Student Solutions Manual for University Calculus

<div class="c-un-numbered-headers-and-contents-list__container"> <ol start="1"> <li><strong>Functions</strong> <ul> <li>1.1 Functions and Their Graphs</li> <li>1.2 Combining Functions; Shifting and Scaling Graphs</li> <li>1.3 Trigonometric Functions</li> <li>1.4 Graphing with Software</li> <li>1.5 Exponential Functions</li> <li>1.6 Inverse Functions and Logarithms</li> </ul> </li> <li><strong>Limits and Continuity</strong> <ul> <li>2.1 Rates of Change and Tangent Lines to Curves</li> <li>2.2 Limit of a Function and Limit Laws</li> <li>2.3 The Precise Definition of a Limit</li> <li>2.4 One-Sided Limits</li> <li>2.5 Continuity</li> <li>2.6 Limits Involving Infinity; Asymptotes of Graphs</li> </ul> <ul> <li>Questions to Guide Your Review</li> <li>Practice Exercises</li> <li>Additional and Advanced Exercises</li> </ul> </li> <li><strong>Derivatives</strong> <ul> <li>3.1 Tangent Lines and the Derivative at a Point</li> <li>3.2 The Derivative as a Function</li> <li>3.3 Differentiation Rules</li> <li>3.4 The Derivative as a Rate of Change</li> <li>3.5 Derivatives of Trigonometric Functions</li> <li>3.6 The Chain Rule</li> <li>3.7 Implicit Differentiation</li> <li>3.8 Derivatives of Inverse Functions and Logarithms</li> <li>3.9 Inverse Trigonometric Functions</li> <li>3.10 Related Rates</li> <li>3.11 Linearization and Differentials</li> </ul> <ul> <li>Questions to Guide Your Review</li> <li>Practice Exercises</li> <li>Additional and Advanced Exercises</li> </ul> </li> <li><strong>Applications of Derivatives</strong> <ul> <li>4.1 Extreme Values of Functions on Closed Intervals</li> <li>4.2 The Mean Value Theorem</li> <li>4.3 Monotonic Functions and the First Derivative Test</li> <li>4.4 Concavity and Curve Sketching</li> <li>4.5 Indeterminate Forms and L’Hôpital’s Rule</li> <li>4.6 Applied Optimization</li> <li>4.7 Newton’s Method</li> <li>4.8 Antiderivatives</li> </ul> <ul> <li>Questions to Guide Your Review</li> <li>Practice Exercises</li> <li>Additional and Advanced Exercises</li> </ul> </li> <li><strong>Integrals</strong> <ul> <li>5.1 Area and Estimating with Finite Sums</li> <li>5.2 Sigma Notation and Limits of Finite Sums</li> <li>5.3 The Definite Integral</li> <li>5.4 The Fundamental Theorem of Calculus</li> <li>5.5 Indefinite Integrals and the Substitution Method</li> <li>5.6 Definite Integral Substitutions and the Area Between Curves</li> </ul> <ul> <li>Questions to Guide Your Review</li> <li>Practice Exercises</li> <li>Additional and Advanced Exercises</li> </ul> </li> <li><strong>Applications of Definite Integrals</strong> <ul> <li>6.1 Volumes Using Cross-Sections</li> <li>6.2 Volumes Using Cylindrical Shells</li> <li>6.3 Arc Length</li> <li>6.4 Areas of Surfaces of Revolution</li> <li>6.5 Work</li> <li>6.6 Moments and Centers of Mass</li> </ul> <ul> <li>Questions to Guide Your Review</li> <li>Practice Exercises</li> <li>Additional and Advanced Exercises</li> </ul> </li> <li><strong>Integrals and Transcendental Functions</strong> <ul> <li>7.1 The Logarithm Defined as an Integral</li> <li>7.2 Exponential Change and Separable Differential Equations</li> <li>7.3 Hyperbolic Functions</li> </ul> <ul> <li>Questions to Guide Your Review</li> <li>Practice Exercises</li> <li>Additional and Advanced Exercises</li> </ul> </li> <li><strong>Techniques of Integration</strong> <ul> <li>8.1 Integration by Parts</li> <li>8.2 Trigonometric Integrals</li> <li>8.3 Trigonometric Substitutions</li> <li>8.4 Integration of Rational Functions by Partial Fractions</li> <li>8.5 Integral Tables and Computer Algebra Systems</li> <li>8.6 Numerical Integration</li> <li>8.7 Improper Integrals</li> </ul> <ul> <li>Questions to Guide Your Review</li> <li>Practice Exercises</li> <li>Additional and Advanced Exercises</li> </ul> </li> <li><strong>Infinite Sequences and Series</strong> <ul> <li>9.1 Sequences</li> <li>9.2 Infinite Series</li> <li>9.3 The Integral Test</li> <li>9.4 Comparison Tests</li> <li>9.5 Absolute Convergence; The Ratio and Root Tests</li> <li>9.6 Alternating Series and Conditional Convergence</li> <li>9.7 Power Series</li> <li>9.8 Taylor and Maclaurin Series</li> <li>9.9 Convergence of Taylor Series</li> <li>9.10 Applications of Taylor Series</li> </ul> <ul> <li>Questions to Guide Your Review</li> <li>Practice Exercises</li> <li>Additional and Advanced Exercises</li> </ul> </li> <li><strong>Parametric Equations and Polar Coordinates</strong> <ul> <li>10.1 