Find the product of 6xy and -3x^{2}y^{3}−3x
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Transcript
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 [to be supplied by publisher] CONTENTS Chapter 1 1.1 1.2 1.3 1.4 1.5 1.6 Functions, Graphs, and Limits Functions 1 The Graph of a Function 6 Linear Functions 14 Functional Models 19 Limits 26 One-Sided Limits and Continuity Checkup for Chapter 1 33 Review Problems 36 1 Chapter 2 2.1 2.2 2.3 2.4 2.5 2.6 Differentiation: Basic Concepts 43 The Derivative 43 Techniques of Differentiation 52 Product and Quotient Rules; Higher-Order Derivatives The Chain Rule 64 Marginal Analysis; Approximations Using Increments Implicit Differentiation and Related Rates 75 Checkup for Chapter 2 82 Review Problems 84 Chapter 3 3.1 3.2 3.3 3.4 3.5 Additional Applications of the Derivative 93 Increasing and Decreasing Functions; Relative Extrema Concavity and Points of Inflection 103 Curve Sketching 114 Optimization 124 Additional Applied Optimization 132 Checkup for Chapter 3 141 Review Problems 148 Chapter 4 4.1 4.2 4.3 4.4 Exponential and Logarithmic Functions 159 Exponential Functions 159 Logarithmic Functions 165 Differentiation of Logarithmic and Exponential Functions Additional Exponential Models 182 Checkup for Chapter 4 199 Review Problems 205 30 iii 57 72 93 173 iv Contents Chapter 5 5.1 5.2 5.3 5.4 5.5 5.6 Integration 219 Antidifferentiation; the Indefinite Integral 219 Integration by Substitution 226 The Definite Integral and the Fundamental Theorem of Calculus 233 Applying Definite Integration: Area Between Curves and Average Value Additional Applications to Business and Economics 245 Additional Applications to the Life and Social Sciences 252 Checkup for Chapter 5 259 Review Problems 262 Chapter 6 6.1 6.2 6.3 6.4 Additional Topics in Integration 273 Integration by Parts; Integral Tables 273 Introduction to Differential Equations 284 Improper Integrals; Continuous Probability 292 Numerical Integration 300 Checkup for Chapter 6 307 Review Problems 312 Chapter 7 7.1 7.2 7.3 7.4 7.5 7.6 Calculus of Several Variables 325 Functions of Several Variables 325 Partial Derivatives 329 Optimizing Functions of Two Variables 336 The Method of Least Squares 346 Constrained Optimization: The Method of Lagrange Multipliers Double Integrals 362 Checkup for Chapter 7 371 Review Problems 375 353 238 Chapter 1 Functions, Graphs, and Limits 1.1 1. Functions 9. f (x) = 3x + 5, f (0) = 3(0) + 5 = 5 f (−1) = 3(−1) + 5 = 2 f (2) = 3(2) + 5 = 11 1 , f (t) = (2t − 1)−3/2 = √ ( 2t − 1)3 1 f (1) = √ = 1, [ 2(1) − 1]3 1 1 1 f (5) = √ = √ = , 3 3 27 [ 2(5) − 1] [ 9] 1 1 1 f (13) = √ . = √ = 3 3 125 [ 2(13) − 1] [ 25] 3. 11. f (x) = x − |x − 2|, f (1) = 1 − |1 − 2| = 1 − | − 1| = 1 − 1 = 0, f (2) = 2 − |2 − 2| = 2 − |0| = 2, f (3) = 3 − |3 − 2| = 3 − |1| = 3 − 1 = 2. 13. −2x + 4 if x ≤ 1 h(x) = if x > 1 x2 + 1 f (x) = 3x 2 + 5x − 2, f (0) = 3(0)2 + 5(0) − 2 = −2, f (−2) = 3(−2)2 + 5(−2) − 2 = 0, f (1) = 3(1)2 + 5(1) − 2 = 6. 5. h(3) = (3)2 + 1 = 10 h(1) = −2(1) + 4 = 2 h(0) = −2(0) + 4 = 4 h(−3) = −2(−3) + 4 = 10 1 g(x) = x + , x g(−1) = −1 + 1 = −2, −1 1 = 2, 1 1 5 g(2) = 2 + = . 2 2 15. g(x) = g(1) = 1 + 7. h(t) = h(2) = h(0) = h(−4) = x . 1 + x2 Since 1 + x 2 = 0 for any real number, the domain is the set of all real numbers. √ 17. f (t) = 1 − t. Since negative numbers do not have real square roots, the domain is all real numbers such that 1 − t ≥ 0, or t ≤ 1. Therefore, the domain is not the set of all real numbers. t 2 + 2t + 4, √ 22 + 2(2) + 4 = 2 3, 02 + 2(0) + 4 = 2, √ (−4)2 + 2(−4) + 4 = 2 3 19. g(x) = 1 x2 + 5 . x+2 2 Chapter 1. Functions, Graphs, and Limits Since denominators cannot be 0, the domain consists of all real numbers such that x = −2. 35. f (x) = 4x − x 2 f (x + h) − f (x) 4(x + h) − (x + h)2 − (4x − x 2) = h h √ 21. f (x) = 2x + 6. Since negative numbers do not have real square roots, the domain is all real numbers such that 2x + 6 ≥ 0, or x ≥ −3. t +2 23. f (t) = √ . 9 − t2 Since negative numbers do not have real square roots and denominators cannot be zero, the domain is the set of all real numbers such that 9 − t 2 > 0, namely −3 < t < 3. 25. f (u) = 3u2 + 2u − 6 and g(x) = x + 2, so f (g(x)) = f (x + 2) = 3(x + 2)2 + 2(x + 2) − 6 27. f (u) = (u − 1)3 + 2u2 and g(x) = x + 1 , so f (g(x)) = f (x + 1) = [(x + 1) − 1]3 + 2(x + 1)2 = x 3 + 2x 2 + 4x + 2. 1 and . ,
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