1 Functions and Models 1.1 Four Ways to Represent a Function Exercise 4 Exercise 13 Exercise 17 Exercise 27 Exercise 35 Exercise 53 Exercise 61 Exercise 67 1.2 Mathematical Models: A Catalog of Essential Functions Exercise 3 Exercise 17 Exercise 19 1.3 New Functions from Old Functions Exercise 1a Exercise 1d Exercise 1h Exercise 5a Exercise 5d Exercise 7 Exercise 19 Exercise 29a Exercise 29c Exercise 31a Exercise 37a Exercise 37d Exercise 48 Exercise 55 Exercise 65 1.4 Exponential Functions Exercise 9 Exercise 13 Exercise 15 Exercise 17 Exercise 21 Exercise 27 1.5 Inverse Functions and Logarithms Exercise 3 Exercise 6 Exercise 13 Exercise 17 Exercise 19 Exercise 22 Exercise 25 Exercise 45 Exercise 47 Exercise 53a Exercise 53b Exercise 71 2 Limits and Derivatives 2.1 The Tangent and Velocity Problems Exercise 3 Exercise 5 Exercise 9 2.2 The Limit of a Function Exercise 5 Exercise 13 Exercise 17 Exercise 33 Exercise 47 Exercise 55 2.3 Calculating Limits Using the Limit Laws Exercise 6 Exercise 15 Exercise 18 Exercise 19 Exercise 37 Exercise 41 Exercise 53 Exercise 62 Exercise 65 2.4 The Precise Definition of a Limit Exercise 3 Exercise 17 Exercise 25 Exercise 29 Exercise 31 Exercise 37 Exercise 43 2.5 Continuity Exercise 3 Exercise 9 Exercise 11 Exercise 20 Exercise 33 Exercise 36 Exercise 43 Exercise 45 Exercise 53 Exercise 59 Exercise 69 2.6 Limits at Infinity; Horizontal Asymptotes Exercise 2 Exercise 7 Exercise 19 Exercise 27 Exercise 35 Exercise 49 Exercise 55 Exercise 67 Exercise 75a Exercise 75b 2.7 Derivatives and Rates of Change Exercise 7 Exercise 9 Exercise 13 Exercise 17 Exercise 22 Exercise 23 Exercise 29 Exercise 33 Exercise 41 Exercise 45 Exercise 51 Exercise 53 Exercise 59 2.8 The Derivative as a Function Exercise 3b Exercise 3c Exercise 5 Exercise 11 Exercise 19a Exercise 19b Exercise 29 Exercise 41 Exercise 49 Exercise 55 Exercise 59 Exercise 61 3 Differentiation Rules 3.1 Derivatives of Polynomials and Exponential Functions Exercise 23 Exercise 31 Exercise 37 Exercise 46 Exercise 49 Exercise 55 Exercise 63 Exercise 77 Exercise 83 3.2 The Product and Quotient Rules Exercise 11 Exercise 25 Exercise 33 Exercise 36 Exercise 43 Exercise 45 Exercise 52 Exercise 53 Exercise 61 3.3 Derivatives of Trigonometric Functions Exercise 9 Exercise 22 Exercise 29 Exercise 33 Exercise 37 Exercise 41 Exercise 45 Exercise 51 Exercise 53 Exercise 57 3.4 The Chain Rule Exercise 5 Exercise 17 Exercise 31 Exercise 37 Exercise 55 Exercise 59 Exercise 61 Exercise 65 Exercise 69 Exercise 77 Exercise 79 Exercise 88 Exercise 97 3.5 Implicit Differentiation Exercise 15 Exercise 29 Exercise 43 Exercise 51 Exercise 67 Exercise 73 Exercise 77a Exercise 77b Exercise 79 3.6 Derivatives of Logarithmic Functions Exercise 19 Exercise 27 Exercise 43 Exercise 45 Exercise 52 Exercise 55 3.7 Rates of Change in the Natural and Social Sciences Exercise 15 Exercise 19 Exercise 23 Exercise 30 Exercise 33 Exercise 39 3.8 Exponential Growth and Decay Exercise 