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Calculus

Introduction

Part 1: Differential Calculus
Definition of derivative:
\[ f^{′}\left(x\right)=\underset{b→0}{\lim}\frac{f\left(x+b\right)−f\left(x\right)}{b}=\underset{b→x}{\lim}\frac{f\left(x\right)−f\left(b\right)}{x−b} \]\[ f\left(x\right)=x^2⇒\underset{b→x}{\lim}\frac{x^2−b^2}{x−b}=\underset{b→x}{\lim}\frac{\left(x+b\right)\left(x−b\right)}{x−b}=\underset{b→x}{\lim}\left(x+b\right)=2x \]\[ f(x)=\frac{1}{x}\Rightarrow \lim_{b→x}\frac{\frac{1}{x}-\frac{1}{b}}{x-b} = \lim_{b→x}\frac{b-x}{xb(x-b)} = \lim_{b→x}\frac{-(x-b)}{xb(x-b)} = \lim_{b→x}\frac{-1}{xb} = \frac{-1}{x^2} = -x^{-2} \]

Part 2: Integral Calculus
Fundamental theorem of calculus:
Let f be a continuous real-valued function defined on a closed interval [a, b]. Let F be the function defined, for all x in [a, b], by
\[ F\left(x\right)=∫_a^xf\left(t\right)dt \]Then F is continuous on [a, b] , differentiable on the open interval (a, b) , and
\[ F^{′}\left(x\right)=f\left(x\right) \]for all x in (a, b) .
Let f and F be real-valued functions defined on a closed interval [a, b] such that the derivative of F is f. That is, f and F are functions such that for all x in [a, b],
\[ F^′\left(x\right)=f\left(x\right) \]If f is Reimann integrable on [a, b] , then
\[ ∫_a^bf\left(x\right)dx=F^′\left(b\right)−F^′\left(a\right) \]

Part 3
Summary

Part 4: Multivariable Calculus
Summary

Part 1: Differential Calculus

Chapter 1: Introduction to Calculus
Summary

Chapter 2: Derivatives
Summary

Chapter 3: Applications of the Derivative
Summary

Chapter 4: The Chain Rule
Summary

Part 2: Integral Calculus

Chapter 5: Integrals
Summary

Chapter 6: Exponentials and Logarithms
Summary

Chapter 7: Techniques of Integration
Summary

Chapter 8: Applications of the Integral
Summary

Part 3

Chapter 9: Polar Coordinates and Complex Numbers
Summary

Chapter 10: Infinite Series
Summary

Chapter 11: Vectors and Matrices
Summary

Chapter 12: Motion along a Curve
Summary

Part 4: Multivariable Calculus

Chapter 13: Partial Derivatives
Summary

Chapter 14: Multiple Integrals
Summary

Chapter 15: Vector Calculus
Summary

Chapter 16: Mathematics after Calculus
Summary

Chapter 1: Introduction to Calculus

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?

