Student Solutions Manual
to accompany
Calculus for Business,
Economics, and the
Social and Life Sciences
Tenth Edition, Brief
Laurence D. Hoffman
Smith Barney
Gerald L. Bradley
Claremon McKenna College
Prepared by
Devilyna Nichols
Purdue University
,CONTENTS
Chapter 1 Functions, Graphs, and Limits 1
1.1 Functions 1
1.2 The Graph of a Function 6
1.3 Linear Functions 14
1.4 Functional Models 19
1.5 Limits 26
1.6 One-Sided Limits and Continuity 30
Checkup for Chapter 1 33
Review Problems 36
Chapter 2 Differentiation: Basic Concepts 43
2.1 The Derivative 43
2.2 Techniques of Differentiation 52
2.3 Product and Quotient Rules; Higher-Order Derivatives 57
2.4 The Chain Rule 64
2.5 Marginal Analysis; Approximations Using Increments 72
2.6 Implicit Differentiation and Related Rates 75
Checkup for Chapter 2 82
Review Problems 84
Chapter 3 Additional Applications of the Derivative 93
3.1 Increasing and Decreasing Functions; Relative Extrema 93
3.2 Concavity and Points of Inflection 103
3.3 Curve Sketching 114
3.4 Optimization 124
3.5 Additional Applied Optimization 132
Checkup for Chapter 3 141
Review Problems 148
Chapter 4 Exponential and Logarithmic Functions 159
4.1 Exponential Functions 159
4.2 Logarithmic Functions 165
4.3 Differentiation of Logarithmic and Exponential Functions 173
4.4 Additional Exponential Models 182
Checkup for Chapter 4 199
Review Problems 205
iii
,iv Contents
Chapter 5 Integration 219
5.1 Antidifferentiation; the Indefinite Integral 219
5.2 Integration by Substitution 226
5.3 The Definite Integral and the Fundamental Theorem of Calculus 233
5.4 Applying Definite Integration: Area Between Curves and Average Value 238
5.5 Additional Applications to Business and Economics 245
5.6 Additional Applications to the Life and Social Sciences 252
Checkup for Chapter 5 259
Review Problems 262
Chapter 6 Additional Topics in Integration 273
6.1 Integration by Parts; Integral Tables 273
6.2 Introduction to Differential Equations 284
6.3 Improper Integrals; Continuous Probability 292
6.4 Numerical Integration 300
Checkup for Chapter 6 307
Review Problems 312
Chapter 7 Calculus of Several Variables 325
7.1 Functions of Several Variables 325
7.2 Partial Derivatives 329
7.3 Optimizing Functions of Two Variables 336
7.4 The Method of Least Squares 346
7.5 Constrained Optimization: The Method of Lagrange Multipliers 353
7.6 Double Integrals 362
Checkup for Chapter 7 371
Review Problems 375
, Chapter 1
Functions, Graphs, and Limits
9. 1
1.1 Functions f (t ) = (2t − 1)−3/2 =
√
,
( 2t − 1)3
f(x) = 3x + 5, 1
1. f (1) = √ = 1,
f (0) = 3(0) + 5 = 5 [ 2(1) − 1]3
f (−1) = 3(−1) + 5 = 2 f (5) = √ 1 = 1 =1 ,
f (2) = 3(2) + 5 = 11 [ 2(5) − 1]3 √
1 [ 9]3 27
1 1
f (13) = √ = √ = .
[ 2(13) − 1]3 [ 25]3 125
3. f(x) = 3x2 + 5x − 2, 11. f(x) = x − |x − 2|,
f (0) = 3(0)2 + 5(0) − 2 = −2, f (1) = 1 − |1 − 2|= 1 − | − 1|= 1 − 1 = 0,
f (−2) = 3(−2)2 + 5(−2) − 2 = 0, f (2) = 2 − |2 − 2|= 2 − |0|= 2,
f (3) = 3 − |3 − 2|= 3 − |1|= 3 − 1 = 2.
f (1) = 3(1)2 + 5(1) − 2 = 6.
13. −2x + 4 if x ≤ 1
h(x) =
x2 + 1 if x > 1
1 h(3) = (3)2 + 1 = 10
5. g(x) = x + , h(1) = −2(1) + 4 = 2
x h(0) = −2(0) + 4 = 4
1
g(−1) = −1 + = −2, h(−3) = −2(−3) + 4 = 10
−1 x
1
g(1) = 1 + = 2, 15. g(x) .
=
1 1 +2 x2
g(2) = 2 1 5 . Since 1 + x /= 0 for any real number, the domain is
+ =
2 2 the set of all real numbers.
√
17. f (t ) = 1 − t.
, Since negative numbers do not have real square
7. h(t) = t 2 + 2t + 4,
roots, the domain is all real numbers such that
, √ 1 —t ≥ 0, or t ≤ 1. Therefore, the domain is not the
h(2) = 22 + 2(2) + 4 = 2 3,
, set of all real numbers.
2h(0) = 0
, √ x2 + 5
h(−4) = (−4) + 2(−4) + 4 = 2 3
2 19. g(x) = .
