Instructor Manual for Calculus with Applications, 12th Edition.
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CONTENTS PREFACE...................................................................................................................................................................
ix
HINTS FOR TEACHING CALCULUS WITH APPLICATIONS ............................................................................
xi
PRETESTS ................................................................................................................................................................. xix ANSWERS TO PRETESTS ....................................................................................................................................... xxvii FINAL EXAMINATIONS ......................................................................................................................................... xxix ANSWERS TO FINAL EXAMINATIONS............................................................................................................... xliii
SOLUTIONS TO ALL EXERCISES CHAPTER R ALGEBRA REFERENCE R.1 Polynomials ..................................................................................................................................................
1
R.2 Factoring .......................................................................................................................................................
3
R.3 Rational Expressions ....................................................................................................................................
4
R.4 Equations ......................................................................................................................................................
8
R.5 Inequalities ...................................................................................................................................................
14
R.6 Exponents .....................................................................................................................................................
24
R.7 Radicals ........................................................................................................................................................
28
CHAPTER 1 LINEAR FUNCTIONS 1.1 Slopes and Equations of Lines ......................................................................................................................
33
1.2 Linear Functions and Applications................................................................................................................
46
1.3 The Least Squares Line .................................................................................................................................
54
Chapter 1 Review Exercises..................................................................................................................................
66
Extended Application: Predicting Life Expectancy ..............................................................................................
74
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CHAPTER 2 NONLINEAR FUNCTIONS 2.1 Properties of Functions..................................................................................................................................
77
2.2 Quadratic Functions; Translation and Reflection ..........................................................................................
86
2.3 Polynomial and Rational Functions .............................................................................................................. 102 2.4 Exponential Functions ................................................................................................................................... 115 2.5 Logarithmic Functions .................................................................................................................................. 125 2.6 Applications: Growth and Decay; Mathematics of Finance .......................................................................... 137 Chapter 2 Review Exercises.................................................................................................................................. 146 Extended Application: Power Functions ............................................................................................................... 162
CHAPTER 3 THE DERIVATIVE 3.1 Limits ............................................................................................................................................................ 163 3.2 Continuity...................................................................................................................................................... 177 3.3 Rates of Change ............................................................................................................................................ 184 3.4 Definition of the Derivative .......................................................................................................................... 195 3.5 Graphical Differentiation .............................................................................................................................. 219 Chapter 3 Review Exercises.................................................................................................................................. 223 Extended Application: A Model for Drugs Administered Intravenously .............................................................. 235
CHAPTER 4 CALCULATING THE DERIVATIVE 4.1 Techniques for Finding Derivatives .............................................................................................................. 237 4.2 Derivatives of Products and Quotients .......................................................................................................... 247 4.3 The Chain Rule ............................................................................................................................................. 258 4.4 Derivatives of Exponential Functions ........................................................................................................... 269 4.5 Derivatives of Logarithmic Functions ........................................................................................................... 283 Chapter 4 Review Exercises.................................................................................................................................. 296 Extended Application: Electric Potential and Electric Field ................................................................................. 308
iv
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CONTENTS
CHAPTER 5 GRAPHS AND THE DERIVATIVE 5.1 Increasing and Decreasing Functions ............................................................................................................ 309 5.2 Relative Extrema ........................................................................................................................................... 324 5.3 Higher Derivatives, Concavity, and the Second Derivative Test .................................................................. 339 5.4 Curve Sketching
............................................................................................................................... 362
Chapter 5 Review Exercises.................................................................................................................................. 383 Extended Application: A Drug Concentration Model for Orally Administered Medications ............................... 400
CHAPTER 6 APPLICATIONS OF THE DERIVATIVE 6.1 Absolute Extrema .......................................................................................................................................... 401 6.2 Applications of Extrema................................................................................................................................ 411 6.3 Further Business Applications....................................................................................................................... 432 6.4 Implicit Differentiation ................................................................................................................................. 440 6.5 Related Rates ................................................................................................................................................. 457 6.6 Differentials: Linear Approximation ............................................................................................................. 466 Chapter 6 Review Exercises.................................................................................................................................. 472 Extended Application: A Total Cost Model for a Training Program .................................................................... 483
