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

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Part 4: Multivariable Calculus

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Chapter 1: Introduction to Calculus

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Chapter 2: Derivatives

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Chapter 3: Applications of the Derivative

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Chapter 4: The Chain Rule

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Chapter 5: Integrals

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Chapter 6: Exponentials and Logarithms

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Chapter 7: Techniques of Integration

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Chapter 8: Applications of the Integral

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Chapter 9: Polar Coordinates and Complex Numbers

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Chapter 10: Infinite Series

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Chapter 11: Vectors and Matrices

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Chapter 12: Motion along a Curve

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Chapter 13: Partial Derivatives

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Chapter 14: Multiple Integrals

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Chapter 15: Vector Calculus

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Chapter 16: Mathematics after Calculus

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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

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1.2 Calculus Without Limits

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1.3 The Velocity at an Instant

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1.4 Circular Motion

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1.5 A Review of Trigonometry

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1.6 A Thousand Points of Light

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1.7 Computing in Calculus

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2.1 The Derivative of a Function

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2.2 Powers and Polynomials

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2.3 The Slope and the Tangent Line

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2.4 Derivative of the Sine and Cosine

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2.5 The Product and Quotient and Power Rules

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2.6 Limits

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2.7 Continuous Functions

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3.1 Linear Approximation

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3.2 Maximum and Minimum Problems

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3.3 Second Derivatives: Minimum vs. Maximum

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3.4 Graphs

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3.5 Ellipses, Parabolas, and Hyperbolas

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3.6 Iterations x_{n+1} = F(x_{n})

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3.7 Newton's Method and Chaos

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

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4.1 Derivatives by the Chain Rule

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4.2 Implicit Differentiation and Related Rates

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4.3 Inverse Functions and Their Derivatives

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4.4 Inverses of Trigonometric Functions

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5.1 The Idea of the Integral

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5.2 Antiderivatives

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5.3 Summation vs. Integration

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5.4 Indefinite Integrals and Substitutions

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5.5 The Definite Integral

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5.6 Properties of the Integral and the Average Value

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5.7 The Fundamental Theorem and Its Consequences

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5.8 Numerical Integration

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6.1 An Overview

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6.2 The Exponential e^{x}

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6.3 Growth and Decay in Science and Economics

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6.4 Logarithms

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6.5 Separable Equations Including the Logistic Equation

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6.6 Powers Instead of Exponentials

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6.7 Hyperbolic Functions

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7.1 Integration by Parts

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7.2 Trigonometric Integrals

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7.3 Trigonometric Substitutions

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7.4 Partial Fractions

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7.5 Improper Integrals

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8.1 Areas and Volumes by Slices

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8.2 Length of a Plane Curve

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8.3 Area of a Surface of Revolution

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8.4 Probability and Calculus

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8.5 Masses and Moments

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8.6 Force, Work, and Energy

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9.1 Polar Coordinates

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9.2 Polar Equations and Graphs

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9.3 Slope, Length, and Area for Polar Curves

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9.4 Complex Numbers

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10.1 The Geometric Series

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10.2 Convergence Tests: Positive Series

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10.3 Convergence Tests: All Series

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10.4 The Taylor Series for e^{x}, sin x, and cos x

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10.5 Power Series

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11.1 Vectors and Dot Products

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11.2 Planes and Projections

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11.3 Cross Products and Determinants

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11.4 Matrices and Linear Equations

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11.5 Linear Algebra in Three Dimensions

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12.1 The Position Vector

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12.2 Plane Motion: Projectiles and Cycloids

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12.3 Tangent Vector and Normal Vector

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12.4 Polar Coordinates and Planetary Motion

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13.1 Surfaces and Level Curves

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13.2 Partial Derivatives

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13.3 Tangent Planes and Linear Approximations

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13.4 Directional Derivatives and Gradients

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13.5 The Chain Rule

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13.6 Maxima, Minima, and Saddle Points

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13.7 Constraints and Lagrange Multipliers

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14.1 Double Integrals

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14.2 Changing to Better Coordinates

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14.3 Triple Integrals

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14.4 Cylindrical and Spherical Coordinates

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15.1 Vector Fields

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15.2 Line Integrals

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15.3 Green's Theorem

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15.4 Surface Integrals

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15.5 The Divergence Theorem

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15.6 Stokes' Theorem and the Curl of F

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16.1 Linear Alegbra

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16.2 Differential Equations

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16.3 Discrete Mathematics

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Introduction

Velocity and Distance

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Constant Velocity

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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

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.