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 INEQUALITIES
Solve by comparing the inequality to zero, factor
if possible, and solve.
2x  7x  4
2
2x2  7x  4  0
(2 x  1)( x  4)  0
(2 x  1  0) AND( x  4  0), x  1 / 2  x  4
OR (2 x  1  0)AND( x  4  0), x  1 / 2  x  4
(,4)  (1 / 2, )
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
Ax  Bx  Cy  D  0
2
y  a ( x  h)  k
2
Ay  By  Cx  D  0
2
x  a( y  k )  h
2
CUBIC FUNCTIONS

Reflects through the origin
y  ax  bx  cx  d
3
2
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)

sin ( x)  cos ( x)  1
2
2