ∫−2 ∫1 ( 4 6 )|2 2 xy | 32x4 4 5 (x y )dy to yield 5. First, evaluate . | = ∫0 6 ||0 3 Therefore, ] ) ( )4 4[ 2 4( 10,912 32x5 || 32x4 4 5 x y dy dx = dx = . | = ∫1 ∫0 ∫1 3 15 ||1 5 x 7. First, evaluate (x + y + 2)dy to yield ∫0 )|x ( ( ) y2 3 2 | + 2y | = x + 2x . xy + | 2 2 |0 Therefore, ] ) ( 3 )|2 2[ x 2( x 3x2 2 | + 2x dx = + x | = 8. (x + y + 2)dy dx = | ∫0 ∫0 ∫0 2 2 |0 ( 2y−x )|x∕2 ( x∕2 ) −x e 1 e | 9. First, evaluate (e2y−x )dy to yield − . | = | ∫0 2 2 2 |0 Therefore, ] 3[ x∕2 ( 3( ) ) 1 e−x 2y−x e dy dx = − dx ∫0 ∫0 ∫0 2 2 )3 ( e−3 e−x || 1 =1+ x+ . = | 2 2 |0 2 3. First, evaluate

y−1

11. First, evaluate (5x − 2y)dx to yield ∫y2 ( 2 )|y−1 5y4 y2 5x 5 | − 2xy | = − + 2y3 + − 3y + . | 2 2 2 2 |y2

EXERCISES 8.6

91

Therefore, ] ) [ y−1 1( 5y4 y2 5 3 + 2y + − 3y + dy − (5x − 2y) dx dy = ∫0 ∫y2 ∫0 2 2 2 )1 ( 5 y4 y3 3y2 5 || −y 7 + + − + y | = . = 2 2 6 2 2 ||0 6 1

SUPPLEMENTARY EXERCISES CHAPTER 8 7 5

1. The first partial derivatives are fx = 7x6 y5 ex y + 36x3 y3 + 32 and 7 5 fy = 5x7 y4 ex y + 27x4 y2 3. .f (mx, my) = k(mx)𝛼 (my)𝛽 = km𝛼+𝛽 x𝛼 y𝛽 = m𝛼+𝛽 ⌊kx𝛼 y𝛽 ⌋ = m1 ⌊kx𝛼 y𝛽 ⌋ = m(f (x, y)). 5. fx = 10(2x3 − 3x + y4 )9 [6x2 − 3] and fy = 10(2x3 − 3x + y4 )9 [4y3 ] 7. The first partial derivatives are fx = 3x2 − 2xy + y2 and fy = −x2 + 2xy − 3y2 . The second partial derivatives are fxx = 6x − 2y, fyy = 2x − 6y, fxy = −2x + 2y, and fyx = −2x + 2y. The mixed derivatives fxy and fyx are equal. 9. First, E = (7 − B − 3A)2 + (15 − B − 4A)2 + (23 − B − 7A)2 . The first partial derivatives are EA = 2(7 − B − 3A)(−3) + 2(15 − B − 4A)(−4) + 2(23 − B − 7A)(−7) = 148A + 28B − 484 EB = 2(7 − B − 3A)(−1) + 2(15 − B − 4A)(−1) + 2(23 − B − 7A)(−1) = 28A + 6B − 90 Solving the system 148A + 28B = 484 28A + 6B = 90 yields A = 48∕13 and B = −29∕13, so the least squares line is 29 48 ̂ y= x− . 13 13

92

FUNCTIONS OF SEVERAL VARIABLES

11. Using E =

n ∑

(yi − 𝛽0 − 𝛽1 xi )2 , the first partial derivatives are

i=1

E𝛽1 = 2(y1 − 𝛽0 − 𝛽1 x1 )(−x1 ) + 2(y2 − 𝛽0 − 𝛽1 x2 )(−x2 ) + · · · + 2(yn − 𝛽0 − 𝛽1 xn )(−xn ) E𝛽0 = 2(y1 − 𝛽0 − 𝛽1 x1 )(−1) + 2(y2 − 𝛽0 − 𝛽1 x2 )(−1) + · · · + 2(yn − 𝛽0 − 𝛽1 xn )(−1) rearranging terms yields 2𝛽1 (x12 + x22 + · · · xn2 ) + 2𝛽0 (x1 + x2 + · · · + xn ) = 2(x1 y1 + x2 y2 + · · · + xn yn ) and 2𝛽1 (x1 + x2 + · · · + xn ) + 2n𝛽0 = 2(y1 + y2 + · · · + yn ) Solving the system of equations as ⌊ ⌋ ∑ ∑ ∑ n 2𝛽1 x2 + 2𝛽0 x = 2 xy ( ∑ )[ ∑ ∑ ] x + 2n𝛽0 = 2 y − x 2𝛽1 yields n𝛽1

