1 Functions and Limits 1.1 Functions and Their Representations Exercise 4 Exercise 9 Exercise 11 Exercise 13 Exercise 21 Exercise 29 Exercise 37 Exercise 41 Exercise 45 Exercise 51 Exercise 53 1.2 A Catalog of Essential Functions Exercise 1 Exercise 5 Exercise 11 Exercise 13 Exercise 17a Exercise 17d Exercise 21a Exercise 21d Exercise 29 Exercise 37 Exercise 43 Exercise 57 Exercise 65 1.3 The Limit of a Funciton Exercise 3 Exercise 9 Exercise 23 Exercise 31 Exercise 39 1.4 Calculating Limits Exercise 6 Exercise 15 Exercise 18 Exercise 19 Exercise 33 Exercise 37 Exercise 45 Exercise 51 Exercise 55 Exercise 63 Exercise 65 1.5 Continuity Exercise 3 Exercise 9 Exercise 13 Exercise 18 Exercise 28 Exercise 31 Exercise 33 Exercise 39 Exercise 47 1.6 Limits Involving Infinity Exercise 5 Exercise 15 Exercise 21 Exercise 25 Exercise 35 Exercise 45 Exercise 47 Exercise 53 Exercise 57a Exercise 57b 2 Derivatives 2.1 Derivatives and Rates of Change Exercise 5 Exercise 7 Exercise 11 Exercise 15 Exercise 18 Exercise 19 Exercise 23 Exercise 27 Exercise 35 Exercise 37 Exercise 41 Exercise 43 Exercise 49 2.2 The Derivative as a Function Exercise 3b Exercise 3c Exercise 5 Exercise 11 Exercise 17a Exercise 17b Exercise 25 Exercise 31 Exercise 33 Exercise 39 Exercise 44 Exercise 47 2.3 Basic Differentiation Rules Exercise 19 Exercise 32 Exercise 35 Exercise 37 Exercise 43 Exercise 45g Exercise 45h Exercise 53 Exercise 59 Exercise 61 Exercise 63 Exercise 69 2.4 The Product and Quotient Rules Exercise 5 Exercise 19 Exercise 25 Exercise 32 Exercise 35 Exercise 41 Exercise 46 Exercise 47 Exercise 55 2.5 The Chain Rule Exercise 5 Exercise 17 Exercise 35 Exercise 51 Exercise 53 Exercise 57 Exercise 63 Exercise 70 Exercise 73 2.6 Implicit Differentiation Exercise 11 Exercise 21 Exercise 33 Exercise 37 Exercise 43 Exercise 49 2.7 Related Rates Exercise 11 Exercise 15 Exercise 19 Exercise 27 Exercise 33 Exercise 37a Exercise 37b Exercise 41 2.8 Linear Approximations and Differentials Exercise 3 Exercise 5 Exercise 9 Exercise 15 Exercise 17 Exercise 21 Exercise 27 Exercise 29 3 Applications of Differentiation 3.1 Maximum and Minimum Values Exercise 9 Exercise 11 Exercise 13 Exercise 21 Exercise 31 Exercise 33 Exercise 37 Exercise 49 Exercise 60 3.2 The Mean Value Theorem Exercise 5 Exercise 9 Exercise 19 Exercise 23 Exercise 25 Exercise 33 3.3 Derivatives and the Shapes of Graphs Exercise 3a Exercise 3b Exercise 3c Exercise 15 Exercise 19 Exercise 27 Exercise 29a Exercise 29b Exercise 29c Exercise 39 Exercise 44 3.4 Curve Sketching Exercise 5 Exercise 9 Exercise 17 Exercise 29 Exercise 53 3.5 Optimization Problems Exercise 11 Exercise 13 Exercise 15 Exercise 17 Exercise 24 Exercise 25 Exercise 27 Exercise 37 Exercise 39 Exercise 43 Exercise 45 Exercise 51 3.6 Newton's Method Exercise 4 Exercise 18 Exercise 23 Exercise 27 Exercise 29 3.7 Antiderivatives Exercise 7 Exercise 15 Exercise 17 Exercise 31 Exercise 37 Exercise 39 Exercise 45 Exercise 51 