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### Calculus Applications and Basics

Engineering Physics

Engineering Mathematics

Calculus Applications and Basics

Gilbert Strang

Massachusetts Institute of Technology

671 pages

Open: Calculus Applications and Basics

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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

Powers Instead of Exponentials7

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

Directional Derivatives and Gradients3.5

The Chain Rule

Maxima, Minima, and Saddle Points

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