CHAPTER 7 INTEGRATION 7.1 Antiderivatives .............................................................................................................................................. 485 7.2 Substitution ................................................................................................................................................... 497 7.3 Area and the Definite Integral ....................................................................................................................... 507 7.4 The Fundamental Theorem of Calculus ........................................................................................................ 521 7.5 The Area Between Two Curves .................................................................................................................... 538 7.6 Numerical Integration ................................................................................................................................... 556 Chapter 7 Review Exercises.................................................................................................................................. 570 Extended Application: Estimating Depletion Dates for Minerals ......................................................................... 588
CONTENTS
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CHAPTER 8 FURTHER TECHNIQUES AND APPLICATIONS OF INTEGRATION 8.1 Integration by Parts ....................................................................................................................................... 589 8.2 Volume and Average Value .......................................................................................................................... 601 8.3 Continuous Money Flow ............................................................................................................................... 612 8.4 Improper Integrals ......................................................................................................................................... 619 Chapter 8 Review Exercises.................................................................................................................................. 633 Extended Application: Estimating Learning Curves in Manufacturing with Integrals ......................................... 643
CHAPTER 9 MULTIVARIABLE CALCULUS 9.1 Functions of Several Variables...................................................................................................................... 645 9.2 Partial Derivatives ........................................................................................................................................ 655 9.3 Maxima and Minima ..................................................................................................................................... 671 9.4 Lagrange Multipliers ..................................................................................................................................... 686 9.5 Total Differentials and Approximations ........................................................................................................ 702 9.6 Double Integrals ............................................................................................................................................ 711 Chapter 9 Review Exercises.................................................................................................................................. 730 Extended Application: Using Multivariable Fitting to Create a Response Surface Design .................................. 749
CHAPTER 10 DIFFERENTIAL EQUATIONS 10.1 Solutions of Elementary and Separable Differential Equations .................................................................. 751 10.2 Linear First-Order Differential Equations ................................................................................................... 769 10.3 Euler’s Method ............................................................................................................................................ 778 10.4 Applications of Differential Equations........................................................................................................ 788 Chapter 10 Review Exercises ............................................................................................................................... 800 Extended Application: Pollution of a Lake ........................................................................................................... 813
CHAPTER 11 PROBABILITY AND CALCULUS 11.1 Continuous Probability Models ................................................................................................................... 815 11.2 Expected Value and Variance of Continuous Random Variables ............................................................... 828 11.3 Special Probability Density Functions ........................................................................................................ 845 Chapter 11 Review Exercises ................................................................................................................................ 857 Extended Application: Exponential Waiting Times .............................................................................................. 871 vi
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CONTENTS
Section R.5
15
To graph this interval on a number line, place a closed circle at -3 and draw a heavy arrow pointing to the right. –3
3.
1£ x < 2
0
1
The endpoints at -2 and 3 are both included. This inequality is written in interval notation as [-2, 3]. To graph this interval, place an open circle at -2 and another at 3 and draw a heavy line segment between them.
5.
0
11. Notice that the endpoint -2 is included, but 6 is not. The interval shown in the graph can be written as the inequality -2 £ x < 6. 12. Notice that neither endpoint is included. The interval shown in the graph can be written as 0 < x < 8. 13. Notice that both endpoints are included. The interval shown in the graph can be written as x £ -4 or x ³ 4.
2
-2 £ x £ 3
Ð2
14. Notice that the endpoint 0 is not included, but 3 is included. The interval shown in the graph can be written as x < 0 or x ³ 3. 15.
p£2
This inequality may be rewritten as x < -9, and is written in interval notation as (-¥, -9). Note that the endpoint at -9 is not included. To graph this interval, place an open circle at -9 and draw a heavy arrow pointing to the left.
The solution in interval notation is (-¥, 2].
16. 6.
0
6k < 3k + 3 3k < 3
This inequality may be written as x ³ 6, and is written in interval notation as [6, ¥). Note that the endpoint at 6 is included. To graph this interval, place a closed circle at 6 and draw a heavy arrow pointing to the right.
k <1
0
1
m - (3m - 2) + 6 < 7m - 19 m - 3m + 2 + 6 < 7m - 19 -2m + 8 < 7m - 19
[-7, -3]
[4, 10)
This represents all the numbers between 4 and 10, including 4 but not including 10. This interval can be written as the inequality 4 £ x < 10. 9.
The solution in interval notation is (-¥,1).
17.
6
This represents all the numbers between -7 and -3, including both endpoints. This interval can be written as the inequality -7 £ x £ -3. 8.
6k - 4 < 3k - 1
6£ x
0
7.
6 p + 7 £ 19 6 p £ 12 æ 1 ö÷ æ ö çç ÷ (6 p) £ çç 1 ÷÷ (12) çè 6 ÷ø èç 6 ÷ø
3
-9 > x
–9
(3, ¥)
This represents all the numbers to the right of 3, and does not include the endpoint. This interval can be written as the inequality x > 3.
0
The endpoint at 1 is included, but the endpoint at 2 is not. This inequality is written in interval notation as [1, 2). To graph this interval, place a closed circle at 1 and an open circle at 2; then draw a heavy line segment between them.
4.
10.
-9m + 8 < -19 -9m < -27
1 1 - (-9m) > - (-27) 9 9 m>3
The solution is (3, ¥).
(-¥, -1]
This represents all the numbers to the left of -1 on the number line and includes the endpoint. This interval can be written as the inequality x £ -1.