∑

x2 − 𝛽1

∑

∑ ∑ x y

∑ ∑ xy − xy − x y SSxy n 𝛽1 = ∑ = (∑ )2 (∑ )2 = SS xx n x2 − x x ∑ x2 − n ∑ ∑ y x and 𝛽0 = − 𝛽1 = y − 𝛽1 x n n 13. Here, to minimize F(x, y) = x2 + y2 subject to x + y = f , form the Lagrangian as F(x, y, 𝜆) = x2 + y2 + 𝜆(f − x − y). The partial derivatives are Fx = 2x − 𝜆 and Fy = 2y − 𝜆 indicating that 𝜆 = 2x = 2y or that x = y. To minimize the sum of squares of two numbers which add to a fixed number, the numbers equal f/2. n

∑

(∑ )2 ∑ ∑ ∑ x y and, therefore, x = n xy −

EXERCISES 8.6

1

15. First, evaluate 2

∫1

[

ex+y dx = (ex+y )|10 = el+y − ey . Next,

]

1

∫0

∫0

e

x+y

2

dx dy =

∫1

e1+y − ey )dy

= (e1+y − ey )|21 = (e3 − e2 ) − (e2 − e) = e3 − 2e2 + e.

93

CHAPTER 9

SERIES AND SUMMATIONS

EXERCISES 9.1 1 1. Here, a = 1 and r = . Therefore, S = 2

3.

5.

7.

9.

1

= 2. 1 1− 2 2 2 1 Here, a = and r = . Therefore, S = 3 = 1. 1 3 3 1− 3 4 2 4 −1 5 Here, a = and r = . Therefore, S = ( )= . −1 5 5 3 1− 5 5 5 1 Here, a = and r = . Therefore, S = 2 = 5. 1 2 2 1− 2 −1 5 Here, a = 5 and r = . Therefore, S = ( ) = 4. −1 4 1− 4

Solutions Manual to Accompany Fundamentals of Calculus, First Edition. Carla C. Morris and Robert M. Stark. © 2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

94

EXERCISES 9.1

1 1 1 + +··· + 10 102 103 1 1 1 1 Therefore, a = and r = and S = 10 = . 1 10 10 9 1− 10 16 16 16 Here, 0.161616 … is rewritten as 2 + 4 + 6 + · · · 10 10 10 16 16 16 1 100 = . Therefore, a = and r = and S = 1 100 100 99 1− 100 135 135 135 Here, 0.135135 … is rewritten as 3 + 6 + 9 + · · · 10 10 10 135 1 Therefore, a = and r = and 1000 1000 135 135 5 1000 S= = = . 1 999 37 1− 1000 4 4 4 Here, 1.44444 … is rewritten as (1) + + +··· + 10 102 103 4 1 Therefore, a = and r = and 10 10 4 4 13 10 = (1) + = . S = (1) + 1 9 9 1− 10 43 43 43 Here, 3.4343 … is rewritten as (3) + 2 + 4 + 6 + · · · 10 10 10 1 43 and r = and Therefore, a = 100 100 43 43 340 100 = (3) + = . S = (3) + 1 99 99 1− 100 ∞ ( )x ∑ 1 1 1 1 1 1 = 3 + 4 + 5 + · · · Therefore, a = and r = and 3 27 3 3 3 3 x=3

11. Here, 0.11111 … is rewritten as

13.

15.

17.

19.

21.

95

S=

1 27 1 1− 3

=

1 . 18

96

23.

SERIES AND SUMMATIONS

∞ ( )x ∑ 3 x=1

5

=

3 32 33 3 3 + + · · · Therefore, a = and r = and + 5 52 53 5 5

3 5

3 = . 3 2 1− 5 25. For the sum 15 + 15(0.80) + 15(0.80)2 + 15(0.80)3 + · · · 15 = 75. a = 15 and r = 0.2, so S = 1 − 0.80 There is a 75 million dollar multiplier effect by introducing 15 million dollars to the economy. S=

EXERCISES 9.2 1. The function and the first three derivatives are f (x) = ex , f ′ (x) = ex , f ′′ (x) = ex , and f ′′′ (x) = ex . Evaluated at x = 0: f (0) = 1, f ′ (0) = 1, f ′′ (0) = 1, and f ′′′ (0) = 1. The third-degree Maclaurin Polynomial is ( 3) ( 2) x x2 x3 x +1 =1+x+ + . 1 + 1(x) + 1 2! 3! 2 6 3. The function and the first three derivatives are f (x) = 2x3 − 3x2 + 1, f ′ (x) = 6x2 − 6x, f ′′ (x) = 12x − 6, f ′′′ (x) = 12. Evaluated at x = 0: f (0) = 1, f ′ (0) = 0, f ′′ (0) = −6, and f ′′′ (0) = 12. The third-degree Maclaurin Polynomial is ( 2) ( 3) x x 1 + 0(x) − 6 + 12 = 1 − 3x2 + 2x3 . 2! 3! 5. The function and the first three derivatives are 1 −1 f (x) = ln(x + 1), f ′ (x) = , and , f ′′ (x) = x+1 (x + 1)2 2 . Evaluated at x = 0: f ′′′ (x) = (x + 1)3 f (0) = 0, f ′ (0) = 1, f ′′ (0) = −1, and f ′′′ (0) = 2. The third-degree Maclaurin Polynomial is( ) ( 2) x x3 x2 x3 0 + 1(x) − 1 +2 =x− + . 2! 3! 2 3