1.1 Velocity and Distance
Summary

1.2 Calculus Without Limits
Summary

1.3 The Velocity at an Instant
Summary

1.4 Circular Motion
Summary

1.5 A Review of Trigonometry
Summary

 1.6 A Thousand Points of Light
Summary

1.7 Computing in Calculus
Summary

Chapter 2: Derivatives

Content

2.1 The Derivative of a Function
Summary

2.2 Powers and Polynomials
Summary

2.3 The Slope and the Tangent Line
Summary

2.4 Derivative of the Sine and Cosine
Summary

2.5 The Product and Quotient and Power Rules
Summary

2.6 Limits
Summary

2.7 Continuous Functions
Summary

Chapter 3: Applications of the Derivative

Content

3.1 Linear Approximation
Summary

3.2 Maximum and Minimum Problems
Summary

3.3 Second Derivatives: Minimum vs. Maximum
Summary

3.4 Graphs
Summary

3.5 Ellipses, Parabolas, and Hyperbolas
Summary

3.6 Iterations xn+1 = F(xn)
Summary

3.7 Newton's Method and Chaos
Summary

3.8 The Mean Value Theorem and l'Hopital's Rule
Summary

Chapter 4: The Chain Rule

Content

4.1 Derivatives by the Chain Rule
Summary

4.2 Implicit Differentiation and Related Rates
Summary

4.3 Inverse Functions and Their Derivatives
Summary

4.4 Inverses of Trigonometric Functions
Summary

Chapter 5: Integrals

Content

5.1 The Idea of the Integral
Summary

5.2 Antiderivatives
Summary

5.3 Summation vs. Integration
Summary

5.4 Indefinite Integrals and Substitutions
Summary

5.5 The Definite Integral
Summary

5.6 Properties of the Integral and the Average Value
Summary

5.7 The Fundamental Theorem and Its Consequences
Summary

5.8 Numerical Integration
Summary

Chapter 6: Exponentials and Logarithms

Content

6.1 An Overview
Summary

6.2 The Exponential ex
Summary

6.3 Growth and Decay in Science and Economics
Summary

6.4 Logarithms
Summary

6.5 Separable Equations Including the Logistic Equation
Summary

6.6 Powers Instead of Exponentials
Summary

6.7 Hyperbolic Functions
Summary

Chapter 7: Techniques of Integration

Content

7.1 Integration by Parts
Summary

7.2 Trigonometric Integrals
Summary

7.3 Trigonometric Substitutions
Summary

7.4 Partial Fractions
Summary

7.5 Improper Integrals
Summary

Chapter 8: Applications of the Integral

Content

8.1 Areas and Volumes by Slices
Summary

8.2 Length of a Plane Curve
Summary

8.3 Area of a Surface of Revolution
Summary

8.4 Probability and Calculus
Summary

8.5 Masses and Moments
Summary

8.6 Force, Work, and Energy
Summary

Chapter 9: Polar Coordinates and Complex Numbers

Content

9.1 Polar Coordinates
Summary

9.2 Polar Equations and Graphs
Summary

9.3 Slope, Length, and Area for Polar Curves
Summary

9.4 Complex Numbers
Summary

Chapter 10: Infinite Series

Content

10.1 The Geometric Series
Summary

10.2 Convergence Tests: Positive Series
Summary

10.3 Convergence Tests: All Series
Summary

10.4 The Taylor Series for ex, sin x, and cos x
Summary

10.5 Power Series
Summary

Chapter 11: Vectors and Matrices

Content

11.1 Vectors and Dot Products
Summary

11.2 Planes and Projections
Summary

11.3 Cross Products and Determinants
Summary

11.4 Matrices and Linear Equations
Summary

11.5 Linear Algebra in Three Dimensions
Summary

Chapter 12: Motion along a Curve

Content

12.1 The Position Vector
Summary

12.2 Plane Motion: Projectiles and Cycloids
Summary

12.3 Tangent Vector and Normal Vector
Summary

12.4 Polar Coordinates and Planetary Motion
Summary

Chapter 13: Partial Derivatives

Content

13.1 Surfaces and Level Curves
Summary

13.2 Partial Derivatives
Summary

13.3 Tangent Planes and Linear Approximations
Summary

13.4 Directional Derivatives and Gradients
Summary

13.5 The Chain Rule
Summary

13.6 Maxima, Minima, and Saddle Points
Summary

13.7 Constraints and Lagrange Multipliers
Summary

Chapter 14: Multiple Integrals

Content

14.1 Double Integrals
Summary

14.2 Changing to Better Coordinates
Summary

14.3 Triple Integrals
Summary

14.4 Cylindrical and Spherical Coordinates
Summary

Chapter 15: Vector Calculus

Content

15.1 Vector Fields
Summary

15.2 Line Integrals
Summary

15.3 Green's Theorem
Summary

15.4 Surface Integrals
Summary

15.5 The Divergence Theorem
Summary

15.6 Stokes' Theorem and the Curl of F
Summary

Chapter 16: Mathematics after Calculus

Content

16.1 Linear Alegbra
Summary

16.2 Differential Equations
Summary

16.3 Discrete Mathematics
Summary

1.1 Velocity and Distance

Introduction

Velocity and Distance
Summary

Constant Velocity
Summary

1.2 Calculus Without Limits

Content

1.3 The Velocity at an Instant

Content

1.4 Circular Motion

Content

1.5 A Review of Trigonometry

Content

 1.6 A Thousand Points of Light

Content

1.7 Computing in Calculus

Content

2.1 The Derivative of a Function

Content

2.2 Powers and Polynomials

Content

2.3 The Slope and the Tangent Line

Content

2.4 Derivative of the Sine and Cosine

Content

2.5 The Product and Quotient and Power Rules

Content

2.6 Limits

Content

2.7 Continuous Functions

Content

3.1 Linear Approximation

Content

3.2 Maximum and Minimum Problems

Content

3.3 Second Derivatives: Minimum vs. Maximum

Content

3.4 Graphs

Content

3.5 Ellipses, Parabolas, and Hyperbolas

Content

3.6 Iterations xn+1 = F(xn)

Content

3.7 Newton's Method and Chaos

Content

3.8 The Mean Value Theorem and l'Hopital's Rule

Content

4.1 Derivatives by the Chain Rule

Content

4.2 Implicit Differentiation and Related Rates

Content

4.3 Inverse Functions and Their Derivatives

Content

4.4 Inverses of Trigonometric Functions

Content

5.1 The Idea of the Integral

Content

5.2 Antiderivatives

Content

5.3 Summation vs. Integration

Content

5.4 Indefinite Integrals and Substitutions

Content

5.5 The Definite Integral

Content

