1 Functions and Models 1.1 Four Ways to Represent a Function Exercise 2 Exercise 4 Exercise 12 Exercise 26 Exercise 30 Exercise 42 Exercise 46 Exercise 59 Exercise 65 1.2 Mathematical Models: A Catalog of Essential Functions Exercise 3 Exercise 31 Exercise 46 Exercise 48 1.3 New Functions from Old Functions Exercise 1a Exercise 1d Exercise 1h Exercise 5a Exercise 5d Exercise 25 Exercise 35a Exercise 35c Exercise 37a Exercise 43a Exercise 43d Exercise 54 Exercise 62 Exercise 72 1.4 Exponential Functions Exercise 13 Exercise 17 Exercise 19 Exercise 23 Exercise 35 Exercise 44 1.5 Inverse Functions and Logarithms Exercise 3 Exercise 6 Exercise 13 Exercise 17 Exercise 19 Exercise 22 Exercise 25 Exercise 49 Exercise 82 1.6 Parametric Curves Exercise 4 Exercise 9 Exercise 13 Exercise 20 Exercise 29 Exercise 31 Exercise 32 Exercise 38 Exercise 43 Exercise 47 2 Limits 2.1 The Tangent and Velocity Problems Exercise 6 Exercise 9 Exercise 14 2.2 The Limit of a Function Exercise 4 Exercise 10 Exercise 18 Exercise 36 Exercise 42 2.3 Calculating Limits Using the Limit Laws Exercise 6 Exercise 18 Exercise 21 Exercise 22 Exercise 41 Exercise 45 Exercise 57 Exercise 68 Exercise 71 2.4 Continuity Exercise 3 Exercise 12 Exercise 20 Exercise 32 Exercise 34 Exercise 37 Exercise 44 Exercise 46 Exercise 55 Exercise 61 Exercise 70 2.5 Limits Involving Infinity Exercise 2 Exercise 7 Exercise 17 Exercise 39 Exercise 58 Exercise 76 2.6 Derivatives and Rates of Change Exercise 8 Exercise 10 Exercise 14 Exercise 19 Exercise 27 Exercise 28 Exercise 34 Exercise 38 Exercise 48 Exercise 54 Exercise 60 Exercise 62 Exercise 66 Exercise 69 2.7 The Derivative as a Function Exercise 3b Exercise 3c Exercise 5 Exercise 11 Exercise 20a Exercise 20b Exercise 45 Exercise 53 Exercise 64 Exercise 68 3 Differentiation Rules 3.1 Derivatives of Polynomials and Exponential Functions Exercise 31 Exercise 37 Exercise 48 Exercise 51 Exercise 60 Exercise 69 Exercise 81 Exercise 91 3.2 The Product and Quotient Rules Exercise 12 Exercise 26 Exercise 34 Exercise 41 Exercise 48 Exercise 54 Exercise 60 Exercise 62 Exercise 71 3.3 Derivatives of Trigonometric Functions Exercise 9 Exercise 24 Exercise 32 Exercise 38 Exercise 44 Exercise 46 Exercise 48 Exercise 52 Exercise 54 Exercise 56 3.4 The Chain Rule Exercise 5 Exercise 37 Exercise 55 Exercise 59 Exercise 62 Exercise 77 Exercise 82 Exercise 90 Exercise 93 Exercise 102 Exercise 107 Exercise 111a Exercise 111b Exercise 118 3.5 Implicit Differentiation Exercise 18 Exercise 35 Exercise 57 Exercise 65 Exercise 71 Exercise 75 3.6 Inverse Trigonometric Functions and their Derivatives Exercise 11 Exercise 19 Exercise 31 Exercise 39 Exercise 41a Exercise 41b 3.7 Derivatives of Logarithmic Functions Exercise 20 Exercise 32 Exercise 46 Exercise 66 Exercise 69 3.8 Rates of Change in the Natural and Social Sciences Exercise 1g Exercise 1h Exercise 1i Exercise 23 Exercise 30 Exercise 33 Exercise 39 3.9 Linear Approximations and Differentials Exercise 9 