Parametrizations of Plane Curves</li> <li>10.2 Calculus with Parametric Curves</li> <li>10.3 Polar Coordinates</li> <li>10.4 Graphing Polar Coordinate Equations</li> <li>10.5 Areas and Lengths in Polar Coordinates</li> </ul> <ul> <li>Questions to Guide Your Review</li> <li>Practice Exercises</li> <li>Additional and Advanced Exercises</li> </ul> </li> <li><strong>Vectors and the Geometry of Space</strong> <ul> <li>11.1 Three-Dimensional Coordinate Systems</li> <li>11.2 Vectors</li> <li>11.3 The Dot Product</li> <li>11.4 The Cross Product</li> <li>11.5 Lines and Planes in Space</li> <li>11.6 Cylinders and Quadric Surfaces</li> </ul> <ul> <li>Questions to Guide Your Review</li> <li>Practice Exercises</li> <li>Additional and Advanced Exercises</li> </ul> </li> <li><strong>Vector-Valued Functions and Motion in Space</strong> <ul> <li>12.1 Curves in Space and Their Tangents</li> <li>12.2 Integrals of Vector Functions; Projectile Motion</li> <li>12.3 Arc Length in Space</li> <li>12.4 Curvature and Normal Vectors of a Curve</li> <li>12.5 Tangential and Normal Components of Acceleration</li> <li>12.6 Velocity and Acceleration in Polar Coordinates</li> </ul> <ul> <li>Questions to Guide Your Review</li> <li>Practice Exercises</li> <li>Additional and Advanced Exercises</li> </ul> </li> <li><strong>Partial Derivatives</strong> <ul> <li>13.1 Functions of Several Variables</li> <li>13.2 Limits and Continuity in Higher Dimensions</li> <li>13.3 Partial Derivatives</li> <li>13.4 The Chain Rule</li> <li>13.5 Directional Derivatives and Gradient Vectors</li> <li>13.6 Tangent Planes and Differentials</li> <li>13.7 Extreme Values and Saddle Points</li> <li>13.8 Lagrange Multiplier</li> </ul> <ul> <li>Questions to Guide Your Review</li> <li>Practice Exercises</li> <li>Additional and Advanced Exercises</li> </ul> </li> <li><strong>Multiple Integrals</strong> <ul> <li>14.1 Double and Iterated Integrals over Rectangles</li> <li>14.2 Double Integrals over General Regions</li> <li>14.3 Area by Double Integration</li> <li>14.4 Double Integrals in Polar Form</li> <li>14.5 Triple Integrals in Rectangular Coordinates</li> <li>14.6 Applications</li> <li>14.7 Triple Integrals in Cylindrical and Spherical Coordinates</li> <li>14.8 Substitutions in Multiple Integrals</li> </ul> <ul> <li>Questions to Guide Your Review</li> <li>Practice Exercises</li> <li>Additional and Advanced Exercises</li> </ul> </li> <li><strong>Integrals and Vector Fields</strong> <ul> <li>15.1 Line Integrals of Scalar Functions</li> <li>15.2 Vector Fields and Line Integrals: Work, Circulation, and Flux</li> <li>15.3 Path Independence, Conservative Fields, and Potential Functions</li> <li>15.4 Green’s Theorem in the Plane</li> <li>15.5 Surfaces and Area</li> <li>15.6 Surface Integrals</li> <li>15.7 Stokes’ Theorem</li> <li>15.8 The Divergence Theorem and a Unified Theory</li> </ul> <ul> <li>Questions to Guide Your Review</li> <li>Practice Exercises</li> <li>Additional and Advanced Exercises</li> </ul> </li> <li><strong>First-Order Differential Equations (online at <a href="http://bit.ly/2pzYlEq">bit.ly/2pzYlEq</a>)</strong> <ul> <li>16.1 Solutions, Slope Fields, and Euler’s Method</li> <li>16.2 First-Order Linear Equations</li> <li>16.3 Applications</li> <li>16.4 Graphical Solutions of Autonomous Equations</li> <li>16.5 Systems of Equations and Phase Planes</li> </ul> </li> <li><strong>Second-Order Differential Equations (online at <a href="http://bit.ly/2IHCJyE">bit.ly/2IHCJyE</a>)</strong> <ul> <li>17.1 Second-Order Linear Equations</li> <li>17.2 Non-homogeneous Linear Equations</li> <li>17.3 Applications</li> <li>17.4 Euler Equations</li> <li>17.5 Power-Series Solutions</li> </ul> </li> </ol> <h3>Appendix</h3> <ul> <li>A.1 Real Numbers and the Real Line</li> <li>A.2 Mathematical Induction</li> <li>A.3 Lines and Circles</li> <li>A.4 Conic Sections</li> <li>A.5 Proofs of Limit Theorems</li> <li>A.6 Commonly Occurring Limits</li> <li>A.7 Theory of the Real Numbers</li> <li>A.8 Complex Numbers</li> <li>A.9 The Distributive Law for Vector Cross Products</li> <li>A.10 The Mixed Derivative Theorem and the increment Theorem</li> </ul> <h3>Additional Topics (online)</h3> <ul> <li>B.1 Relative Rates of Growth</li> <li>B.2 Probability</li> <li>B.3 Conics in Polar Coordinates</li> <li>B.4 Taylor’s Formula for Two Variables</li> <li>B.5 Partial Derivatives with Constrained Variables</li> </ul> <h4 class="h5">Odd Answers</h4> </div>