EXERCISES 9.2

97

7. The function and the first three derivatives are 3 3 f (x) = (x + 1)3∕2 , f ′ (x) = (x + 1)1∕2 , f ′′ (x) = (x + 1)−1∕2 , and 2 4 −3 3 ′′′ −3∕2 (x + 1) . Evaluated at x = 0: f (0) = 1, f ′ (0) = , f (x) = 8 2 3 −3 . The third-degree Maclaurin Polynomial f ′′ (0) = , and f ′′′ (0) = 4 ( ) 8( 3) 3 x2 3 x 3 3 1 3 − = 1 + x + x2 − x3 . is 1 + (x) + 2 4 2! 8 3! 2 8 16 9. The function and the first three derivatives are 1 2 1 , f ′′ (x) = , and , f ′ (x) = f (x) = 1−x (1 − x)2 (1 − x)3 6 . Evaluated at x = 0: f (0) = 1, f ′ (0) = 1, f ′′ (0) = 2, f ′′′ (x) = (1 − x)4 and f ′′′ (0) = 6. The third-degree Maclaurin Polynomial is ( 2) ( 3) x x 1 + 1(x) + 2 +6 = 1 + x + x2 + x3 . 2! 3! 11. The function and the first three derivatives are 1 1 2 f (x) = ln x, f ′ (x) = , f ′′ (x) = 2 , and f ′′′ (x) = 3 . Evaluated at x x x x = 1: f (1) = 0, f ′ (1) = 1, f ′′ (1) = −1, and f ′′′ (1) = 2. The third-degree Taylor Polynomial is ( ) ( ) (x − 1)2 (x − 1)3 0 + 1(x − 1) − 1 +2 2! 3! 1 1 = (x − 1) − (x − 1)2 + (x − 1)3 . 2 3 13. The function and first four derivatives are 1 2 6 1 , f ′ (x) = , f ′′ (x) = , f ′′′ (x) = f (x) = 2 3 3−x (3 − x) (3 − x) (3 − x)4 24 and f iv (x) = . Evaluated at x = 2: (3 − x)5 f (1) = 1, f ′ (1) = 1, f ′′ (1) = 2, f ′′′ (0) = 6, and f iv (1) = 24. The fourth-degree Taylor Polynomial is ( ) ( ) ( ) (x − 2)2 (x − 2)3 (x − 2)4 1 + 1(x − 2) + 2 +6 + 24 2! 3! 4! = 1 + (x − 2) + (x − 2)2 + (x − 2)3 + (x − 2)4 .

98

SERIES AND SUMMATIONS

15. The function and the first six derivatives are f (x) = ex , f ′ (x) = ex , f ′′ (x) = ex , f ′′′ (x) = ex , f iv (x) = ex , f v (x) = ex , and f vi (x) = ex . Evaluated at x = 1: f (1) = e, f ′ (1) = e, f ′′ (1) = e, f ′′′ (1) = e, f iv (1) = e, f v (1) = e, and f vi (1) = e. The sixth-degree Taylor Polynomial is (

) ( ) ( ) (x − 1)2 (x − 1)3 (x − 1)4 e + e(x − 1) + e +e +e 2! 3! 4! ( ) ( ) (x − 1)5 (x − 1)6 +e +e 5! 6! (x − 1) 2 (x − 1)3 (x − 1)4 +e +e 2 6 24 5 6 (x − 1) (x − 1) +e +e . 120 720 = e + e(x − 1) + e

17. The function and the first four derivatives are f (x) = e2x , f ′ (x) = 2e2x , f ′′ (x) = 4e2x , f ′′′ (x) = 8e2x , and f iv (x) = 16e2x . Evaluated at x = 3: f (3) = e6 , f ′ (3) = 2e6 , f ′′ (3) = 4e6 , f ′′′ (3) = 8e6 , and f iv (3) = 16e6 . The fourth-degree Taylor Polynomial is ( 6

6

e + 2e (x − 3) + 4e ( + 16e6

(x − 3)4 4!

)

6

(x − 3)2 2!

)

( + 8e

= e6 + 2e6 (x − 3) + 2e6 (x − 3)2 +

6

(x − 3)3 3!

)

4e6 (x − 3)3 2e6 (x − 3)4 + . 3 3

19. The function and the first three derivatives are 3 3 −3 −3 ∕ 2 . x f (x) = x3 ∕ 2 , f ′ (x) = x1 ∕ 2 , f ′′ (x) = x−1 ∕ 2 , and f ′′′ (x) = 2 4 8 3 −3 Evaluated at x = 4: f (4) = 8, f ′ (4) = 3, f ′′ (4) = , and f ′′′ (4) = . 8 64 The third-degree Taylor Polynomial is ( ( 2) 3) 3 (x − 4) 3 (x − 4) − 8 + 3(x − 4) + 8 2! 64 3! 1 3 2 = 8 + 3(x − 4) + (x − 4) − (x − 4)3 . 16 128