3 Exercise 5 Exercise 9 Exercise 15 Exercise 21 3.9 Related Rates Exercise 14 Exercise 17 Exercise 21 Exercise 27 Exercise 29 Exercise 37 Exercise 43a Exercise 43b Exercise 49 3.10 Linear Approximations and Differentials Exercise 3 Exercise 5 Exercise 7 Exercise 13 Exercise 15 Exercise 29 Exercise 33 Exercise 40 Exercise 43 3.11 Hyperbolic Functions Exercise 9 Exercise 15 Exercise 17 Exercise 37 Exercise 43 Exercise 51 Exercise 55 4 Applications of Differentiation 4.1 Maximum and Minimum Values Exercise 7 Exercise 11 Exercise 13 Exercise 25 Exercise 39 Exercise 41 Exercise 49 Exercise 67 Exercise 80 4.2 The Mean Value Theorem Exercise 9 Exercise 11 Exercise 21 Exercise 25 Exercise 27 Exercise 37 4.3 How Derivatives Affect the Shape of a Graph Exercise 5a Exercise 5b Exercise 7c Exercise 11a Exercise 11b Exercise 11c Exercise 17 Exercise 27 Exercise 35 Exercise 43 Exercise 45a Exercise 45b Exercise 45c Exercise 55 Exercise 59 Exercise 69 Exercise 70 Exercise 73 Exercise 83 4.4 Indeterminate Forms and L'Hospital's Rule Exercise 1 Exercise 27 Exercise 47 Exercise 52 Exercise 59 Exercise 73 Exercise 89 4.5 Summary of Curve Sketching Exercise 5 Exercise 9 Exercise 15 Exercise 35 Exercise 43 Exercise 61 4.6 Graphing with Calculus and Calculators Exercise 11 Exercise 13 Exercise 25 Exercise 32 Exercise 33 Exercise 36 4.7 Optimization Problems Exercise 15 Exercise 20 Exercise 21 Exercise 23 Exercise 26 Exercise 34 Exercise 37 Exercise 53 Exercise 54 Exercise 59 Exercise 61 Exercise 71 Exercise 77 4.8 Newton's Method Exercise 4 Exercise 17 Exercise 31 Exercise 37 Exercise 41 4.9 Antiderivatives Exercise 15 Exercise 23 Exercise 25 Exercise 41 Exercise 51 Exercise 55 Exercise 59 Exercise 67 Exercise 75 5 Integrals 5.1 Areas and Distances Exercise 2 Exercise 5 Exercise 13 Exercise 17 Exercise 25 5.2 The Definite Integral Exercise 5 Exercise 9 Exercise 19 Exercise 23 Exercise 33 Exercise 37 Exercise 47 Exercise 49 Exercise 57 5.3 The Fundamental Theorem of Calculus Exercise 3 Exercise 9 Exercise 13 Exercise 53 Exercise 59 Exercise 72 Exercise 73 Exercise 82 Exercise 83 Exercise 84 5.4 Indefinite Integrals and the Net Change Theorem Exercise 2 Exercise 9 Exercise 31 Exercise 45 Exercise 50 Exercise 53 Exercise 59 Exercise 61 5.5 The Substitution Rule Exercise 3 Exercise 13 Exercise 21 Exercise 25 Exercise 35 Exercise 45 Exercise 59 Exercise 64 Exercise 69 Exercise 79 Exercise 87 Exercise 91 6 Applications of Integration 6.1 Areas between Curves Exercise 3 Exercise 9 Exercise 11 Exercise 13 Exercise 33 Exercise 53 Exercise 59 Exercise 61 6.2 Volumes Exercise 7 Exercise 9 Exercise 11 Exercise 41 Exercise 47 Exercise 49 Exercise 55 Exercise 63 Exercise 67 6.3 Volumes by Cylindrical Shells Exercise 5 Exercise 13 Exercise 17 Exercise 25a Exercise 29 Exercise 41 Exercise 47 6.4 Work Exercise 7 Exercise 9 Exercise 13 Exercise 17 Exercise 21 Exercise 29 6.5 Average Value of a Function Exercise 7 Exercise 9 Exercise 13 Exercise 17 Exercise 25 7 