Functions and Derivatives: The Graphical View Functions, Calculus Style Graphs | |
A Field Guide to Elementary Functions | |
Amount Functions and Rate Functions: The Idea of the Derivative | |
Estimating Derivatives: A Closer Look | |
The Geometry of Derivatives | |
The Geometry of Higher-Order Derivatives | |
Interlude: Zooming in on Differences | |
Functions and Derivatives: The Symbolic View Defining the Derivative | |
Derivatives of Power Functions and Polynomials Limits | |
Using Derivative and Antiderivative Formulas | |
Differential Equations; | |
Modeling Motion Derivatives of Exponential and Logarithm Functions; | |
Modeling Growth Derivatives of Trigonometric Functions: Modeling Oscillation | |
Interlude: Tangent Lines in History | |
Interlude: Limit--The Formal Definition | |
New Derivatives from Old Algebraic Combinations: The Product and Quotient Rules | |
Composition and the Chain Rule Implicit | |
Functions and Implicit Differentiation | |
Inverse Functions and Their Derivatives; | |
Inverse Trigonometric Functions Miscellaneous | |
Derivatives and Antiderivatives Interlude: Vibrations--Simple and Damped | |
Interlude: Hyperbolic Functions | |
Using the Derivative Slope Fields; | |
More Differential Equation Models More on Limits: Limits Involving Infinity and l'Hocirc;pital's Rule Optimization | |
Parametric Equations, Parametric Curves Related Rates | |
Newton's Method: Finding Roots Building Polynomials to Order; | |
Taylor Polynomials Why Continuity Matters | |
Why Differentiability Matters: The Mean Value Theorem | |
Interlude: Growth with Interest | |
Interlude: Logistic Growth | |
Interlude: Digging Deeper for Roots | |
The Integral Areas and Integrals | |
The Area Function | |
The Fundamental Theorem of Calculus Finding Antiderivatives; | |
The Method of Substitution Integral Aids: Tables and Computers | |
Approximating Sums: The Integral as a Limit Working with Sums | |
Interlude: Mean Value Theorems and Integrals | |
Numerical Integration Approximating Integrals | |
Numerically Error Bounds for approximating Sums | |
Euler's Method: Solving DEs Numerically | |
Interlude: Simpson's Rule | |
Interlude: Gaussian Quadrature: Approximating Integrals Efficiently | |
Using the Integral Measurement and the Definite Integral; | |
Arc Length Finding Volumes by Integration Work | |
Separating Variables: Solving DEs Symbolically Present Value | |
Interlude: Mass and Center of Mass | |
Symbolic Antidifferentiation Techniques Integration by Parts | |
Partial Fractions Trigonometric Antiderivatives | |
Miscellaneous Antiderivatives | |
Interlude: Beyond Elementary Functions | |
Interlude: First-Order Linear Differential Equations | |
Function Approximation Taylor Polynomials | |
Taylor's Theorem: Accuracy Guarantees for Taylor Polynomials | |
Fourier Polynomials: Approximating Periodic Functions | |
Interlude: Splines--Connecting the Dots | |
Improper Integrals Improper Integrals: Ideas and Definitions Detecting Convergence, Estimating Limits | |
Improper Integrals and Probability | |
Infinite Series Sequences and Their Limits Infinite Series, Convergence, and Divergence Testing for Convergence; | |
Estimating Limits Absolute Convergence; | |
Alternating Series Power Series Power Series as Functions Taylor Series Interlude: Fourier Series V. Vectors and Polar Coordinates | |
Vectors and Vector-Valued Functions | |
Polar Coordinates and Polar Curves Calculus in Polar Coordinates M. | |
Multivariable Calculus: A First Look | |
Three-Dimensional Space Functions of Several Variables | |
Partial Derivatives Optimization and Partial Derivatives: A First Look | |
Multiple Integrals and Approximating Sums | |
Calculating Multiple Integrals by Iteration | |
Double Integrals in Polor Coordinates | |
Curves and Vectors | |
Three-dimensional Space Curves and Parametric Equations Vectors | |
Vector-valued Functions, Derivatives, and Integrals | |
Derivatives, Antiderivatives, and Motion | |
The Dot Product Lines and Planes in Three Dimensions | |
The Cross Product | |
Derivatives Functions of Several Variables | |
Partial Derivatives | |
Partial Derivatives and Linear Approximation | |
The Gradient and Directional Derivatives | |
Local Linearity: Theory of the Derivative | |
Higher Order Derivatives and Quadratic Approximation | |
Maxima, Minima, and Quadratic Approximation | |
The Chain Rule | |
Integrals Multiple Integrals and Approximating Sums | |
Calculating Integrals by Iteration | |
Double Integrals in Polar Coordinates | |
More on Triple Integrals; | |
Cylindrical and Spherical Coordinates | |
Multiple Integrals Overviewed; | |
Change of Variables | |
Other Topics Linear, Circular, and Combined Motion Using the Dot Product: More on Curves Curvature Lagrange Multipliers and Constrained Optimization | |
Vector Calculus Line Integrals | |
More on Line Integrals; | |
A Fundamental Theorem Relating Line and Area Integrals: Green's Theorem Surfaces and Their Parametrizations | |
Surface Integrals | |
Derivatives and Integrals of Vector Fields | |
Back to Fundamentals: Stokes' Theorem and the Divergence Theorem | |
Appendices | |
Machine Graphics | |
Real Numbers and the Coordinate Plane | |
Lines and Linear Functions | |
Polynomials and Rational Functions | |
Algebra of Exponentials and Logarithms | |
Trigonometric Functions | |
Real-World Calculus: From Words to Mathematics | |
Selected Proofs | |
A Graphical Glossary of Functions | |
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