4 Integrals 4.1 Areas and Distances Exercise 2 Exercise 5 Exercise 9 Exercise 13 Exercise 17 4.2 The Definite Integral Exercise 7 Exercise 11 Exercise 17 Exercise 21 Exercise 29 Exercise 33 Exercise 39 Exercise 41 Exercise 47 4.3 Evaluating Definite Integrals Exercise 11 Exercise 29 Exercise 37 Exercise 40 Exercise 45 Exercise 50 Exercise 53 Exercise 59 Exercise 61 4.4 The Fundamental Theorem of Calculus Exercise 1 Exercise 7 Exercise 9 Exercise 13 Exercise 19 Exercise 25 Exercise 30 Exercise 31 4.5 The Substitution Rule Exercise 3 Exercise 23 Exercise 47 Exercise 53 Exercise 57 5 Inverse Functions 5.1 Inverse Functions Exercise 3 Exercise 13 Exercise 17 Exercise 19 Exercise 22 Exercise 39 Exercise 41 5.2 The Natural Logarithmic Function Exercise 21 Exercise 35 Exercise 39 Exercise 59 Exercise 60 Exercise 73 5.3 The Natural Exponential Function Exercise 7 Exercise 13 Exercise 25 Exercise 29 Exercise 38 Exercise 39 Exercise 43 Exercise 53 Exercise 55 Exercise 63 Exercise 67 5.4 General Logarithmic and Exponential Functions Exercise 12 Exercise 17 Exercise 19 Exercise 31 Exercise 33 Exercise 41 Exercise 45 Exercise 48 5.5 Exponential Growth and Decay Exercise 3 Exercise 5 Exercise 9 Exercise 13 Exercise 19 5.6 Inverse Trigonometric Functions Exercise 9 Exercise 19 Exercise 33 Exercise 37 Exercise 43 5.7 Hyperbolic Functions Exercise 9 Exercise 13 Exercise 31 Exercise 39 Exercise 47 Exercise 51 5.8 Indeterminate Forms and l'Hospital's Rule Exercise 13 Exercise 25 Exercise 30 Exercise 35 Exercise 41 Exercise 55 6 Techniques of Integration 6.1 Integration by Parts Exercise 3 Exercise 13 Exercise 18 Exercise 29 Exercise 35 Exercise 43 Exercise 46 6.2 Trigonometric Integrals and Substitutions Exercise 3 Exercise 5 Exercise 9 Exercise 19 Exercise 25 Exercise 41 Exercise 45 Exercise 51 Exercise 55 Exercise 56 6.3 Partial Fractions Exercise 6 Exercise 11 Exercise 17 Exercise 23 Exercise 27 Exercise 29 Exercise 37 Exercise 39 6.4 Integration with Tables and Computer Algebra Systems Exercise 6 Exercise 11 Exercise 13 Exercise 17 Exercise 18 Exercise 21 Exercise 23 6.5 Approximate Integration Exercise 1 Exercise 3 Exercise 4 Exercise 27 Exercise 33 Exercise 39 6.6 Improper Integrals Exercise 1 Exercise 7 Exercise 13 Exercise 17 Exercise 25 Exercise 29 Exercise 41 Exercise 49 Exercise 51 Exercise 57 7 Applications of Integration 7.1 Areas Between Curves Exercise 3 Exercise 9 Exercise 11 Exercise 21 Exercise 28 Exercise 35 Exercise 41 7.2 Volumes Exercise 5 Exercise 7 Exercise 9 Exercise 31 Exercise 33 Exercise 39 Exercise 47 Exercise 51 7.3 Volumes by Cylindrical Shells Exercise 5 Exercise 13 Exercise 17 Exercise 25 Exercise 29 Exercise 37 Exercise 41 7.4 Arc Length Exercise 7 Exercise 11 Exercise 13 Exercise 27 Exercise 29 Exercise 35 Exercise 36 7.5 Area of a Surface of Revolution Exercise 1ai Exercise 1aii Exercise 5 Exercise 11 Exercise 15 Exercise 23 7.6 Applications to Physics and Engineering Exercise 5 Exercise 9 Exercise 