AP CALCULUS

Pre-requisites

.1

Real Numbers, Estimation, & Logic

IN CALCULUS, THE PRINCIPLE NUMBERS ARE REAL NUMBERS.

Be able to calculate with rational numbers (expressed as either repeating or terminating decimals) or irrational numbers (decimals that do NOT terminate or repeat)

Be able to ESTIMATE answers before pushing a button on a calculator! Use good mental mathematics.

Much done in math must be proven, and different methods of proof can be employed.

0.2

Inequalities and Absolute Value

SOLVING INEQUALITIESSolve by comparing the inequality to

zero, factor if possible, and solve.

),2/1()4,(

42/1),04()012(

42/1),04()012(

0)4)(12(

0472

4722

2

xxxANDxOR

xxxANDx

xx

xx

xx

SOLVING ABSOLUTE VALUE

Consider absolute value as distance, if the distance is greater than a constant, you must get further away in both directions. If the distance is less than a constant, the solution values must be within a certain range of values.

0.3

The Rectangular Coordinate System

CARTESIAN COORDINATE SYSTEM

Graphs are done in the x-y system. You can find distance between any 2 points using Pythagorean theorem and midpoint of 2 any 2 points simply as the average.

In both instances, a graph is often helpful in understanding the situation, prior to calculating.

LINEAR EQUATIONS

General form: Ax + By + C = 0 Slope-intercept form: y = mx + b Point-slope form y – y1 = m(x – x1)

0.4

Graphs of Equations

QUADRATIC FUNCTIONS

Graphs to a parabola Vertex at (h,k) Graph has reflection symmetry

hkyax

DCxByAy

khxay

DCyBxAx

2

2

2

2

)(

0

)(

0

CUBIC FUNCTIONS Reflects through the origin

dcxbxaxy 23

0.5

Functions & Their Graphs

FUNCTIONS

Domain (x-values): real numbers which can be placed for x

Range (y-values): real numbers which are created from the values for x

Even functions: Reflect through the y-axis, f(x) = f(-x)

Odd functions: Reflect through the origin, f(x) = -f(-x)

0.6

Operations on Functions

FUNCTIONS CAN BE ADDED, SUBTRACTED, MULTIPLIED OR DIVIDED

Only consideration? Operations cannot result in a zero denominator

Composition of functions: When g is composed on f, the range of f becomes the domain for g.

0.7

Trigonometric Functions

FOR ALL PTS, (X,Y) ON THE UNIT CIRCLE:SIN T = Y, COS T = X, TAN T = Y/X

t = real number (length of arc on unit circle) that corresponds to pt (x,y)

y = sin x y = cos x

OTHER TRIG FUNCTIONS

sec x = 1/cos x csc x = 1/sin x cot x = 1/tan x Pythagorean identity (main one,

others may be developed from this one)

1)(cos)(sin 22 xx