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Chapter 1
Functions
1.1 | Functions and Their Graphs | Exercises | p.11 |
1.2 | Combining Functions: Shifting and Scaling Graphs | Exercises | p.19 |
1.3 | Trigonometric Functions | Exercises | p.28 |
1.4 | Graphing with Calculators and Computers | Exercises | p.34 |
Questions to Guide Your Review | p.34 | ||
Practice Exercises | p.35 | ||
Additional and Advanced Exercises | p.37 |
Chapter 2
Limits And Continuity
2.1 | Rates of Change and Tangents to Curves | Exercises | p.44 |
2.2 | Limit of a Function and Limit Laws | Exercises | p.54 |
2.3 | The Precise Definition of a Limit | Exercises | p.63 |
2.4 | One-Sided Limits | Exercises | p.71 |
2.5 | Continuity | Exercises | p.82 |
2.6 | Limits Involving Infinity; Asymptotes of Graphs | Exercises | p.94 |
Questions to Guide Your Review | p.96 | ||
Practice Exercises | p.97 | ||
Additional and Advanced Exercises | p.98 |
Chapter 3
Differentiation
3.1 | Tangents and the Derivative at a Point | Exercises | p.105 |
3.2 | The Derivative as a Function | Exercises | p.112 |
3.3 | Differentiation Rules | Exercises | p.122 |
3.4 | The Derivative as a Rate of Change | Exercises | p.132 |
3.5 | Derivatives of Trigonometric Functions | Exercises | p.139 |
3.6 | The Chain Rule | Exercises | p.147 |
3.7 | Implicit Differentiation | Exercises | p.153 |
3.8 | Related Rates | Exercises | p.160 |
3.9 | Linearization and Differentials | Exercises | p.173 |
Questions to Guide Your Review | p.175 | ||
Practice Exercises | p.176 | ||
Additional and Advanced Exercises | p.180 |
Chapter 4
Applications Of Derivatives
4.1 | Extreme Values of Functions | Exercises | p.189 |
4.2 | The Mean Value Theorem | Exercises | p.196 |
4.3 | Monotonic Functions and the First Derivative Test | Exercises | p.201 |
4.4 | Concavity and Curve Sketching | Exercises | p.211 |
4.5 | Applied Optimization | Exercises | p.219 |
4.6 | Newton's Method | Exercises | p.228 |
4.7 | Antiderivatives | Exercises | p.236 |
Questions to Guide Your Review | p.239 | ||
Practice Exercises | p.240 | ||
Additional and Advanced Exercises | p.243 |
Chapter 5
Integration
5.1 | Area and Estimating with Finite Sums | Exercises | p.253 |
5.2 | Sigma Notation and Limits of Finite Sums | Exercises | p.261 |
5.3 | The Definite Integral | Exercises | p.270 |
5.4 | The Fundamental Theorem of Calculus | Exercises | p.282 |
5.5 | Indefinite Integrals and the Substitution Method | Exercises | p.290 |
5.6 | Substitution and Area Between Curves | Exercises | p.297 |
Questions to Guide Your Review | p.300 | ||
Practice Exercises | p.301 | ||
Additional and Advanced Exercises | p.304 |
Chapter 6
Applications Of Definite Integrals
6.1 | Volumes Using Cross-Sections | Exercises | p.316 |
6.2 | Volumes Using Cylindrical Shells | Exercises | p.324 |
6.3 | Arc Length | Exercises | p.330 |
6.4 | Areas of Surfaces of Revolution | Exercises | p.335 |
6.5 | Work and Fluid Forces | Exercises | p.342 |
6.6 | Moments and Centers of Mass | Exercises | p.355 |
Questions to Guide Your Review | p.357 | ||
Practice Exercises | p.357 | ||
Additional and Advanced Exercises | p.359 |
Chapter 7
Transcendental Functions
7.1 | Inverse Functions and Their Derivatives | Exercises | p.367 |
7.2 | Natural Logarithms | Exercises | p.375 |
7.3 | Exponential Functoins | Exercises | p.385 |
7.4 | Exponential Change and Separable Differential Equations | Exercises | p.394 |
7.5 | Indeterminate Forms and L'Hopital's Rule | Exercises | p.402 |
7.6 | Inverse Trigonometric Functions | Exercises | p.413 |
7.7 | Hyperbolic Functions | Exercises | p.421 |
7.8 | Relative Rates of Growth | Exercises | p.428 |
Questions to Guide Your Review | p.429 | ||
Practice Exercises | p.430 | ||
Additional and Advanced Exercises | p.433 |
Chapter 8
Techniques Of Integration