Exercise 24 Exercise 31 Exercise 37 Exercise 39 4 Applications of Differentiation 4.1 Related Rates Exercise 17 Exercise 20 Exercise 28 Exercise 36 Exercise 44 Exercise 50a Exercise 50b Exercise 56 4.2 Maximum and Minimum Values Exercise 11 Exercise 13 Exercise 24 Exercise 37 Exercise 39 Exercise 47 Exercise 71 Exercise 82 4.3 Derivatives and the Shapes of Curves Exercise 14 Exercise 16c Exercise 21a Exercise 21b Exercise 21c Exercise 28 Exercise 43 Exercise 59a Exercise 59b Exercise 59c Exercise 66 Exercise 66d Exercise 71 Exercise 80 Exercise 84 Exercise 85 Exercise 93 Exercise 100 Exercise 102 Exercise 104 4.4 Graphing with Calculus and Calculators Exercise 12 Exercise 24 Exercise 26 Exercise 27 Exercise 31 Exercise 32 4.5 Indeterminate Forms and l'Hospital's Rule Exercise 1 Exercise 28 Exercise 46 Exercise 51 Exercise 60 Exercise 72 Exercise 76 Exercise 78 Exercise 85 Exercise 97 4.6 Optimization Problems Exercise 18 Exercise 23 Exercise 26 Exercise 30 Exercise 41 Exercise 44 Exercise 61 Exercise 62 Exercise 67 Exercise 69 Exercise 78 Exercise 84 4.7 Newton's Method Exercise 4 Exercise 41 Exercise 51 4.8 Antiderivatives Exercise 23 Exercise 44 Exercise 57 Exercise 63 Exercise 72 Exercise 79 5 Integrals 5.1 Areas and Distances Exercise 2 Exercise 6 Exercise 15 Exercise 19 Exercise 27 5.2 The Definite Integral Exercise 9 Exercise 43 Exercise 49 Exercise 61 Exercise 63 Exercise 73 5.3 Evaluating Definite Integrals Exercise 11 Exercise 25 Exercise 36 Exercise 39 Exercise 61 Exercise 66 Exercise 73 Exercise 75 Exercise 92 5.4 The Fundamental Theorem of Calculus Exercise 3 Exercise 15 Exercise 26 Exercise 37 Exercise 38 5.5 The Substitution Rule Exercise 4 Exercise 24 Exercise 28 Exercise 40 Exercise 50 Exercise 67 Exercise 71 Exercise 76 Exercise 86 Exercise 95 Exercise 99 5.6 Integration by Parts Exercise 3 Exercise 16 Exercise 23 Exercise 43 Exercise 55 Exercise 61 Exercise 65 5.7 Additional Techniques of Integration Exercise 3 Exercise 6 Exercise 13 Exercise 28 Exercise 30 Exercise 45 5.8 Integration Using Tables and Computer Algebra Systems Exercise 6 Exercise 11 Exercise 13 Exercise 17 Exercise 18 Exercise 21 Exercise 23 5.9 Approximate Integration Exercise 1 Exercise 29 Exercise 31 Exercise 39 5.10 Improper Integrals Exercise 16 Exercise 25 Exercise 33 Exercise 55 Exercise 63 Exercise 67 Exercise 75 6 Applications of Integration 6.1 More about Areas Exercise 3 Exercise 11 Exercise 13 Exercise 57 Exercise 63 Exercise 69 6.2 Volumes Exercise 8 Exercise 11 Exercise 14 Exercise 60 Exercise 62 Exercise 68 Exercise 74 Exercise 78 6.3 Volumes by Cylindrical Shells Exercise 5 Exercise 13 Exercise 17 Exercise 25 Exercise 29a Exercise 41 Exercise 47 6.4 Arc Length Exercise 18 Exercise 21 Exercise 35 Exercise 40 Exercise 43 6.5 Average Value of a Function Exercise 9 Exercise 11 Exercise 17 Exercise 23 Exercise 30 6.6 Applications to Physics and Engineering Exercise 5 Exercise 7 Exercise 11 Exercise 15 Exercise 17 Exercise 25 Exercise 33 Exercise 