Student Solutions Manual – Calculus for Business, Economics, and the Social & Life Sciences (10th Edition, Brief | Hoffman, Smith & Bradley) | Latest 2025 Update

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
i Functions, Graphs, and Limits i i i 1
1.1 Functions 1
1.2 The Graph of a Function
i 6
i i i


1.3 Linear Functions
i 14
1.4 Functional Models 19 i


1.5 Limits 26
1.6 One-Sided Limits and Continuity
i i i 30
Checkup for Chapter 1
i 33 i i

Review Problems i 36


Chapter 2
i ii i Differentiation: Basic Concepts 43 i i


2.1 The Derivative
i 43
2.2 Techniques of Differentiation i 52 i

2.3 Product and Quotient Rules; Higher-Order Derivatives
i i i i i 57
2.4 The Chain Rule
i 64i


2.5 Marginal Analysis; Approximations Using Increments
i i i i 72
2.6 Implicit Differentiation and Related Rates
i 75 i i i


Checkup for Chapter 2
i 82 i i

Review Problems i 84


Chapter 3
i ii i Additional Applications of the Derivative
i 93 i i i


3.1 Increasing and Decreasing Functions; Relative Extrema
i i i i i 93
3.2 Concavity and Points of Inflection
i 103 i i i


3.3 Curve Sketching
i 114
3.4 Optimization 124
3.5 Additional Applied Optimization
i 132 i

Checkup for Chapter 3
i 141 i i


Review Problems i 148


Chapter 4
i ii i Exponential and Logarithmic Functions 159
i i i

4.1 Exponential Functions 159 i


4.2 Logarithmic Functions 165 i


4.3 Differentiation of Logarithmic and Exponential Functionsi i i i i 173
4.4 Additional Exponential Models
i 182 i


Checkup for Chapter 4
i 199 i i


Review Problems i 205
iii

,iv Contents




Chapter 5
i ii i Integration 219
5.1 Antidifferentiation; the Indefinite Integral 219 i i i

5.2 Integration by Substitution i 226 i


5.3 The Definite Integral and the Fundamental Theorem of Calculus
i i 233 i i i i i i

5.4 Applying Definite Integration: Area Between Curves and Average Value
i i i i i i i i 238
5.5 Additional Applications to Business and Economics
i 245 i i i i

5.6 Additional Applications to the Life and Social Sciences
i 252 i i i i i i


Checkup for Chapter 5
i 259 i i


Review Problems
i 262

Chapter 6 Additional Topics in Integration
i ii i 273 i i i


6.1 Integration by Parts; Integral Tables 273
i i i i


6.2 Introduction to Differential Equations 284 i i i


6.3 Improper Integrals; Continuous Probability
i 292 i i

6.4 Numerical Integration 300 i


Checkup for Chapter 6 i 307 i i


Review Problems i 312

Chapter 7 Calculus of Several Variables
i ii i i 325 i i


7.1 Functions of Several Variables i 325 i i


7.2 Partial Derivatives
i 329
7.3 Optimizing Functions of Two Variables 336
i i i i

7.4 The Method of Least Squares
i 346 i i i


7.5 Constrained Optimization: The Method of Lagrange Multipliers
i i i i i i 353
7.6 Double Integrals i 362
Checkup for Chapter 7 i 371 i i


Review Problems i 375

, Chapter 1 i




Functions, Graphs, and Limits i i i




9. 1
1.1 Functions f (t ) = (2t − 1)−3/2 = i i i i i
√
ii i ii i
,
( 2t − 1)3 i i


f(x) = 3x + 5, 1
1. i i i i
f(1) =
√
i =1, i i

f (0) = 3(0) + 5 = 5
i i i i i i i [ 2(1) − 1]3 i i

f(−1) = 3(−1) + 5 = 2
i i i i i i i
f (5) = √ 1
i = 1 =1 , i i
i i ii




f (2) = 3(2) + 5 = 11 i i i [ 2(5) − √ i i
i i i i
[ 9]3 27
1]3 1 1 i

1
f (13) = √
i = √ = i . i i i

[ 2(13) − 1]3 [ 25]3 125 i i i
i
i


3. f(x) = 3x2 + 5x − 2,
i i i i i i 11. f(x) = x − |x − 2|,
i
i i i i i i



f (0) = 3(0) + 5(0) − 2 = −2,
i i i
2
i i i i i i
f (1) = 1 − |1 − 2|= 1 − | − 1|= 1 − 1 = 0,f (2)
i i i i i i i i i i i i i i i i i




f (−2) = 3(−2)2 + 5(−2) − 2 = 0,
i i i i i i i i i
= 2 − |2 − 2|= 2 − |0|= 2,
i i i i i i i i i i



f (3) = 3 − |3 − 2|= 3 − |1|= 3 − 1 = 2.
f (1) = 3(1)2 + 5(1) − 2 = 6.
i i i i i i i i i i i i i i i
i i i i i i i i i

13. −2x + 4 if x ≤ 1
h(x) =
i i iiii i i i


x2 + 1 if x > 1 i i i i i


1 h(3) = (3) + 1 = 10
2
i i i i i i


5. g(x) = x + i i i , h(1) = −2(1) + 4 = 2 i i i i i i
x h(0) = −2(0) + 4 = 4
1 i i i i i i



g(−1) = −1+ = −2, i i i h(−3) = −2(−3) + 4 = 10
−1
i i i i i i i i

1
i
x
g(1) = 1 + = 2, 15. g(x) .
=
i
i i i i
1 i 1+ x 2 i i

g(2) = 2 1 5. i i Since 1+ x2 /= 0 for any real number, the domain is i i i i i i i i i i i i

+ = i i
2 2 i
the set of all real numbers. i i i i i


√
17. f (t) = 1− t .
ii i i i i i


, Since negative numbers do not have real square
h(t) = t2 + 2t + 4,
i i i i i i i
7. i i
i
i i i
roots, the domain is all real numbers such that
√
i i i i i i i i i
, 1 —t ≥ 0, or t ≤ 1. Therefore, the domain is not the
h(2) = i 3, i 22 i + i2(2) i+ i4 i= i2 i i i i i i i i


, set of all real numbers.
i i i i i

2h(0) = 0
, √ x2 + 5
h(−4) = (−4)2 + 2(−4) + 4 = 2 3 i
i
i i i 19. g(x) = .
+ 2(0) + 4 = 2,
i i i

x +2 i i



1