EXERCISES 9.3

99

21. Let f (x) = x1 ∕ 2 , then the first few derivatives are 1 −1 −3 ∕ 2 3 , and f ′′′ (x) = x−5∕2 . x f ′ (x) = x−1∕2 , f ′′ (x) = 2 4 8 1 −1 3 Evaluated at x = 1: f (1) = 1, f ′ (1) = , f ′′ (1) = , and f ′′′ (1) = . 2 4 8 The Polynomial takes the form 1 1 1 f (x) = 1 + (x − 1) − (x − 1)2 + (x − 1)3 + · · · 2 8 16 One seeks f (1.04) to three decimal places. Add terms until their difference is less than 0.0005. The estimate is accurate to three decimal places. 1 1 Therefore, f (1.04) = 1 + (0.04) − (0.04)2 ≈ 1.020. 2 8 1 23. Let f (x) = x1 ∕ 3 , then the first few derivatives are f ′ (x) = x−2 ∕ 3 , and 3 −2 −5 ∕ 3 ′′ . The function and its derivatives at x = 27 yield x f (x) = 9 1 −2 f (27) = 3, f ′ (27) = , f ′′ (27) = . The Taylor Polynomial takes 27 2187 the form f (x) = 3 +

1 1 (x − 27) − (x − 27)2 + · · · 27 2,187

One seeks f (26.98) to three decimal places. Add terms until their difference is less than 0.0005. The estimate is accurate to three decimal places. 1 Therefore, f (26.98) = 3 + (−0.02) ≈ 2.999. 27 25. The function and the first three derivatives are f (x) = ex , f ′ (x) = ex , f ′′ (x) = ex , and f ′′′ (x) = ex . Evaluated at x = 0: are f (0) = 1, f ′ (0) = 1, f ′′ (0) = 1, and f ′′′ (0) = 1. The third-degree Taylor (Maclaurin) x2 x3 + and Polynomial is f (x) = 1 + x + 2 6 (0.1)2 (0.1)3 f (0.1) = 1 + (0.1) + + ≈ 1.105 2 6 EXERCISES 9.3 1 , 1+x ′ −2 ′′ −3 ′′′ −4 f (x) = −(1 + x) , f (x) = 2(1 + x) , f (x) = −6(1 + x) , and f iv (x) = 24(1 + x)−5 . Evaluated at x = 0, respectively: 1, −1, 2, −6,

1. The function and its first few derivatives are f (x) =

100

SERIES AND SUMMATIONS

and 24. Therefore, the expansion yields ( 3) ( 4) ( ) ( 2) (−x)n x x x −6 + 24 + · · · n! +··· 1 − 1(x) + 2 2! 3! 4! n! Simplifying yields 1 = 1 − x + x2 − x3 + x4 + · · · + xn + · · · 1+x ∞ ∑ = (−x)n for − 1 < x < 1. n=0

1 , 1 + 2x f ′ (x) = −2(1 + 2x)−2 , f ′′ (x) = 8(1 + 2x)−3 , f ′′′ (x) = −48(1 + 2x)−4 , and f iv (x) = 384(1 + 2x)−5 . Evaluated at x = 0, respectively: 1, −2, 8, −48, and 384. Therefore, the expansion yields ( 2) ( 3) ( 4) ( n) x x x x 1 − 2(x) + 8 − 48 + 384 + · · · n!(−2)n +··· 2! 3! 4! n!

3. The function and its first few derivatives are f (x) =

Simplifying yields 1 = 1 + (−2x) + (−2x)2 + (−2x)3 + (−2x)4 1 + 2x + · · · + (−2x)n + · · · =

∞ ∑

(−2x)n .

n=0

(Notice that it is simpler to replace x in Exercise 1 by (2x), and the −1 1 series is only valid for < x < .) 2 2 1 , 5. The function and its first few derivatives are f (x) = 1 + x2 6x2 − 2 24x − 24x3 , f ′′′ (x) = , and f ′ (x) = −2x(1 + x2 )−2 , f ′′ (x) = 2 3 (1 + x ) (1 + x2 )4 120x4 − 240x2 + 24 . Evaluated at x = 0, respectively: 0, −2, f iv (x) = (1 + x2 )5 0, and 24. Therefore, the expansion yields (( )n ) ( 2) ( 3) ( 4) −x2 x x x 1 + 0(x) − 2 +0 + 24 + · · · n! +··· 2! 3! 4! n!

EXERCISES 9.3

101

Simplifying yields ∑ 1 2 2 2 2 n = 1 + (−x ) + (−x ) + · · · + (−x ) + · · · = (−x2 )n . 1 + x2 n=0 ∞

(Notice that it is simpler to replace x by x2 in Exercise 1 for −1 < x < 1.) 7. The function and its first few derivatives are f (x) = ln(1 + x), 1 , f ′′ (x) = −(1 + x)−2 , f ′′′ (x) = 2(1 + x)−3 , and f ′ (x) = 1+x f iv (x) = −6(1 + x)−4 . Evaluated at x = 0, respectively: 0, 1, −1, 2, and −6. Therefore, the expansion yields ( 3) ( 4) ( 2) ( n) x x x x +2 −6 + · · · (−1)n+1 (n − 1)! +··· 0 + 1(x) − 2! 3! 4! n! Simplifying,

∞ n ∑ xn x2 x3 x4 n+1 x (−1)n+1 + − + · · · + (−1) +···= ln(1 + x) = x − 2 3 4 n n n=1 ∞ n+1 ∑ x or (−1)n n+1 n=0 1 dx = ln(1 + x), Exercise 1 could be integrated term Since ∫ 1+x by term to yield the series. 9. The function and its first few derivatives are 1 f (x) = 4ex ∕ 2 , f ′ (x) = 2ex ∕ 2 , f ′′ (x) = ex ∕ 2 , f ′′′ (x) = ex ∕ 2 , and 2 1 1 1 f iv (x) = ex ∕ 2 . Evaluated at x = 0, respectively: 4, 2, 1, , and . 4 2 4 Therefore, the expansion yields ( ) ( ) ( 2) ( )n ( n ) x 1 x3 1 x4 1 x + + + · · ·4 +··· 4 + 2(x) + 2! 2 3! 4 4! 2 n!