+ 2(0) + 4 = 2, x +2
1
to accompany
Calculus for Business,
Economics, and the
Social and Life Sciences
Tenth Edition, Brief
Laurence D. Hoffman
Smith Barney
Gerald L. Bradley
Claremon McKenna College
Prepared by
Devilyna Nichols
Purdue University
,CONTENTS
Chapter 1 Functions, Graphs, and Limits 1
1.1 Functions 1
1.2 The Graph of a Function 6
1.3 Linear Functions 14
1.4 Functional Models 19
1.5 Limits 26
1.6 One-Sided Limits and Continuity 30
Checkup for Chapter 1 33
Review Problems 36
Chapter 2 Differentiation: Basic Concepts 43
2.1 The Derivative 43
2.2 Techniques of Differentiation 52
2.3 Product and Quotient Rules; Higher-Order Derivatives 57
2.4 The Chain Rule 64
2.5 Marginal Analysis; Approximations Using Increments 72
2.6 Implicit Differentiation and Related Rates 75
Checkup for Chapter 2 82
Review Problems 84
Chapter 3 Additional Applications of the Derivative 93
3.1 Increasing and Decreasing Functions; Relative Extrema 93
3.2 Concavity and Points of Inflection 103
3.3 Curve Sketching 114
3.4 Optimization 124
3.5 Additional Applied Optimization 132
Checkup for Chapter 3 141
Review Problems 148
Chapter 4 Exponential and Logarithmic Functions 159
4.1 Exponential Functions 159
4.2 Logarithmic Functions 165
4.3 Differentiation of Logarithmic and Exponential Functions 173
4.4 Additional Exponential Models 182
Checkup for Chapter 4 199
Review Problems 205
iii
,iv Contents
Chapter 5 Integration 219
5.1 Antidifferentiation; the Indefinite Integral 219
5.2 Integration by Substitution 226
5.3 The Definite Integral and the Fundamental Theorem of Calculus 233
5.4 Applying Definite Integration: Area Between Curves and Average Value 238
5.5 Additional Applications to Business and Economics 245
5.6 Additional Applications to the Life and Social Sciences 252
Checkup for Chapter 5 259
Review Problems 262
Chapter 6 Additional Topics in Integration 273
6.1 Integration by Parts; Integral Tables 273
6.2 Introduction to Differential Equations 284
6.3 Improper Integrals; Continuous Probability 292
6.4 Numerical Integration 300
Checkup for Chapter 6 307
Review Problems 312
Chapter 7 Calculus of Several Variables 325
7.1 Functions of Several Variables 325
7.2 Partial Derivatives 329
7.3 Optimizing Functions of Two Variables 336
7.4 The Method of Least Squares 346
7.5 Constrained Optimization: The Method of Lagrange Multipliers 353
7.6 Double Integrals 362
Checkup for Chapter 7 371
Review Problems 375
, Chapter 1
Functions, Graphs, and Limits
9. 1
1.1 Functions f (t ) = (2t − 1)−3/2 =
√
,
( 2t − 1)3
f(x) = 3x + 5, 1
1. f (1) = √ = 1,
f (0) = 3(0) + 5 = 5 [ 2(1) − 1]3
f (−1) = 3(−1) + 5 = 2 f (5) = √ 1 = 1 =1 ,
f (2) = 3(2) + 5 = 11 [ 2(5) − 1]3 √
1 [ 9]3 27
1 1
f (13) = √ = √ = .
[ 2(13) − 1]3 [ 25]3 125
3. f(x) = 3x2 + 5x − 2, 11. f(x) = x − |x − 2|,
f (0) = 3(0)2 + 5(0) − 2 = −2, f (1) = 1 − |1 − 2|= 1 − | − 1|= 1 − 1 = 0,
f (−2) = 3(−2)2 + 5(−2) − 2 = 0, f (2) = 2 − |2 − 2|= 2 − |0|= 2,
f (3) = 3 − |3 − 2|= 3 − |1|= 3 − 1 = 2.
f (1) = 3(1)2 + 5(1) − 2 = 6.
13. −2x + 4 if x ≤ 1
h(x) =
x2 + 1 if x > 1
1 h(3) = (3)2 + 1 = 10
5. g(x) = x + , h(1) = −2(1) + 4 = 2
x h(0) = −2(0) + 4 = 4
1
g(−1) = −1 + = −2, h(−3) = −2(−3) + 4 = 10
−1 x
1
g(1) = 1 + = 2, 15. g(x) .
=
1 1 +2 x2
g(2) = 2 1 5 . Since 1 + x /= 0 for any real number, the domain is
+ =
2 2 the set of all real numbers.
√
17. f (t ) = 1 − t.
, Since negative numbers do not have real square
7. h(t) = t 2 + 2t + 4,
roots, the domain is all real numbers such that
, √ 1 —t ≥ 0, or t ≤ 1. Therefore, the domain is not the
h(2) = 22 + 2(2) + 4 = 2 3,
, set of all real numbers.
2h(0) = 0
, √ x2 + 5
h(−4) = (−4) + 2(−4) + 4 = 2 3
2 19. g(x) = .
+ 2(0) + 4 = 2, x +2
1