Techniques of Integration 7.1 Integration by Parts Exercise 3 Exercise 15 Exercise 17 Exercise 24 Exercise 39 Exercise 51 Exercise 61 Exercise 69 Exercise 72 7.2 Trigonometric Integrals Exercise 3 Exercise 7 Exercise 11 Exercise 23 Exercise 27 Exercise 41 Exercise 55 Exercise 61 7.3 Trigonometric Substitution Exercise 3 Exercise 7 Exercise 13 Exercise 17 Exercise 22 Exercise 31a Exercise 31b 7.4 Integration of Rational Functions by Partial Fractions Exercise 6 Exercise 11 Exercise 17 Exercise 23 Exercise 29 Exercise 31 Exercise 43 Exercise 47 Exercise 57 7.5 Strategy for Integration Exercise 7 Exercise 17 Exercise 23 Exercise 31 Exercise 41 Exercise 45 Exercise 49 Exercise 57 Exercise 63 Exercise 71 7.6 Integration Using Tables and Computer Algebra Systems Exercise 10 Exercise 17 Exercise 19 Exercise 26 Exercise 29 Exercise 31 Exercise 35 7.7 Approximate Integration Exercise 1 Exercise 3 Exercise 4 Exercise 35 Exercise 37 Exercise 47 7.8 Improper Integrals Exercise 1 Exercise 7 Exercise 13 Exercise 21 Exercise 29 Exercise 31 Exercise 49 Exercise 57 Exercise 61 Exercise 71 8 Further Applications of Integration 8.1 Arc Length Exercise 9 Exercise 13 Exercise 15 Exercise 33 Exercise 35 Exercise 43 Exercise 46 8.2 Area of a Surface of Revolution Exercise 1 Exercise 1 Exercise 7 Exercise 13 Exercise 17 Exercise 27 Exercise 33 8.3 Applications to Physics and Engineering Exercise 7 Exercise 13 Exercise 27 Exercise 31 Exercise 41 8.4 Applications to Economics and Biology Exercise 5 Exercise 12 Exercise 21 8.5 Probability Exercise 1 Exercise 7 Exercise 8 Exercise 15 9 Differential Equations 9.1 Modeling with Differential Equations Exercise 3 Exercise 7 Exercise 9 Exercise 11 Exercise 15 9.2 Direction Fields and Euler's Method Exercise 3 Exercise 11 Exercise 13 Exercise 18 Exercise 19 Exercise 21 Exercise 23 9.3 Separable Equations Exercise 10 Exercise 13 Exercise 25 Exercise 31 Exercise 39 Exercise 43 Exercise 49 9.4 Models for Population Growth Exercise 3 Exercise 5 Exercise 11 Exercise 13 Exercise 18 Exercise 19 Exercise 21 9.5 Linear Equations Exercise 7 Exercise 9 Exercise 19 Exercise 25 Exercise 31 Exercise 33 9.6 Predator-Prey Systems Exercise 1 Exercise 5 Exercise 7 Exercise 9 10 Parametric Equations and Polar Coordinates 10.1 Curves Defined by Parametric Equations Exercise 4 Exercise 9 Exercise 13 Exercise 21 Exercise 31 Exercise 33 Exercise 34 Exercise 41 Exercise 47 Exercise 51 10.2 Calculus with Parametric Curves Exercise 5 Exercise 11 Exercise 23 Exercise 25 Exercise 31 Exercise 41 Exercise 45 Exercise 63 Exercise 65 10.3 Polar Coordinates Exercise 11 Exercise 25 Exercise 33 Exercise 37 Exercise 47 Exercise 53 Exercise 57 Exercise 61 Exercise 65 10.4 Areas and Lengths in Polar Coordinates Exercise 7 Exercise 21 Exercise 27 Exercise 31 Exercise 41 Exercise 47 10.5 Conic Sections Exercise 5 Exercise 15 Exercise 19 Exercise 27 Exercise 33 Exercise 37 Exercise 47 10.6 Conic Sections in Polar Coordinates Exercise 1 Exercise 13 Exercise 