Simplifying, 4ex ∕ 2

( )n x 2 3 4 x x x + + + · · · + (4) 2 + · · · = 4 + 2x + 2 12 96 n! ∞ n n ∑ (1 ∕ 2) x =4 . n! n=0

(Also in the series for ex replace x by x ∕ 2 and multiply by 4.)

102

SERIES AND SUMMATIONS

11. The function and its first few derivatives are f (x) = xex − x, f ′ (x) = (x + 1)ex − 1, f ′′ (x) = (x + 2)ex , f ′′′ (x) = (x + 3)ex , and f iv (x) = (x + 4)ex . The function and these derivatives evaluated at x = 0 are 3, 0, 2, 3, and 4. Therefore, the expansion yields ( 2) ( 3) ( 4) (x)n x x x 0 + (0)(x) + 2 +3 +4 + · · ·n +··· 2! 3! 4! n! =

∞ ∑ n=2

xn . (n − 1)!

13. Use the expansion for ex so [ ] 1 x 1 x2 x3 xn −x (e + e ) = 1+x+ + +···+ +··· 2 2 2! 3! n! [ ] (−x)2 (−x)3 (x)n 1 1 + (−x) + + +···+ +··· + 2 2! 3! n! [ ( 2) ( 3) ( 4) ] 1 x x x = 2 + 0 (x) + 2 +0 +2 +··· 2 2! 3! 4! =1+

x2 x4 xn + +··· 2! 4! n!

n = 0, 2, 4 …

∑ (x)2n 1 or (ex + e−x ) = 2 (2n)! n=0 ∞

15. The ) ( and ) its first four derivatives of ln x at x = 1∕ 2 are ( function 1 1 = ln f 2 2 ( ) ( ) 1 1 1 −1 1 = f ′′ (x) = 2 , f ′′ = −4 f ′ (x) = , f ′ x 2 2 2 x ( ) ( ) 2 −6 1 1 f ′′′ (x) = 3 , f ′′′ = 16 f iv (x) = 4 , f iv = −96 2 2 x x Substituting yields, ) ) ) ( ( ( 1 2 1 3 1 4 x− x− x− ( ) ( ) 1 1 2 + 16 2 − 96 2 +··· +2 x− −4 ln 2 2 2! 3! 4! ( ) 1 n ∞ (−1)n+1 (2)n x − ( ) ∑ 1 2 . ln + 2 n n=1

EXERCISES 9.4

103

17. Exercise 9.3.1 yielded (for −1 < x < 1) ∑ 1 (−x)n = 1 − x + x2 − x3 + x4 + · · · + xn + · · · = 1+x n=0 ∞

1 dx = (1 − x + x2 − x3 + x4 + · · · + (−x)n + · · ·)dx ∫ 1+x ∫ =

∞ ∑

∫

(−x)n dx

n=0

ln | x + 1 | + C = x −

(−1)n xn+1 x2 x3 x4 x5 + − + −···+ +··· 2 3 4 5 n+1

If x = 0 then ln 1 + C = 0, so C = 0. ln | x + 1 | =

∞ ∑ (−1)n xn+1 n=0

n+1

EXERCISES 9.4 1. The integral test yields ∞

t

1 1 dx = lim dx = lim (ln t − ln 3) → ∞. t→∞ ∫3 x t→∞ ∫3 x The series is divergent. ∞ t ln x ln x dx = lim dx. 3. The integral test yields t→∞ ∫1 ∫1 x ] [ x 2 (ln 1)2 (ln t) Using substitution to integrate yields lim − → ∞. t→∞ 2 2 The series is divergent. ∞ t 1 1 dx = lim dx. Using 5. The integral test yields t→∞ ∫2 x(ln x)3 ∫2 x(ln x)[3 ] 1 1 1 substitution to integrate yields lim − = + . 2 2 t→∞ 2(ln 2) 2(ln 2)2 2(ln t) The limit is

1 so the series is convergent. 2(ln 2)2 ∞

7. The integral test yields

∫2

t

2

2xe−x dx = lim

t→∞ ∫2 2

2

2xe−x dx.

Using substitution to integrate yields lim [−e−t + e−4 ] = e−4 . t→∞

The limit is e−4 so the series is convergent.