8.1 | Integration by Parts | Exercises | p.441 |
8.2 | Trigonometric Integrals | Exercises | p.448 |
8.3 | Trigonometric Substitutions | Exercises | p.452 |
8.4 | Integration of Rational Functions by Partial Fractions | Exercises | p.461 |
8.5 | Integral Tables and Computer Algebra Systems | Exercises | p.467 |
8.6 | Numerical Integration | Exercises | p.475 |
8.7 | Improper Integrals | Exercises | p.487 |
Questions to Guide Your Review | p.489 | ||
Practice Exercises | p.489 | ||
Additional and Advanced Exercises | p.491 |
Chapter 9
First-Order Differential Equations
9.1 | Solutions, Slope Fields, and Euler's Method | Exercises | p.502 |
9.2 | First-Order Linear Equations | Exercises | p.508 |
9.3 | Applications | Exercises | p.515 |
9.4 | Graphical Solutions of Autonomous Equations | Exercises | p.522 |
9.5 | Systems of Equations and Phase Planes | Exercises | p.527 |
Questions to Guide Your Review | p.529 | ||
Practice Exercises | p.529 | ||
Additional and Advanced Exercises | p.530 |
Chapter 10
Infinite Sequences And Series
10.1 | Sequences | Exercises | p.541 |
10.2 | Infinite Series | Exercises | p.551 |
10.3 | The Integral Test | Exercises | p.557 |
10.4 | Comparison Tests | Exercises | p.562 |
10.5 | The Ratio and Root Tests | Exercises | p.567 |
10.6 | Alternating Series, Absolute and Conditional Convergence | Exercises | p.573 |
10.7 | Power Series | Exercises | p.582 |
10.8 | Taylor and Maclaurin Series | Exercises | p.588 |
10.9 | Convergence of Taylor Series | Exercises | p.595 |
10.10 | The Binomial Series and Applications of Taylor Series | Exercises | p.602 |
Questions to Guide Your Review | p.605 | ||
Practice Exercises | p.605 | ||
Additional and Advanced Exercises | p.607 |
Chapter 11
Parametric Equations And Polar Coordinates
11.1 | Parametrizations of Plane Curves | Exercises | p.616 |
11.2 | Calculus with Parametric Curves | Exercises | p.625 |
11.3 | Polar Coordinates | Exercises | p.630 |
11.4 | Graphing in Polar Coordinates | Exercises | p.634 |
11.5 | Areas and Lengths in Polar Coordinates | Exercises | p.638 |
11.6 | Conic Sections | Exercises | p.645 |
11.7 | Conics in Polar Coordinates | Exercises | p.653 |
Questions to Guide Your Review | p.654 | ||
Practice Exercises | p.655 | ||
Additional and Advanced Exercises | p.657 |
Chapter 12
Vectors And The Geometry Of Space
12.1 | Three-Dimensional Coordinate Systems | Exercises | p.663 |
12.2 | Vectors | Exercises | p.672 |
12.3 | The Dot Product | Exercises | p.680 |
12.4 | The Cross Product | Exercises | p.686 |
12.5 | Lines and Plans in Space | Exercises | p.694 |
12.6 | Cylinders and Quadric Surfaces | Exercises | p.700 |
Questions to Guide Your Review | p.701 | ||
Practice Exercises | p.702 | ||
Additional and Advanced Exercises | p.704 |
Chapter 13
Vector-Valued Functions And Motion In Space
13.1 | Curves in Space and Their Tangents | Exercises | p.713 |
13.2 | Integrals of Vector Functions; Projectile Motion | Exercises | p.720 |
13.3 | Arc Length in Space | Exercises | p.727 |
13.4 | Curvature and Normal Vectors of a Curve | Exercises | p.733 |
13.5 | Tangential and Normal Components of Acceleration | Exercises | p.738 |
13.6 | Velocity and Acceleration in Polar Coordinates | Exercises | p.742 |
Questions to Guide Your Review | p.742 | ||
Practice Exercises | p.743 | ||
Additional and Advanced Exercises | p.745 |
Chapter 14
Partial Derivatives
14.1 | Functions of Several Variables | Exercises | p.753 |
14.2 | Limits and Continuity in Higher Dimensions | Exercises | p.761 |
14.3 | Partial Derivatives | Exercises | p.772 |
14.4 | The Chain Rule | Exercises | p.782 |
14.5 | Directional Derivatives and Gradient Vectors | Exercises | p.790 |
14.6 | Tangent Planes and Differentials | Exercises | p.799 |
14.7 | Extreme Values and Saddle Points | Exercises | p.808 |
