Review Notes of AP Calculus BC. Based on Barron’s AP Calculus 14th Edition.

## Chapter 1 Functions

### Function, Domain, Range

A function f is a correspondence that associates with each element a of a set called the domain one and only one element b of a set called the range.

We write which means that b is the value of f at a. A vertical line cuts the graph of a function in most one point.

### Composition

A composition of f with g, written as f(g(x)) and read as “f of g of x”. We also write .

The domain of is the set of all x in the domain of g for which g(x) is the domain of f.

### Symmetry

A function is odd if, for all x in the domain of f, .

A function is even if, for all x in the domain of f, .

### Inverse

If a function f yields a single output for each input and also yields a single input for every output, then f is said to be one-to-one. Any horizontal line cuts the graph of f in at most one point.

If f is one-to-one with domain X and range Y, then there is a function , with domain Y and range X.

For example, if and only if .

The graph of a function and its inverse are symmetric with respect to the line y=x.

### Polynomial Functions

A polynomial function is of the form , where n is a positive integer or zero.

A rational function is of the form , where P and Q are polynomials and .

### Trigonometric Functions

#### Basic Indentities

#### Periodicity and Amplitude

The trigonometric functions are periodic. A function is periodic if there is a positive number p such that f(x+p)=f(x) or each x in the domain of f.

The function sinx, cosx, cscx, and secx have period ; tanx and cotx have period .

The function has amplitude A and period .

#### Invserses of Trigonometric Functions

We obtain iverse of the trigonometric functions by limiting the domains of the latter so each trigonometric function is one-to-one over its restriced domain.

### Exponential and Logarithmic Functions

#### Exponential Functions

There are the laws of exponents or all rational m and n, provided that .

#### Logarithmic Functions

Since is one-to-one, it has an inverse, which is , called the logarithmic function with base a.

It has the following properties.

The logarithmic base e has special symbol: , which is called natrual logarithms.

### Parametrically Defined Functions

If the x- and y- coordinates of a point on a graph are given as functions f and g of a third variable, say t, then x=f(t), y=g(t) are called parametric equations and t is called the parameter.

For example, when , since , we have .

#### Polar Functions

Polar coordinates of the form identify the location of a point by specifying , an angle of rotation from the positive x-axis, and r, a distance from the origin.

A polar function defines a curve with an equation of the form .

A polar function may also be expressed parametrically: .