104

SERIES AND SUMMATIONS

√ √ 9. First, n3 + 1 > n3 and n3 + 1 > n3 . It follows that ∞ ∞ ∑ ∑ 1 1 1 1 < √ . Therefore, < √ √ √ . The summan3 + 1 n3 + 1 n=1 n3 n=1 n3 ∞ ∑ 1 ∑ 1 tion is a p series that converges since 3∕2 > 1. √ = n3∕2 n=1 n3 A series which is term for term less than a convergent series is ∞ ∑ 1 is convergent. convergent. Therefore, √ n3 + 1 n=1 √ √ 11. First, n2 − 1 < n2 and n2 − 1 < n2 . It follows that ∞ ∞ ∑ ∑ 1 1 1 1 > √ . Therefore, > √ √ √ . n=1 n2 − 1 n2 n2 − 1 n=1 n2 ∞ ∞ ∑ ∑ 1 1 The summation is the divergent harmonic series √ = n=1 n2 n=1 n (p = 1). Any series that is term for term greater than a divergent series ∞ ∑ 1 is also divergent. Therefore, is divergent. √ n=2 n2 − 1 1 1 < 3 . Multiplying by n yields 13. First, n3 + 1 > n3 and 3 n +1 n n 1 n = . Therefore, < n∞3 + 1 n3 ∞ n2 ∞ ∑ ∑ n ∑ n 1 = a convergent p series (p = 2). < 3 3 n + 1 n=1 n n2 n=1 n=1 1 2 3 4 n Therefore, + + + · · · is convergent. + +···+ 3 2 9 28 65 n +1 ln n n1∕2 1 15. First, ln n < n1 ∕ 2 and therefore, 2 < 2 = 3∕2 . n n n ∞ ∞ ∑ ln n ∑ 1 It follows that < . n2 n3∕2 n=2 n=2 A convergent p series with p = 3 ∕ 2 > 1. A series term for term less than a convergent series also converges. ln n ln 2 ln 3 ln 4 + + + · · · + 2 + · · · is convergent. Therefore, 4 9 16 n ∞ 1 17. Using an integral test to evaluate dx. Using a substitution p ∫2 x(ln ( ) x) ln 2 t yields lim − . p t→∞ (1 − p) t (1 − p)(ln 2)p

EXERCISES 9.4

105

When p < 1 the limit is infinite, so the integral is divergent and when ∞ ln 2 1 1 p > 1, dx = and is convergent. ∫2 x(ln x)p 1 − p (ln 2)p 3n 3n+1 19. Here, an = and an+1 = . The ratio test relies on the limit: n! (n + 1)! | 3n+1 | | | | (n + 1)! | | an+1 | | 3n+1 n! | | | | = lim | lim || ⋅ n || | = lim || | n | n→∞ | (n + 1)! 3 | n→∞ | an | n→∞ | | 3 | | n! | | | | 3 | | = 0 < 1. = lim || n→∞ | n + 1 || ∞ ∑ 3n 3 9 27 The series = + + + · · · is convergent. n! 1 2 6 n=0 4n 4n+1 . The ratio test relies on the limit: 21. Here, an = 3 and an+1 = n (n + 1)3 | 4n+1 | | | | (n + 1)3 | | 4n+1 | an+1 | n3 || | | | | = lim | lim || ⋅ | = lim | | | n | n→∞ | (n + 1)3 4n | n→∞ | an | n→∞ | | 4 | | | | n3 | | | | n3 || | = lim |4 | = 4 > 1. n→∞ | (n + 1)3 | | | ∞ ∑ 4n

4 16 64 + + + · · · is divergent. 1 8 27 n=1 (−1)n 6n (−1)n+1 6n+1 and an+1 = . The ratio test relies on 23. Here, an = n! (n + 1)! the limit: The series

n3

=

| (−1)n+1 6n+1 | | | | (n + 1)! | | an+1 | | (−1) (6)n! | | | | = lim | | lim || | = lim || | n n | n→∞ | (n + 1)! || n→∞ | an | n→∞ | (−1) 6 | | | | n! | | | −6 | | = 0 < 1. = lim || n→∞ | n + 1 || The series

∞ ∑ (−1)n 6n n=0

n!