37 Exercise 47 6.7 Applications to Economics and Biology Exercise 3 Exercise 5 Exercise 10 Exercise 17 6.8 Probability Exercise 1 Exercise 10 Exercise 15 7 Differential Equations 7.1 Modeling with Differential Equations Exercise 4 Exercise 8 Exercise 10 Exercise 12 Exercise 16 7.2 Direction Fields and Euler's Method Exercise 6 Exercise 14 Exercise 16 Exercise 21 Exercise 22 Exercise 24 Exercise 27 7.3 Separable Equations Exercise 12 Exercise 17 Exercise 31 Exercise 37 Exercise 47 Exercise 52 Exercise 58 7.4 Exponential Growth and Decay Exercise 5 Exercise 7 Exercise 12 Exercise 16 Exercise 21 Exercise 24 7.5 The Logistic Equation Exercise 3 Exercise 13 Exercise 17 Exercise 19 Exercise 23 Exercise 25 7.6 Predator-Prey Systems Exercise 1 Exercise 5 Exercise 7 Exercise 9 8 Infinite Sequences and Series 8.1 Sequences Exercise 3 Exercise 11 Exercise 29 Exercise 40 Exercise 55 Exercise 59 Exercise 65 Exercise 67 8.2 Series Exercise 9 Exercise 15 Exercise 40 Exercise 44 Exercise 52 Exercise 70 Exercise 76 Exercise 82 Exercise 87 Exercise 89 8.3 The Integral and Comparison Tests; Estimating Sums Exercise 3 Exercise 18 Exercise 22 Exercise 26 Exercise 40 Exercise 49 Exercise 55 Exercise 57 8.4 Other Convergence Tests Exercise 8 Exercise 16 Exercise 20 Exercise 41 Exercise 43 Exercise 44 Exercise 51 8.5 Power Series Exercise 20 Exercise 23 Exercise 28 Exercise 34 8.6 Representations of Functions as Power Series Exercise 6 Exercise 9 Exercise 12a Exercise 12b Exercise 14 Exercise 22 Exercise 24 Exercise 41 Exercise 43 8.7 Taylor and Maclaurin Series Exercise 5 Exercise 27 Exercise 33 Exercise 35 Exercise 39 Exercise 45 Exercise 59 Exercise 63 Exercise 69 8.8 Applications of Taylor Polynomials Exercise 6 Exercise 12 Exercise 21 Exercise 22 Exercise 27 Exercise 33 Exercise 35 9 Vectors and the Geometry of Space 9.1 Three-Dimensional Coordinate Systems Exercise 5 Exercise 12 Exercise 22 Exercise 28 Exercise 34 Exercise 40 Exercise 44 9.2 Vectors Exercise 3 Exercise 10 Exercise 27 Exercise 29 Exercise 47 Exercise 51 Exercise 55 9.3 The Dot Product Exercise 15 Exercise 23 Exercise 39 Exercise 49 Exercise 51 Exercise 57 Exercise 59 Exercise 65 9.4 The Cross Product Exercise 1 Exercise 4 Exercise 16 Exercise 30 Exercise 48 Exercise 52 Exercise 56 9.5 Equations of Lines and Planes Exercise 5 Exercise 7 Exercise 14 Exercise 23 Exercise 35 Exercise 51 Exercise 63 Exercise 75 9.6 Functions and Surfaces Exercise 1 Exercise 6 Exercise 17 Exercise 29 9.7 Cylindrical and Spherical Coordinates Exercise 6 Exercise 9 Exercise 18 Exercise 20 Exercise 25 Exercise 37 10 Vector Functions 10.1 Vector Functions and Space Curves Exercise 19 Exercise 31 Exercise 35 Exercise 51 Exercise 55 10.2 Derivatives and Integrals of Vector Functions Exercise 1 Exercise 3 Exercise 19 Exercise 23 Exercise 31 Exercise 63 10.3 Arc Length and Curvature Exercise 3 Exercise 5 Exercise 19 Exercise 35 Exercise 37 Exercise 41 Exercise 49 Exercise 56 Exercise 62 10.4 Motion in Space: Velocity