13b Exercise 13c Exercise 13d Exercise 21 Exercise 27 11 Infinite Sequences and Series 11.1 Sequences Exercise 17 Exercise 23 Exercise 42 Exercise 43 Exercise 53 Exercise 64 Exercise 71 Exercise 73 Exercise 81 11.2 Series Exercise 9 Exercise 15 Exercise 23 Exercise 39 Exercise 43 Exercise 51 Exercise 57 Exercise 67 Exercise 75 Exercise 81 Exercise 87 Exercise 89 11.3 The Integral Test and Estimates of Sums Exercise 7 Exercise 11 Exercise 17 Exercise 21 Exercise 37 Exercise 43 11.4 The Comparison Tests Exercise 1 Exercise 5 Exercise 7 Exercise 17 Exercise 31 Exercise 37 Exercise 41 11.5 Alternating Series Exercise 3 Exercise 7 Exercise 11 Exercise 17 Exercise 23 Exercise 32 11.6 Absolute Convergence and the Ratio and Root Tests Exercise 7 Exercise 13 Exercise 17 Exercise 25 Exercise 39 Exercise 43 Exercise 45 11.8 Power Series Exercise 5 Exercise 7 Exercise 15 Exercise 23 Exercise 24 Exercise 29 Exercise 37 11.9 Representations of Functions as Power Series Exercise 5 Exercise 8 Exercise 13a Exercise 13b Exercise 15 Exercise 23 Exercise 25 Exercise 37 Exercise 39 11.10 Taylor and Maclaurin Series Exercise 11 Exercise 21 Exercise 33 Exercise 39 Exercise 41 Exercise 45 Exercise 51 Exercise 63 Exercise 67 Exercise 73 11.11 Applications of Taylor Polynomials Exercise 5 Exercise 9 Exercise 18 Exercise 19 Exercise 25 Exercise 31 Exercise 33 12 Vectors and the Geometry of Space 12.1 Three-Dimensional Coordinate Systems Exercise 7 Exercise 15 Exercise 23 Exercise 29 Exercise 37 Exercise 41 Exercise 45 12.2 Vectors Exercise 3 Exercise 13 Exercise 25 Exercise 29 Exercise 45 Exercise 47 Exercise 51 12.3 The Dot Product Exercise 11 Exercise 19 Exercise 27 Exercise 45 Exercise 47 Exercise 53 Exercise 55 Exercise 61 12.4 The Cross Product Exercise 7 Exercise 13 Exercise 16 Exercise 19 Exercise 31 Exercise 45 Exercise 49 Exercise 53 12.5 Equations of Lines and Planes Exercise 5 Exercise 7 Exercise 13 Exercise 19 Exercise 31 Exercise 51 Exercise 63 Exercise 75 12.6 Cylinders and Quadric Surfaces Exercise 9 Exercise 19 13 Vector Functions 13.1 Vector Functions and Space Curves Exercise 13 Exercise 23 Exercise 27 Exercise 43 Exercise 47 13.2 Derivatives and Integrals of Vector Functions Exercise 1 Exercise 3 Exercise 15 Exercise 19 Exercise 25 Exercise 55 13.3 Arc Length and Curvature Exercise 3 Exercise 5 Exercise 17 Exercise 31 Exercise 33 Exercise 39 Exercise 47 Exercise 53 Exercise 59 13.4 Motion in Space: Velocity and Acceleration Exercise 11 Exercise 19 Exercise 22 Exercise 25 Exercise 39 14 Partial Derivatives 14.1 Functions of Several Variables Exercise 1 Exercise 7 Exercise 15 Exercise 19 Exercise 25 Exercise 32 Exercise 37 Exercise 49 Exercise 61 Exercise 67 Exercise 71 14.2 Limits and Continuity Exercise 9 Exercise 13 Exercise 21 Exercise 25 Exercise 28 Exercise 37 Exercise 39 14.3 Partial Derivatives Exercise 1 Exercise 5a Exercise 5b Exercise 9 Exercise 21 Exercise 33 Exercise 52 Exercise 73 Exercise 83 Exercise 96 Exercise 97 14.4 