=

1 6 36 − + + · · · is convergent. 1 1 2

106

SERIES AND SUMMATIONS

EXERCISES 9.5 1. The arithmetic sequence 1 + 4 +)7 + · · · + 40 has n = 14, a = 1, and ( 13 d = 3. Its sum is 14 1 + (3) = 287. 2 3. The arithmetic sequence + 19 + · · · + 35 has n = 7, a = 11, ( 11 + 15 ) 6 and d = 4. Its sum is 7 1 + (4) = 161. 2 ( )2 ( )3 ( )4 ( )5 3 3 3 3 3 5. The geometric sequence + + + + has 2 2 2 2 2 ( )5 ⎛1 − 3 ⎞ 3⎜ 2 ⎟ = 633 . a = 3 ∕ 2, r = 3 ∕ 2, and n = 5. Its sum is ⎜ 3 ⎟⎟ 2⎜ 32 1− ⎝ 2 ⎠ 1 1 1 1 1 7. The geometric sequence + 2 + 3 + 4 + · · · + 12 has a = 1 ∕ 5, 5 5 5 5 5 ( )5 ⎛1 − 1 ⎞ 1⎜ 5 ⎟ = 61, 035, 156 . r = 1 ∕ 5, and n = 12. Its sum is ⎜ 1 ⎟⎟ 244, 140, 625 5⎜ 1− ⎝ 5 ⎠ 5 52 53 54 512 9. − 4 + 7 − 10 + · · · − 34 has a = 5 ∕ 2, r = −5 ∕ 8, and n = 12. 2 2 2 2 2 ( )12 ⎛1 − −5 ⎞ ⎟ 5⎜ 8 Its sum is ⎜ ( ) ⎟ ≈ 1.533. 5 ⎟ 2⎜ 1− − ⎝ 8 ⎠ 11. 10 + 12 + 14.4 + · · · + 24.8832 has a = 10, r = 1.2, and n = 6. ) ( 1 − (1.2)6 62, 062 = . Its sum is 10 1 − (1.2) 625 ) ( 1 − (5)6 = 11, 718. 13. Here, a = 3, r = 5, and n = 6. Its sum is 3 1 − (5) 15. Here, a = 3, r = −2, and n = 10. Its sum is ) ( 1 − (−2)10 = −1023. 3 1 − (−2) 17. Here, a = 1 and r = 0.51, so one seeks n such that ) ( 1 − (0.51)n < 2. Therefore, 1 1 − (0.51)

EXERCISES 9.5

107

2.0408163 ⌊1 − (0.51)n ⌋ < 2 ⌊1 − (0.51)n ⌋ < 0.98 (0.51)n > 0.02 n

55) = 0.30 + 0.10 + 0.20 = 0.60 P(x ≤ 58) = 0.20 + 0.15 + 0.05 + 0.30 + 0.10 = 0.80 P(52 ≤ x < 60) = 0.15 + 0.05 + 0.30 + 0.10 = 0.60 P(x ≤ 57) = 0.20 + 0.15 + 0.05 + 0.30 = 0.70 P(x = 59) = 0

Solutions Manual to Accompany Fundamentals of Calculus, First Edition. Carla C. Morris and Robert M. Stark. © 2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

110

EXERCISES 10.2

111

7. There are two properties to check: First, that f (x) is nonnegative on the 8 8 1 8 1 1 | interval [1, 8], which it is. Next, that dx = x|| = − = 1. ∫1 7 7 |1 7 7 9. There are two properties to check: First, that f (x) is nonnegative on the interval [0, 10] which it is. Next, that 10 10 1 100 x2 || 0 = xdx = − = 1. ∫0 50 100 ||0 100 100 11. There are two properties to check: First, that f (x) is nonnegative on the 1

3x2 dx = x3 ||0 = 1 − 0 = 1. ∫0 13. There are two properties to check: First, that f (x) is nonnegative on the interval [0, 1] which it is. Next, that

1

1

1 4x3 dx = x4 ||0 = 1 − 0 = 1. ∫0 15. There are two properties to check: First, that f (x) is nonnegative on the interval [0, ∞) which it is. Next, that

interval [0, 1] which it is. Next, that

∞

t

3e−3x dx = lim

∫0

3e−3x dx = lim (−e−3t + 1) = 1.

t→∞ ∫0

t→∞

7 2 5 1 | 1 17. . dx = x|| = − = . ∫2 7 7 |2 7 7 7 7 7 1 49 1 2 || 9 40 2 19. . xdx = x| = − = = . ∫3 50 100 |3 100 100 100 5 7

7

1

1 21. . 3x2 dx = x3 || = 1 − 1 = 7 . |0.5 ∫0.5 8 8 0.8

|0.8 23. . 4x3 dx = x4 | = 0.4096 − 0.001 = 0.4095 |0.1 ∫0.1 5

| 25. . 3e−3x dx = −e−3x | = e−1 − e−15 ≈ 0.3679 |1∕3 ∫1∕3 5

EXERCISES 10.2 1. .E(x) = 50(0.20) + 100(0.10) + 150(0.30) + 200(0.40) = 145 𝜎 2 = (50 − 145)2 (0.20) + (100 − 145)2 (0.10) + (150 − 145)2 (0.30) + (200 − 145)2 (0.40) = 3225 √ 𝜎 = 3225 = 56.789

112

APPLICATIONS TO PROBABILITY

3. .E(x) = 1(0.15) + 4(0.15) + 7(0.25) + 10(0.20) + 12(0.25) = 7.5 𝜎 2 = (1 − 7.5)2 (0.15) + (4 − 7.5)2 (0.15) + (7 − 7.5)2 (0.25) + (10 − 7.5)2 (0.20) + (12 − 7.5)2 (0.25) = 14.55 √ 𝜎 = 14.55 = 3.814 5. The mean, E(x); variance, 𝜎 2 , and standard deviation, 𝜎 are

∫1

8 1 64 x2 || 1 63 = xdx = − = = 4.5. 7 14 ||1 14 14 14

(

8

8

E(x) =

𝜎2

=

∫1 √

𝜎=

) 8 1 2 x3 || 343 49 − (4.5)2 = x dx − (4.5)2 = = , and | 7 21 |1 84 12

49 = 2.0207. 12

7. The mean, E(x); variance, 𝜎 2 , and standard deviation, 𝜎 are 10

E(x) =

∫0 (

𝜎2

10 0 20 1 2 x3 || 1000 x dx = − = = | 50 150 |0 150 150 3 10

=

) ( )2 10 ) ( 1 3 50 20 x4 || 400 x dx − = , and = − | 50 3 200 |0 9 9

∫0 √ 50 𝜎= = 2.357. 9

9. The mean, E(x); variance, 𝜎 2 , and standard deviation, 𝜎 are 3x3 dx =

∫0 (

𝜎 = 2

√ 𝜎=

)