13 Exercise 15 Exercise 19 Exercise 29 Exercise 31 Exercise 41 Exercise 45 Exercise 51 7.7 Differential Equations Exercise 8 Exercise 11 Exercise 19 Exercise 21 Exercise 29 Exercise 31 Exercise 33 Exercise 35 Exercise 39 Exercise 41 Exercise 47 8 Series 8.1 Sequences Exercise 7 Exercise 11 Exercise 24 Exercise 25 Exercise 29 Exercise 35 Exercise 37 Exercise 43 8.2 Series Exercise 9 Exercise 11 Exercise 21 Exercise 25 Exercise 31 Exercise 35 Exercise 39 Exercise 45 Exercise 49 Exercise 55 Exercise 57 8.3 The Integral and Comparison Tests Exercise 3 Exercise 13 Exercise 17 Exercise 19 Exercise 20 Exercise 22 Exercise 29 Exercise 35 Exercise 37 Exercise 39 Exercise 43 8.4 Other Convergence Tests Exercise 3 Exercise 7 Exercise 9 Exercise 18 Exercise 19 Exercise 27 Exercise 29 Exercise 33 Exercise 43 Exercise 45 8.5 Power Series Exercise 5 Exercise 7 Exercise 8 Exercise 15 Exercise 19 Exercise 22 Exercise 23 Exercise 31 8.6 Representing Functions as Power Series Exercise 5 Exercise 8 Exercise 13a Exercise 13b Exercise 15 Exercise 23 Exercise 25 Exercise 37 Exercise 39 8.7 Taylor and Maclaurin Series Exercise 5 Exercise 13 Exercise 25 Exercise 31 Exercise 33 Exercise 37 Exercise 41 Exercise 53 Exercise 55 Exercise 59 8.8 Applications of Taylor Polynomials Exercise 5 Exercise 7 Exercise 13 Exercise 14 Exercise 19 Exercise 23 Exercise 25 9 Parametric Equations and Polar Coordinates 9.1 Parametric Curves Exercise 4 Exercise 7 Exercise 11 Exercise 17 Exercise 25 Exercise 27 Exercise 28 Exercise 35 Exercise 39 Exercise 43 9.2 Calculus with Parametric Curves Exercise 5 Exercise 9 Exercise 19 Exercise 21 Exercise 27 Exercise 37 Exercise 41 9.3 Polar Coordinates Exercise 11 Exercise 13 Exercise 19 Exercise 27 Exercise 31 Exercise 41 Exercise 49 Exercise 51 Exercise 55 9.4 Areas and Lengths in Polar Coordinates Exercise 7 Exercise 17 Exercise 21 Exercise 25 Exercise 31 Exercise 35 9.5 Conic Sections in Polar Coordinates Exercise 1 Exercise 13a Exercise 13b Exercise 13c Exercise 13d Exercise 19 Exercise 25 10 Vectors and the Geometry of Space 10.1 Three-Dimensional Coordinate Systems Exercise 5 Exercise 11 Exercise 19 Exercise 25 Exercise 29 Exercise 33 Exercise 35 10.2 Vectors Exercise 1 Exercise 7 Exercise 17 Exercise 21 Exercise 35 Exercise 37 Exercise 41 10.3 The Dot Product Exercise 11 Exercise 17 Exercise 23 Exercise 33 Exercise 35 Exercise 41 Exercise 43 Exercise 49 10.4 The Cross Product Exercise 7 Exercise 13 Exercise 16 Exercise 19 Exercise 31 Exercise 45 Exercise 49 Exercise 53 10.5 Equations of Lines and Planes Exercise 5 Exercise 7 Exercise 11 Exercise 17 Exercise 25 Exercise 41 Exercise 53 10.6 Cylinders and Quadric Surfaces Exercise 9 Exercise 19 10.7 Vector Functions and Space Curves Exercise 11 Exercise 19 Exercise 23 Exercise 29 Exercise 31 Exercise 33 Exercise 43 Exercise 45 Exercise 51 Exercise 81 10.8 Arc Length and Curvature Exercise 3 Exercise 13 Exercise 27 Exercise 29 Exercise 35 Exercise 39 Exercise 45 Exercise 