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16 18.
Chapter R ALGEBRA REFERENCE 22.
-2(3 y - 8) ³ 5(4 y - 2)
8 £ 3r + 1 £ 13 8 + (-1) £ 3r + 1 + (-1) £ 13 + (-1)
-6 y + 16 ³ 20 y - 10
7 £ 3r £ 12
-6 y + 16 + (-16) ³ 20 y - 10 + (-16)
1 1 1 (7) £ (3r ) £ (12) 3 3 3 7 £r £4 3 The solution is éê 73 , 4 ùú . ë û
-6 y ³ 20 y - 26 -6 y + (-20 y) ³ 20 y + (-20 y) - 26 -26 y ³ -26 -
1 1 (-26) y £ - (-26) 26 26 y £1
0
19.
23.
1
3 p - 1 < 6 p + 2( p - 1) 3p - 1 < 6 p + 2 p - 2
1 1 1 - (-9) > - (-3k ) ³ - (15) 3 3 3 Rewrite the inequalities in the proper order. -5 £ k < 3
1 1 - (-5 p) > - (-1) 5 5 1 p> 5
The solution is [ -5, 3 ).
( 15 , ¥ ). 01
Ð5
1
5
24.
x + 5( x + 1) > 4 (2 - x) + x 6 x + 5 > 8 - 3x
5y + 2 £4 3
-5 £ 5 y £ 10
9x > 3
-1 £ y £ 2
1 x> 3
The solution is [ -1, 2 ].
( 13 , ¥ ). 0 1
–1
25.
1
3
-11 < y - 7 < -1 -11 + 7 < y - 7 + 7 < -1 + 7 -4 < y < 6
0
0
2
3 1 (2 p + 3) ³ (5 p + 1) 5 10
æ3ö æ1 ö 10 çç ÷÷ (2 p + 3) ³ 10 çç ÷÷ (5 p + 1) ÷ èç 5 ø èç 10 ÷ø 6(2 p + 3) ³ 5 p + 1 12 p + 18 ³ 5 p + 1
The solution is (-4, 6). Ð4
3
-3 £ 5 y + 2 £ 12
6 x > 3 - 3x
21.
-1 £
0
æ 5 y + 2 ÷ö 3(-1) £ 3çç £ 3(4) çè 3 ÷÷ø
x + 5x + 5 > 8 - 4 x + x
The solution is
1 - 3k £4 4 æ 1 - 3k ö÷ 4(-2) < 4 çç £ 4(4) çè 4 ÷÷ø
-2 <
-9 < -3k £ 15
- 5 p - 1 < -2 -5 p < -1
20.
4
-8 < 1 - 3k £ 16
3p - 1 < 8p - 2
The solution is
7 3
0
The solution is (-¥, 1].
7 p ³ -17 17 p³7
6
The solution is éê - 17 , ¥ ). ë 7 – 17 7
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Section R.5 26.
17
8 2 ( z - 4) £ (3z + 2) 3 9 8 2 (9) ( z - 4) £ (9) (3z + 2) 3 9 24( z - 4) £ 2(3z + 2)
29.
y2 - 3y + 2 < 0 ( y - 2)( y - 1) < 0
Solve ( y - 2)( y - 1) = 0. y = 2
24 z - 96 £ 6 z + 4
Intervals: (-¥, 1), (1, 2), (2, ¥) For (-¥, 1), choose y = 0.
24 z £ 6 z + 100 18z £ 100
02 - 3(0) + 2 = 2 < / 0
100 8 50 z £ 9
For (1, 2), choose y = 32 .
z £
æ 3 ÷ö2 æ ö çç ÷ - 3çç 3 ÷÷ + 2 = 9 - 9 + 2 çè 2 ÷ø çè 2 ÷ø 4 2
(
ù. The solution is -¥, 50 9 úû
9 - 18 + 8 4 1 =- <0 4 =
50 9
0
27.
(m - 3)(m + 5) < 0
For (2, ¥), choose 3.
Solve (m - 3)(m + 5) = 0. (m - 3)(m + 5) = 0 m=3 or m = -5
32 - 3(3) + 2 = 2 < / 0
The solution is (1, 2).
Intervals: (-¥, - 5), (-5, 3), (3, ¥) For (-¥, - 5), choose -6 to test for m. (-6 - 3)(-6 + 5) = -9(-1) = 9 < / 0 For (-5, 3), choose 0. (0 - 3)(0 + 5) = -3(5) = -15 < 0 For (3, ¥), choose 4. (4 - 3)(4 + 5) = 1(9) = 9 < / 0
0
30.
28.