and Acceleration Exercise 13 Exercise 23 Exercise 26 Exercise 29 Exercise 43 10.5 Parametric Surfaces Exercise 4 Exercise 15 Exercise 21 Exercise 25 Exercise 28 Exercise 33 11 Partial Derivatives 11.1 Functions of Several Variables Exercise 1 Exercise 7 Exercise 17 Exercise 29 Exercise 41 Exercise 47 Exercise 51 11.2 Limits and Continuity Exercise 7 Exercise 18 Exercise 26 Exercise 30 Exercise 33 Exercise 42 Exercise 44 11.3 Partial Derivatives Exercise 1 Exercise 5a Exercise 5b Exercise 9 Exercise 21 Exercise 37 Exercise 62 Exercise 86 Exercise 98 Exercise 103 Exercise 104 11.4 Tangent Planes and Linear Approximations Exercise 14 Exercise 26 Exercise 38 Exercise 42 Exercise 49 Exercise 55 Exercise 57 11.5 The Chain Rule Exercise 7 Exercise 16 Exercise 24 Exercise 39 Exercise 44 Exercise 48 Exercise 54 Exercise 56 11.6 Directional Derivatives and the Gradient Vector Exercise 1 Exercise 15 Exercise 24 Exercise 28 Exercise 33 Exercise 35 Exercise 39 Exercise 51 Exercise 69 Exercise 75 11.7 Maximum and Minimum Values Exercise 1 Exercise 3 Exercise 20 Exercise 39 Exercise 48 Exercise 50 Exercise 58 11.8 Lagrange Multipliers Exercise 1 Exercise 3 Exercise 15 Exercise 25 Exercise 31 Exercise 40 Exercise 50 12 Multiple Integrals 12.1 Double Integrals over Rectangles Exercise 1 Exercise 9 Exercise 11a Exercise 11b Exercise 15 Exercise 20 12.2 Iterated Integrals Exercise 5 Exercise 11 Exercise 21 Exercise 23 Exercise 29 Exercise 34 Exercise 42 12.3 Double Integrals over General Regions Exercise 7 Exercise 21 Exercise 25 Exercise 29 Exercise 53 Exercise 55 Exercise 61 Exercise 68 12.4 Double Integrals in Polar Coordinates Exercise 1 Exercise 13 Exercise 15 Exercise 18 Exercise 28 Exercise 40 12.5 Applications of Double Integrals Exercise 1 Exercise 5 Exercise 15 Exercise 23 Exercise 25 12.6 Surface Area Exercise 3 Exercise 7 Exercise 9 Exercise 21 Exercise 27 12.7 Triple Integrals Exercise 12 Exercise 20 Exercise 24 Exercise 29 Exercise 37 Exercise 41 Exercise 53 12.8 Triple Integrals in Cylindrical and Spherical Coordinates Exercise 3 Exercise 7 Exercise 11 Exercise 17 Exercise 26 Exercise 31 12.9 Change of Variables in Multiple Integrals Exercise 9 Exercise 19 Exercise 27 13 Vector Calculus 13.1 Vector Fields Exercise 7 Exercise 13 Exercise 21 Exercise 28 Exercise 37 Exercise 43 13.2 Line Integrals Exercise 3 Exercise 7 Exercise 11 Exercise 18 Exercise 22 Exercise 36 Exercise 42 Exercise 46 13.3 The Fundamental Theorem for Line Integrals Exercise 9 Exercise 13 Exercise 18 Exercise 30 Exercise 34 Exercise 40 13.4 Green's Theorem Exercise 3 Exercise 7 Exercise 10 Exercise 19 Exercise 23 Exercise 31 13.5 Curl and Divergence Exercise 1 Exercise 13 Exercise 18 Exercise 23 Exercise 25 Exercise 35 13.6 Surface Integrals Exercise 4 Exercise 9 Exercise 17 Exercise 21 Exercise 27 Exercise 38 Exercise 46 13.7 Stokes' Theorem Exercise 1 Exercise 6 Exercise 8 Exercise 16 Exercise 20 13.8 The Divergence Theorem Exercise 1 Exercise 7 Exercise 20 Exercise 26