5.6 Properties of the Integral and the Average Value

Content

5.7 The Fundamental Theorem and Its Consequences

Content

5.8 Numerical Integration

Content

6.1 An Overview

Content

6.2 The Exponential ex

Content

6.3 Growth and Decay in Science and Economics

Content

6.4 Logarithms

Content

6.5 Separable Equations Including the Logistic Equation

Content

6.6 Powers Instead of Exponentials

Content

6.7 Hyperbolic Functions

Content

7.1 Integration by Parts

Content

7.2 Trigonometric Integrals

Content

7.3 Trigonometric Substitutions

Content

7.4 Partial Fractions

Content

7.5 Improper Integrals

Content

8.1 Areas and Volumes by Slices

Content

8.2 Length of a Plane Curve

Content

8.3 Area of a Surface of Revolution

Content

8.4 Probability and Calculus

Content

8.5 Masses and Moments

Content

8.6 Force, Work, and Energy

Content

9.1 Polar Coordinates

Content

9.2 Polar Equations and Graphs

Content

9.3 Slope, Length, and Area for Polar Curves

Content

9.4 Complex Numbers

Content

10.1 The Geometric Series

Content

10.2 Convergence Tests: Positive Series

Content

10.3 Convergence Tests: All Series

Content

10.4 The Taylor Series for ex, sin x, and cos x

Content

10.5 Power Series

Content

11.1 Vectors and Dot Products

Content

11.2 Planes and Projections

Content

11.3 Cross Products and Determinants

Content

11.4 Matrices and Linear Equations

Content

11.5 Linear Algebra in Three Dimensions

Content

12.1 The Position Vector

Content

12.2 Plane Motion: Projectiles and Cycloids

Content

12.3 Tangent Vector and Normal Vector

Content

12.4 Polar Coordinates and Planetary Motion

Content

13.1 Surfaces and Level Curves

Content

13.2 Partial Derivatives

Content

13.3 Tangent Planes and Linear Approximations

Content

13.4 Directional Derivatives and Gradients

Content

13.5 The Chain Rule

Content

13.6 Maxima, Minima, and Saddle Points

Content

13.7 Constraints and Lagrange Multipliers

Content

14.1 Double Integrals

Content

14.2 Changing to Better Coordinates

Content

14.3 Triple Integrals

Content

14.4 Cylindrical and Spherical Coordinates

Content

15.1 Vector Fields

Content

15.2 Line Integrals

Content

15.3 Green's Theorem

Content

15.4 Surface Integrals

Content

15.5 The Divergence Theorem

Content

15.6 Stokes' Theorem and the Curl of F

Content

16.1 Linear Alegbra

Content

16.2 Differential Equations

Content

16.3 Discrete Mathematics

Content

Constant Velocity

Suppose the velocity is fixed at v =60 (miles per hour). Then f increases at this constant rate. After two hours the distance is f =120 (miles). After four hours f =240 and after t hours f =60t. We say that f increases linearly with time-its graph is a straight line.

Fig. 1.2 Constant velocity v =60 and linearly increasing distance f=60t.

Notice that this example starts the car at full velocity. No time is spent picking up speed. (The velocity is a "step function.") Notice also that the distance starts at zero; the car is new. Those decisions make the graphs of v and f as neat as possible. One is the horizontal line v =60. The other is the sloping line f =60t. This v, f, t relation needs algebra but not calculus:

if v is constant and f starts at zero then f =vt.

The opposite is also true. When f increases linearly, v is constant. The division by time gives the slope. The distance is fl =120 miles when the time is t1 =2 hours. Later f' =240 at t, =4. At both points, the ratio f/t is 60 miles/hour. Geometrically, the velocity is the slope of the distance graph:

slope=change in distancechange in time=vtt=v

Velocity and Distance

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. The two instruments sit together on the dashboard:

Fig. 1.1 Velocity v and total distance f (at one instant of time).

Notice that the units of measurement are different for v and f.The distance f is measured in kilometers or miles (it is easier to say miles). The velocity v is measured in km/hr or miles per hour. A unit of time enters the velocity but not the distance. Every formula to compute v from f will have f divided by time.

Can you find v if you know f, and vice versa, and how? If we know the velocity over the whole history of the car, we should be able to compute the total distance traveled. In other words, if the speedometer record is complete but the odometer is missing, its information could be recovered. One way to do it (without calculus) is to put in a new odometer and drive the car all over again at the right speeds. That seems like a hard way; calculus may be easier. But the point is that the information is there. If we know everything about v, there must be a method to find f.

What happens in the opposite direction, when f is known? If you have a complete record of distance, could you recover the complete velocity? In principle you could drive the car, repeat the history, and read off the speed. Again there must be a better way.

The whole subject of calculus is built on the relation between u and f. The question we are raising here is not some kind of joke, after which the book will get serious and the mathematics will get started. On the contrary, I am serious now-and the mathematics has already started. We need to know how to find the velocity from a record of the distance. (That is called differentiation, and it is the central idea of differential calculus.) We also want to compute the distance from a history of the velocity. (That is integration, and it is the goal of integral calculus.)

Differentiation goes from f to v; integration goes from v to f. We look first at examples in which these pairs can be computed and understood.