0
1
2
2k 2 + 7k - 4 > 0 Solve 2k 2 + 7k - 4 = 0. 2k 2 + 7 k - 4 = 0 (2k - 1)(k + 4) = 0 1 k = or k = -4 2
The solution is (-5, 3). Ð5
y =1
or
Intervals: (-¥, -4), ( -4, 12 ),( 12 , ¥ )
3
For (-¥, -4), choose -5.
(t + 6)(t - 1) ³ 0
2(-5)2 + 7(-5) - 4 = 11 > 0
Solve (t + 6)(t - 1) = 0. (t + 6)(t - 1) = 0 t = -6 or t =1
For -4, 12 , choose 0.
Intervals: (-¥, -6), (-6, 1), (1, ¥) For (-¥, -6), choose -7 to test for t. (-7 + 6)(-7 - 1) = (-1)(-8) = 8 ³ 0 For (-6, 1), choose 0. (0 + 6)(0 - 1) = (6)(-1) = -6 ³ / 0 For (1, ¥), choose 2. (2 + 6)(2 - 1) = (8)(1) = 8 ³ 0 Because the symbol ³ is used, the endpoints -6 and 1 are included in the solution. The solution is (-¥, -6] È [1, ¥).
For
–6
(
)
2(0)2 + 7(0) - 4 = -4 > / 0
0 1
( 12 , ¥ ), choose 1.
2(1) 2 + 7(1) - 4 = 5 > 0
The solution is (-¥, -4) È ( 12 , ¥ ). –4
01 2
31.
x 2 - 16 > 0
Solve x 2 - 16 = 0. x 2 - 16 = 0 ( x + 4)( x - 4) = 0 x = -4 or x = 4
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Chapter R ALGEBRA REFERENCE Intervals: (-¥, -4), (-4, 4), (4, ¥)
For (5, ¥), choose 6.
For (-¥, -4), choose - 5.
(6)2 - 4(6) = 12 ³ 5 The solution is (-¥, -1] È [5, ¥).
(-5) 2 - 16 = 9 > 0
For (-4, 4), choose 0. 02 - 16 = -16 > / 0
Ð1 0
34. 10r + r £ 2
For (4, ¥), choose 5.
Solve 10r 2 + r = 2.
2
5 - 16 = 9 > 0
10r 2 + r = 2
The solution is (-¥, -4) È (4, ¥).
10r 2 + r - 2 = 0
Ð4
32.
0
(5r - 2)(2r + 1) = 0 2 1 or r = r = 5 2
4
2k 2 - 7k - 15 £ 0
(
(2k + 3)(k - 5) = 0 3 2
(
)
For - 12 , 52 , choose 0. 10(0)2 + 0 = 0 £ 2
Intervals: ( -¥, - 23 ), ( - 23 , 5 ), (5, ¥)
For
For ( -¥, - 23 ), choose -2.
)
For - 32 , 5 , choose 0.
–1
2(0)2 - 7(0) - 15 = -15 £ 0
2 5
0
2
35.
For (5, ¥), choose 6.
( 52 , ¥ ), choose 1.
10(1) 2 + 1 = 11 £ / 2 The solution is éê - 12 , 52 ùú . ë û
2(-2) 2 - 7(-2) - 15 = 7 £ / 0
(
)
10(-1) 2 + (-1) = 9 £ / 2
k =5
or
3x 2 + 2 x > 1
Solve 3x 2 + 2 x = 1.
2(6)2 - 7(6) - 15 £ / 0
3x 2 + 2 x = 1
The solution is éê - 23 , 5 ùú . ë û
3x 2 + 2 x - 1 = 0
x2 - 4x ³ 5
(3x - 1)( x + 1) = 0 1 or x = -1 x = 3
Solve x 2 - 4 x = 5.
Intervals: (-¥, -1), -1, 13 ,
0
–3
5
2
(
3(-2)2 + 2(-2) = 8 > 1
x2 - 4x - 5 = 0
(
)
For -1, 13 , choose 0.
( x + 1)( x - 5) = 0 or
) ( 13 , ¥ )
For (-¥, -1), choose -2.
x2 - 4x = 5
x +1= 0
) ( 52 , ¥ )
For -¥, - 12 , choose -1.
2k 2 - 7k - 15 = 0
k =-
)(
(
Intervals: -¥, - 12 , - 12 , 52 ,
Solve 2k 2 - 7k - 15 = 0.
33.
5
2
3(0)2 + 2(0) = 0 > / 1
x-5 = 0
x = -1 or x =5 Intervals: (-¥, -1), (-1, 5), (5, ¥) For (-¥, -1), choose -2. (-2)2 - 4(-2) = 12 ³ 5
For
( 13 , ¥ ), choose 1. 3(1)2 + 2(1) = 5 > 1
The solution is (-¥, -1) È ( 13 , ¥ ).
For (-1, 5), choose 0.
–1
02 - 4(0) = 0 ³ / 5
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