14.8 | Lagrange Multipliers | Exercises | p.818 |
14.9 | Taylor's Formula for Two Variables | Exercises | p.824 |
14.10 | Partial Derivatives with Constrained Variables | Exercises | p.828 |
Questions to Guide Your Review | p.829 | ||
Practice Exercises | p.829 | ||
Additional and Advanced Exercises | p.833 |
Chapter 15
Multiple Integrals
15.1 | Double and Iterated Integrals over Rectangles | Exercises | p.840 |
15.2 | Double Integrals over General Regions | Exercises | p.847 |
15.3 | Area by Double Integration | Exercises | p.852 |
15.4 | Double Integrals in Polar Form | Exercises | p.857 |
15.5 | Triple Integrals in Rectangular Coordinates | Exercises | p.865 |
15.6 | Moments and Centers of Mass | Exercises | p.873 |
15.7 | Triple Integrals in Cylindrical and Spherical Coordinates | Exercises | p.883 |
15.8 | Substitutions in Multiple Integrals | Exercises | p.894 |
Questions to Guide Your Review | p.896 | ||
Practice Exercises | p.896 | ||
Additional and Advanced Exercises | p.898 |
Chapter 16
Integration In Vector Fields
16.1 | Line Integrals | Exercises | p.906 |
16.2 | Vector Fields and Line Integrals: Work, Circulation and Flux | Exercises | p.917 |
16.3 | Path Independence, Conservative Fields, and Potentials Functions | Exercises | p.929 |
16.4 | Green's Theorem in the Plane | Exercises | p.940 |
16.5 | Surfaces and Area | Exercises | p.951 |
16.6 | Surface Integrals | Exercises | p.960 |
16.7 | Stoke's Theorem | Exercises | p.970 |
16.8 | The Divergence Theorem and a Unified Theory | Exercises | p.981 |
Questions to Guide Your Review | p.983 | ||
Practice Exercises | p.983 | ||
Additional and Advanced Exercises | p.986 |
Chapter 17
Second-Order Differential Equations
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- Step 1 of 5
- Step 2 of 5(a)Consider the limitObserve the graph of the function,take a small interval that contain 1 and the values which are closeto 1 from left then the corresponding values of the functionapproaches to 1.Now take small interval that contain 1 and the values which areclose to 1 from right then the corresponding values of the functionapproaches to 0.In both cases the limit of the function exists, but there is nosingle number that all the values get arbitrarily close to as.Thus the limit of the function does not exist at.Therefore the limitdoes not exist.
- Step 3 of 5(b)Consider the limitObserve the graph of the function,take a small interval that contain 2 and the values which are closeto 2 from left then the corresponding values of the functionapproaches to 1.Now take small interval that contain 2 and the values which areclose to 2 from right then the corresponding values of the functionapproaches to 1.From above cases the limit of the function exists and same, sothe limit of the function exists atandis equal to 1.Therefore the limit.
- Step 4 of 5(c)Consider the limitObserve the graph of the function,take a small interval that contain 3 and the values which are closeto 3 from left then the corresponding values of the functionapproaches to 0.Now take small interval that contain 3 and the values which areclose to 3 from right then the corresponding values of the functionapproaches to 0.From above cases the limit of the function exists and same, sothe limit of the function exists atandis equal to 0, but.Therefore the limit.
- Step 5 of 5(d)Consider the limitObserve the graph of the function,take a small interval that contain 2.5 and the values which areclose to 2.5 from left then the corresponding values of thefunction approaches to 0.5.Now take small interval that contain 2.5 and the values whichare close to 2.5 from right then the corresponding values of thefunction approaches to 0.5.From above cases the limit of the function exists and same, sothe limit of the function exists atandis equal to 0.5.Therefore the limit.