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