Tangent Planes and Linear Approximations Exercise 11 Exercise 21 Exercise 31 Exercise 35 Exercise 43 Exercise 45 14.5 The Chain Rule Exercise 5 Exercise 11 Exercise 17 Exercise 35 Exercise 39 Exercise 45 14.6 Directional Derivatives and the Gradient Vector Exercise 1 Exercise 11 Exercise 19 Exercise 23 Exercise 27 Exercise 29 Exercise 33 Exercise 41 Exercise 61 Exercise 69 14.7 Maximum and Minimum Values Exercise 1 Exercise 3 Exercise 15 Exercise 33 Exercise 43 Exercise 45 Exercise 53 14.8 Lagrange Multipliers Exercise 1 Exercise 3 Exercise 11 Exercise 23 Exercise 29 Exercise 37 Exercise 47 15 Multiple Integrals 15.1 Double Integrals over Rectangles Exercise 1 Exercise 5 Exercise 7a Exercise 7b Exercise 11 Exercise 29 Exercise 15 Exercise 21 Exercise 29 Exercise 31 Exercise 35 Exercise 39 Exercise 47 15.2 Double Integrals over General Regions Exercise 5 Exercise 17 Exercise 21 Exercise 25 Exercise 49 Exercise 51 Exercise 57 Exercise 64 15.3 Double Integrals in Polar Coordinates Exercise 11 Exercise 13 Exercise 15 Exercise 25 Exercise 39 15.4 Applications of Double Integrals Exercise 5 Exercise 15 Exercise 27 Exercise 29 15.5 Surface Area Exercise 3 Exercise 9 Exercise 12 15.6 Triple Integrals Exercise 13 Exercise 19 Exercise 23 Exercise 27 Exercise 35 Exercise 41 Exercise 53 15.7 Triple Integrals in Cylindrical Coordinates Exercise 9 Exercise 17 Exercise 21 15.8 Triple Integrals in Spherical Coordinates Exercise 5 Exercise 17 Exercise 21 Exercise 30 Exercise 35 15.9 Change of Variables in Multiple Integrals Exercise 7 Exercise 17 Exercise 25 16 Vector Calculus 16.1 Vector Fields Exercise 5 Exercise 11 Exercise 17 Exercise 23 Exercise 29 Exercise 35 16.2 Line Integrals Exercise 3 Exercise 7 Exercise 11 Exercise 17 Exercise 21 Exercise 33 Exercise 39 Exercise 45 16.3 The Fundamental Theorem for Line Integrals Exercise 7 Exercise 11 Exercise 15 Exercise 25 Exercise 29 Exercise 35 16.4 Green's Theorem Exercise 3 Exercise 7 Exercise 9 Exercise 17 Exercise 21 Exercise 29 16.5 Curl and Divergence Exercise 11 Exercise 13 Exercise 19 Exercise 21 Exercise 31 16.6 Parametric Surfaces and Their Areas Exercise 3 Exercise 13 Exercise 19 Exercise 23 Exercise 26 Exercise 33 Exercise 39 Exercise 45 Exercise 49 Exercise 59 Exercise 61 16.7 Surface Integrals Exercise 4 Exercise 9 Exercise 17 Exercise 23 Exercise 27 Exercise 39 Exercise 47 16.8 Stokes' Theorem Exercise 1 Exercise 5 Exercise 7 Exercise 15 Exercise 19 16.9 The Divergence Theorem Exercise 1 Exercise 7 Exercise 19 Exercise 25 17 Second-Order Differential Equations 17.1 Second-Order Linear Equations Exercise 1 Exercise 9 Exercise 11 Exercise 19 Exercise 31 17.2 Nonhomogeneous Linear Equations Exercise 5 Exercise 9 Exercise 16 Exercise 18 Exercise 21a Exercise 21b Exercise 25 17.3 Applications of Second-Order Differential Equations Exercise 3 Exercise 9 Exercise 13 Exercise 17 17.4 Series Solutions Exercise 3 Exercise 9 Appendixes Appendix G Appendix G Exercise 1a Exercise 3 Exercise 5