**AP Calculus AB Study Guide**

**By: "Lew" Sterling Jr.**

All of the baggage

has to be checked

before it is loaded.

What the AP Calculus AB will be about:

Elementary Functions

Limits

Differential Calculus

Integral Calculus

Elementary

Functions

Properties of Functions

Combinations of

Functions

Inverse

Functions

Even and Odd Functions

The function y = ƒ(x) is even if ƒ(-x) = ƒ(x).

Even functions are symmetric about the y-axis (e.g. y = x^2)

The function y = ƒ(x) is odd if ƒ(-x) = -ƒ(x).

Odd functions are symmetric about the origin (e.g. y = x^3)

Periodic

Functions

Zeros of

a Function

These occur where the function ƒ(x) crosses the x-axis. These points are also called the roots of a function.

Limits

A limit is the value that a function or sequence "approaches" as the input or index approaches some value. Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals.

**One last thing you need to know before Differential Calculus...**

Properties of Limits

and One-Sided Limits

Limits at Infinity

Intermediate Value Theorem

Differential Calculus

Differentiation Rules

Mobile X-rays are large truck carriers that has a complex X-ray system.

Example of

Differentiating

CT scanner mechanism

In the United States, most major airports have a computer tomography (CT) scanner.

The CT scanner rotates around the bag, bombarding it with X-rays and recording the resulting data, creating a tomographic.

Not all baggages go through a CT scan

Usuallylocated near the ticket counter.

A function ƒ is defined as a set of all ordered pairs (x, y), such that for each element x, there corresponds exactly one element y.

The domain of ƒ is the set x.

The range of ƒ is the set y.

If ƒ(x) = 3x + 1 and g(x) = x^2 - 1

a) the sum ƒ(x) + g(x) =

(3x + 1) + (x^2 - 1) = x^2 + 3x

b) the difference ƒ(x) - g(x) =

(3x + 1) - (x^2 - 1) = -x^2 + 3x + 2

c) the product ƒ(x)g(x) =

(3x + 1)(x^2 - 1) = 3x^3 + x^2 - 3x - 1

d) the quotient ƒ(x)/g(x) =

(3x + 1)/(x^2 - 1)

e) the composite (ƒ * g)(x) = ƒ(g(x)) =

3(x^2 - 1) + 1 = 3x^2 - 2

Functions ƒ and g are inverses of each other if

ƒ(g(x)) = x

for each x in the domain of g

g(ƒ(x)) = x

for each x in the domain of ƒ

The inverse of the function ƒ is denoted ƒ-1.

To find ƒ-1, switch x and y in the original equation and solve the equation for y in terms of x.

You should be familiar with the definitions and graphs of these trigonometric functions:

sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc)