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### Calculus Applications and Basics

Calculus Applications and Basics
Gilbert Strang
Massachusetts Institute of Technology
671 pages

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Introduction

The right way to begin a calculus book is with calculus. This chapter will jump directly into the two problems that the subject was invented to solve. You will see what the questions are, and you will see an important part of the answer. There are plenty of good things left for the other chapters, so why not get started? The book begins with an example that is familiar to everybody who drives a car. It is calculus in action-the driver sees it happening. The example is the relation between the speedometer and the odometer. One measures the speed (or velocity); the other measures the distance traveled. We will write v for the velocity, and f for how far the car has gone.

TOC

Chapter i Introduction to Calculus
Velocity and Distance
Calculus Without Limits
The Velocity at an Instant
Circular Motion
A Review of Trigonometry
A Thousand Points of Light
Computing in Calculus

Chapter 2 Derivatives
The Derivative of a Function 2
Powers and Polynomials
The Slope and the Tangent Line
Derivative of the Sine and Cosine
The Product and Quotient and Power Rules6
Limits
Continuous Functions

Chapter 3 Applications of the Derivative1
Linear Approximation
Maximum and Minimum Problems
Second Derivatives: Minimum vs. Maximum
Graphs
Ellipses, Parabolas, and Hyperbolas6
Iterations x n+1 = F(x n )
Newton's Method and Chaos
The Mean Value Theorem and FHopital's Rule

Chapter 4 The Chain Rule
Derivatives by the Chain Rule
Implicit Differentiation and Related Rates
Inverse Functions and Their Derivatives
Inverses of Trigonometric Functions

Chapter 5 Integrals
The Idea of the Integral
Antiderivatives
Summation vs. Integration
Indefinite Integrals and Substitutions
The Definite Integral
Properties of the Integral and the Average Value
The Fundamental Theorem and Its Consequences 8
Numerical Integration

Chapter 6 Exponentials and Logarithms
An Overview
The Exponential e x
Growth and Decay in Science and Economics
Logarithms
Separable Equations Including the Logistic Equation .6
Hyperbolic Functions

Chapter 7 Techniques of Integration
Integration by Parts
Trigonometric Integrals
Trigonometric Substitutions
Partial Fractions
Improper Integrals

Chapter 8 Applications of the Integral
Areas and Volumes by Slices
Length of a Plane Curve
Area of a Surface of Revolution4
Probability and Calculus
Masses and Moments
Force, Work, and Energy

Chapter 9 Polar Coordinates and Complex Numbers .1
Polar Coordinates
Polar Equations and Graphs
Slope, Length, and Area for Polar Curves
Complex Numbers

Chapter 10 to Infinite Series
The Geometric Series
Convergence Tests: Positive Series 0.3
Convergence Tests: All Series
The Taylor Series for e x , sin x, and cos x.5
Power Series

Chapter 11 Vectors and Matrices
Vectors and Dot Products
Planes and Projections
Cross Products and Determinants
Matrices and Linear Equations
Linear Algebra in Three Dimensions

Chapter 12 Motion along a Curve 1
The Position Vector
Plane Motion: Projectiles and Cycloids
Tangent Vector and Normal Vector 4
Polar Coordinates and Planetary Motion

Chapter 13 Partial Derivatives
Surfaces and Level Curves
Partial Derivatives
Tangent Planes and Linear Approximations3.4
The Chain Rule
Constraints and Lagrange Multipliers

Chapter 14 Multiple Integrals
Double Integrals
Changing to Better Coordinates
Triple Integrals
Cylindrical and Spherical Coordinates

Chapter 15 Vector Calculus
Vector Fields
Line Integrals
Green's Theorem
Surface Integrals
The Divergence Theorem
Stokes' Theorem and the Curl of F

Chapter 16 Mathematics after Calculus 1
Linear Algebra
Differential Equations
Discrete Mathematics