

This general calculus book covers a fairly standard course sequence: single variable calculus, infinite series, and multivariable calculus.
• ## 1: Analytic Geometry

Much of the mathematics in this chapter will be review for you. However, the examples will be oriented toward applications and so will take some thought.

• ## 3: Rules for Finding Derivatives

It is tedious to compute a limit every time we need to know the derivative of a function. Fortunately, we can develop a small collection of examples and rules that allow us to compute the derivative of almost any function we are likely to encounter. Many functions involve quantities raised to a constant power, such as polynomials and more complicated combinations . So we start by examining powers of a single variable; this gives us a building block for more complicated examples.
• ## 4: Transcendental Functions

So far we have used only algebraic functions as examples when finding derivatives, that is, functions that can be built up by the usual algebraic operations of addition, subtraction, multiplication, division, and raising to constant powers. Both in theory and practice there are other functions, called transcendental, that are very useful. Most important among these are the trigonometric functions, the inverse trigonometric functions, exponential functions, and logarithms.
• ## 5: Curve Sketching

Whether we are interested in a function as a purely mathematical object or in connection with some application to the real world, it is often useful to know what the graph of the function looks like. We can obtain a good picture of the graph using certain crucial information provided by derivatives of the function and certain limits.

• ## 8: Techniques of Integration

Over the next few sections we examine some techniques that are frequently successful when seeking antiderivatives of functions.

• ## 10: Polar Coordinates and Parametric Equations

Coordinate systems are tools that let us use algebraic methods to understand geometry. While the rectangular (also called Cartesian) coordinates that we have been using are the most common, some problems are easier to analyze in alternate coordinate systems.
• ## 11: Sequences and Series

A power series (in one variable) is an infinite series. Any polynomial can be easily expressed as a power series around any center c, although most of the coefficients will be zero since a power series has infinitely many terms by definition. One can view power series as being like "polynomials of infinite degree," although power series are not polynomials.

• ## 13: Vector Functions

A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could be a scalar or a vector.
• ## 14: Partial Differentiation

In single-variable calculus we were concerned with functions that map the real numbers $$\mathbb{R}$$ to $$\mathbb{R}$$. In the last chapter we considered functions taking a real number to a vector. Now we turn to functions of several variables, meaning several input variables, functions.

• ## 17: Differential Equations

Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time.