Study Notes: AP Calculus BC

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 \small f(a)=b 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 \small (f \circ g)(x).
The domain of \small (f \circ g)(x) 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, \small f(-x)=-f(x).
A function is even if, for all x in the domain of f, \small f(x)=-f(x).

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 \small f^{-1}, with domain Y and range X.
For example, \small f^{-1}(y_{0})=x_{0} if and only if \small f(x_{0})=y_{0}.
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 \small f(x)=a_{0}x^{n}+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-1}x+a_{n}, where n is a positive integer or zero.
A rational function is of the form \small f(x)=\frac{P(x)}{Q(x)}, where P and Q are polynomials and \small Q(x) \neq 0.

Trigonometric Functions

Basic Indentities

\small cscx=\frac{1}{sinx},\,secx=\frac{1}{cosx},\,cotx=\frac{1}{tanx}

\small sin^{2}\theta+cos^{2}\theta=1 \\ \small 1+tan^{2}\theta=sec^{2}\theta \\ \small 1+cot^{2}\theta=csc^{2}\theta

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 \small 2\pi; tanx and cotx have period \small \pi.
The function \small f(x)=Asinbx has amplitude A and period \small \frac{2\pi}{b}.

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.
\small arcsinx=sin^{-1}x\\ \small arccosx=cos^{-1}x \\ \small arctanx=tan^{-1}x

Exponential and Logarithmic Functions

Exponential Functions

There are the laws of exponents or all rational m and n, provided that \small a>0, a \neq n.
\small a^{0}=1,\,a^{1}=a,\,a^{m} \dot a^{n}=a^{m+n},\,a^{m}\div a^{n}= a^{m-n}\\ \small a^{m^{n}}=a^{mn},\,a^{-m}=\frac{1}{a^{m}}

Logarithmic Functions

Since \small f(x)=a^{x} is one-to-one, it has an inverse, which is \small log_{a}x, called the logarithmic function with base a.
It has the following properties.

log_{a}1=0,\,log_{a}a=1 \\ \small log_{a}mn=log_{a}m+log_{a}n \\ \small log_{a}\frac{m}{n}=log_{a}m-log_{a}n \\ \small log_{a}x^{m}=mlog_{a}x
The logarithmic base e has special symbol: \small log_{e}x=lnx, 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 \small x= 4 sint, y=5cost, since \small sin^{2}t+cos^{2}t=1, we have \small \frac{x^{2}}{16}+\frac{y^{2}}{25}=1.

Polar Functions

Polar coordinates of the form \small (r,\theta) identify the location of a point by specifying \theta, 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 \small r=f(\theta).
A polar function may also be expressed parametrically: x=rcos\theta,\,y=rsin\theta.

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Siujoeng Lau

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