AP Calculus Topics by Andy Wickell
Definition of Continuity
Continuity at a Point:
1.
exists,
2.
exists, and
3.
Continuity on an Interval:
Function f is continuous on an interval of x-values
if and only if it is continuous at each value of x in
that interval.
Definition of Cusp
1. A continuous point with a discontinuous derivative
2. Graphically, a sharp point of an abrupt change in direction.
3. A point in which no one tangent line can be drawn.
Equal left and right
limits
if and only if 
In other words, the left
and right limits must be equal for the general limit to exist.
Intermediate Value Theorem (don’t memorize this one)
On a continuous interval
of [a,b] and y is some number between 
and 
, then there exists some value 
in (a,b) that 
.
Definition of
Derivative at a point
Instantaneous rate of
change at 
is:
Symmetric Difference
Quotient
The tolerance, h, is positive.
Definition of Derivative
The numerator represents 
while the denominator
represents
. 
,
.
Derivative of the power
function
If 
, and n is constant, then 
Three Properties of
Differentiation
1. Sum of functions: 
2. A constant multiple of
a function:
c is
constant
3. Derivative of a
constant: 
c is
constant
Chain Rule
Limit of Sin(x)/x
Graph of a sinusoid
C=Vertical shift, sinusoidal axis at y=C
= Amplitude
=Period
D=Horizontal shift/displacement
Product Rule
Shorthand:
Quotient Rule
Shorthand:
Derivatives of the 6
trig functions
Derivatives of the 6
inverse trig functions
Implicit
Differentiation
Implicit differentiation
is the calculation of 
when the equation is
written implicitly (e.g. 
) as opposed to explicitly (e.g. 
). To do this, one must differentiate both sides of the
equation with respect to x, keeping in mind, that y is a function
of x, so the derivative of y with respect to x is 
, not 1. For example,
to differentiate a term of xy, you would use the product rule, which
would turn out 
. Next, you would solve the equation for 
, and thus you would have 
.
Linearization of a
function
Linear function of best
fit at 
:
This is derived from the point-slope linear equation.
Definition of an
indefinite integral
The indefinite integral can be thought as the inverse operation of differentiation:
if and only if 
Four properties of
indefinite integrals
1. Integral of a constant
multiple of a function: 
2. Integral of a sum of functions
Definite integral and
integrability
A function is integrable
on an interval 
if the limits of the
lower and upper Riemann sums are equal as 
. The convergence of this limit is the definite integral of 
. The definite integral can be thought as the area under the
curve from 
, with the units being the product of the axes. A definite
integral takes the form of: 
.
Mean Value Theorem
Preamble:
If 1. f is differentiable
for all values of x in the open interval 
2. f is continuous at 
and at 
then there exists at least
one number 
in 
which satisfies
In other words, the slope of the secant line from a to b is equal to the slope of at least one tangent line between a and b.
Rolle’s Theorem
Rolle’s theorem is a single case of the m.v.t.
Preamble:
If 1. f is
differentiable for all values of x in the open interval 
2. f is continuous at 
and at 
3. 
then there exists at least
one number 
in 
such that 
Fundamental Theorem of
Calculus
If f is integrable, then:
, where 
is the indefinite
integral of 
.
Properties of Definite
Integrals
1.
is positive if 
is positive in 
and is negative if 
is negative in 
2. Reverse of limits:
3. Sum of integrals with
same integrand
4. Symmetric limits:
For an even function: 
For and odd function: 
5. Sum of functions, and constant multiple of function:
6. If 
for all x in 
, then 