47 10.9 Motion in Space: Velocity and Acceleration Exercise 9 Exercise 15 Exercise 18 Exercise 21 Exercise 31 11 Partial Derivatives 11.1 Functions of Several Variables Exercise 7 Exercise 9 Exercise 13 Exercise 23 Exercise 29 Exercise 41 Exercise 47 Exercise 51 11.2 Limits and Continuity Exercise 5 Exercise 9 Exercise 15 Exercise 19 Exercise 27 Exercise 29 11.3 Partial Derivatives Exercise 1 Exercise 3a Exercise 3b Exercise 11 Exercise 23 Exercise 44 Exercise 69 Exercise 74 Exercise 75 11.4 Tangent Planes and Linear Approximations Exercise 11 Exercise 19 Exercise 25 Exercise 29 Exercise 35 Exercise 37 11.5 The Chain Rule Exercise 3 Exercise 7 Exercise 13 Exercise 26 Exercise 29 Exercise 33 Exercise 37 Exercise 39 11.6 Directional Derivatives and the Gradient Vector Exercise 7 Exercise 13 Exercise 15 Exercise 19 Exercise 21 Exercise 25 Exercise 31 Exercise 47 Exercise 53 11.7 Maximum and Minimum Values Exercise 1 Exercise 9 Exercise 25 Exercise 33 Exercise 35 Exercise 43 11.8 Lagrange Multipliers Exercise 1 Exercise 9 Exercise 19 Exercise 21 Exercise 27 Exercise 37 Exercise 47 12 Multiple Integrals 12.1 Double Integrals over Rectangles Exercise 1 Exercise 5 Exercise 9 Exercise 11 Exercise 17 Exercise 21 Exercise 23 Exercise 27 Exercise 31 Exercise 39 Exercise 41 12.2 Double Integrals over General Regions Exercise 5 Exercise 15 Exercise 19 Exercise 23 Exercise 41 Exercise 43 Exercise 49 Exercise 54 12.3 Double Integrals over Polar Coordinates Exercise 1 Exercise 11 Exercise 17 Exercise 21 Exercise 29 12.4 Applications of Double Integrals Exercise 5 Exercise 15 12.5 Triple Integrals Exercise 11 Exercise 17 Exercise 21 Exercise 25 Exercise 33 Exercise 39 Exercise 49 12.6 Triple Integrals in Cylindrical Coordinates Exercise 9 Exercise 17 Exercise 21 12.7 Triple Integrals in Spherical Coordinates Exercise 5 Exercise 17 Exercise 21 Exercise 28 Exercise 33 12.8 Change of Variables in Multiple Integrals Exercise 7 Exercise 17 Exercise 25 13 Vector Calculus 13.1 Vector Fields Exercise 5 Exercise 11 Exercise 17 Exercise 23 Exercise 31 13.2 Line Integrals Exercise 3 Exercise 7 Exercise 11 Exercise 17 Exercise 21 Exercise 31 Exercise 37 Exercise 43 13.3 The Fundamental Theorem for Line Integrals Exercise 7 Exercise 13 Exercise 21 Exercise 25 Exercise 31 13.4 Green's Theorem Exercise 3 Exercise 7 Exercise 9 Exercise 17 Exercise 21 Exercise 29 13.5 Curl and Divergence Exercise 9 Exercise 11 Exercise 17 Exercise 19 Exercise 29 13.6 Parametric Surfaces and Their Areas Exercise 1 Exercise 11 Exercise 15 Exercise 19 Exercise 22 Exercise 29 Exercise 33 Exercise 39 Exercise 43 Exercise 53 Exercise 55 13.7 Surface Integrals Exercise 4 Exercise 9 Exercise 17 Exercise 23 Exercise 27 Exercise 37 Exercise 45 13.8 Stokes' Theorem Exercise 3 Exercise 5 Exercise 13 Exercise 17 13.9 The Divergence Theorem Exercise 1 Exercise 7 Exercise 19 Exercise 25 Appendixes Appendix C The Logarithm Defined as an Integral Exercise 1a Exercise 3 Exercise 5