1

∫0

3x4 || 3 3 = −0= . | 4 |0 4 4 1

1

E(x) =

4

3x dx −

1 ( )2 ( ) 3 9 3 3 3x5 || 9 = − = , and = − 4 5 ||0 16 5 16 80

3 = 0.1936. 80

EXERCISES 10.3

113

11. The mean, E(x); variance, 𝜎 2 , and standard deviation, 𝜎 are 4x5 || 4 = . | ∫0 5 |0 5 ( 1 ) ( )2 1 ( ) 4 2x6 16 2 2 5 𝜎 = 4x dx − = − = , and ∫0 5 3 0 25 75 √ 2 𝜎= = 0.1633. 75 1

1

4x4 dx =

E(x) =

13. The mean, E(x); variance, 𝜎 2 , and standard deviation, 𝜎 are ∞

E(x) =

∫0

t

3xe−3x dx = lim [(

t→∞ ∫0

3xe−3x dx

) t] −3x | 1 e | −xe−3x − = lim | = . | t→∞ 3 |0 3 ( ∞ ) ( )2 ) ( ) ( t 1 1 2 2 −3x 2 −3x 3x e dx − = lim 3x e dx − 𝜎 = t→∞ ∫0 ∫0 3 9 [( ] ) t ( ) 1 2xe−3x 2e−3x || 1 −x2 e−3x − = lim − = and | − | t→∞ 3 9 9 9 |0 1 𝜎= . 3 EXERCISES 10.3 1. A bell curve sketch is useful. In this case, the probabilities are obtained directly from a Standard Normal Table. a) P(0 ≤ z ≤ 1.47) = 0.4292 d) P(−1.24 < z < 0) = 0.3925 b) P(0 ≤ z ≤ 0.97) = 0.3340 e) P(−2.13 ≤ z ≤ 0) = 0.4834 c) P(−2.36 < z < 0) = 0.4909 f) P(−0.19 ≤ z ≤ 0) = 0.0753 3. .a) This includes the entire upper half of the distribution. Add 0.5000 and 0.4394 (P(−1.55 < Z < 0)) to yield 0.9394. b) This is in the upper tail, so 0.5000 − 0.4686 = 0.0314. c) This is the lower tail, so 0.5000 − 0.4292 = 0.0708. d) This is the entire lower half of the curve plus P(0 < Z < 1.30). Therefore, 0.5000 + 0.4032 = 0.9032.

114

APPLICATIONS TO PROBABILITY

(

640 − 550 460 − 550 ≤x≤ 100 100 = P(−0.90 ≤ z ≤ 0.90)

5. .a) .P(460 ≤ x ≤ 640) = P

)

= 0.3159 + 0.3159 = 0.6318. ( ) 730 − 550 b) P . (x ≤ 730) = P z ≤ 100 = P(z ≤ 1.80) = 0.5000 + 0.4641 = 0.9641. ( ) 410 − 550 c) .P(x ≥ 410) = P z ≥ 100 = P(z ≥ −1.40) = 0.5000 + 0.4192 = 0.9192. 7. Here, 𝜇 = 128.4. One seeks P(x ≤ 128) ≤ 0.01. The z score corresponding to this probability is −2.33. Therefore, 128 − 128.4 −2.33 = 𝜎 Solving yields a value of 0.171674 for the standard deviation 𝜎.

SUPPLEMENTARY EXERCISES CHAPTER 10 1. .a) 0.13 + 0.17 + 0.35 + 0.45 = 1.1, which exceeds unity; it is not a probability distribution. b) 0.33 + 0.27 + 0.22 + 0.18 = 1, and each probability is nonnegative, so it is a probability distribution. 3. .a) b) c) d)

P(15) + P(16) + P(18) + P(20) = 0.30 + 0.05 + 0.20 + 0.10 = 0.65 P(18) + P(20) = 0.20 + 0.10 = 0.30 P(12) + P(15) = 0.20 + 0.30 = 0.50 P(x ≤ 18) = 1 − P(20) = 1 − 0.10 = 0.90

5. First, f (x) is nonnegative on [0, 1]. Second, 1

∫0

1 5x4 dx = x5 ||0 = 1 − 0 = 1.

Therefore, f(x) satisfies both criteria for a probability density function.

EXERCISES 10.3

115

7. The mean, E(x); variance, 𝜎 2 , and standard deviation, 𝜎 are 5x6 || = ∫0 6 ||0 1 ( )2 5 5x6 dx − = 𝜎2 = ∫0 6 √ 5 𝜎= = 0.141. 252 1

1

E(x) =

5x5 dx =

5 . 6 1 ( )2 5x7 || 5 5 25 5 − = − = and 7 ||0 6 7 36 252

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