Summary Advanced Placement Calculus AB Cram Sheet with Formulas and Theorems

ABCramSheet.nb 1




AP Calculus AB Cram Sheet
Definition of the Derivative Function:

f +xh/  f +x/
f ' (x) = limh‘0 cccccccccccccccc
h
cccccccccccccc

Definition of Derivative at a Point:

f +ah/ f +a/
f ' (a) = limh‘0 cccccccccccccccc
ccccccccccccc (note: the first definition results in a function, the second definition results in a number. Also
f +ah/ f +a/
h
note that the difference quotient, cccccccccccccccc h
ccccccccccccc , by itself, represents the average rate of change of f from x = a to x = a + h)

Interpretations of the Derivative: f ' (a) represents the instantaneous rate of change of f at x = a, the slope of the tangent
line to the graph of f at x = a, and the slope of the curve at x = a.

Derivative Formulas: (note:a and k are constants)

d
cccc
dx
cc +k/ 0

d
cccc
dx
cc (k·f(x))= k·f ' (x)

d
cc + f +x//n
cccc
dx
n+ f +x//n1  f ' +x/

d
cccc
dx
cc [f(x) ± g(x)] = f ' (x) ± g ' (x)

d
cccc
dx
cc [f(x)·g(x)] = f(x)·g ' (x) + g(x) · f ' (x)

d
cc , ccccf c+x/
cccc
dx g+x/
ccccc 0 g+x/ f ' +x/  f +x/g ' +x/
+g+x//2
cccccccccccccccccccccccccccccccc
ccccccccccccccccc

d
cccc
dx
cc sin(f(x)) = cos (f(x)) ·f ' (x)

d
cccc
dx
cc cos(f(x)) = -sin(f(x))·f ' (x)

d
cc tan(f(x)) = sec2 + f +x// º f ' +x/
cccc
dx

d
cccc
dx
ccccc º f ' +x/
cc ln(f(x)) = ccccf c1+x/

d
cc e f +x/
cccc
dx
e f +x/ º f ' +x/

d
cc a f +x/
cccc
dx
a f +x/ º ln a º f ' +x/

cc sin1  f +x/ f ' +x/
r

2
1+ f +x//
d
cccc
dx
cccccccccccccccc
cccccccc
cccc


cc cos1  f +x/ f ' +x/
r

2
1+ f +x//
d
cccc
dx
 cccccccccccccccc
cccccccc
cccc

, ABCramSheet.nb 2


d
cc tan1  f +x/
cccc
dx
f ' +x/
1+ f +x//2
cccccccccccccccc
cccccc

d
cc + f 1 +x// at x
cccc
dx
f +a/ equals cccccccc
1
ccccc at x
f '+x/
a

L'Hopitals's Rule:

f +x/ f ' +x/
g ' +x/
ˆ
If limx‘a cccc
cccccc
g+x/
cccc00 or cccc
ˆ
cc and if limx‘a cccccccc
ccccc exists then

f +x/ f ' +x/
g ' +x/
limx‘a cccc
cccccc
g+x/
limx‘a cccccccc
ccccc

ˆ f +x/
The same rule applies if you get an indeterminate form ( cccc00 or cccc
ˆ
cc ) for limx‘ˆ cccc
cccccc as well.
g+x/


Slope; Critical Points: Any c in the domain of f such that either f ' (c) = 0 or f ' (c) is undefined is called a critical point or
critical value of f.

Tangents and Normals
The equation of the tangent line to the curve y = f(x) at x = a is

y - f(a) = f ' (a) (x - a)

The tangent line to a graph can be used to approximate a function value at points very near the point of tangency. This is
known as local linear approximations. Make sure you use ž instead of = when you approximate a function.

The equation of the line normal(perpendicular) to the curve y = f(x) at x = a is


f ' +a/
1
y - f(a) =  cccccccc
ccccc +x  a/


Increasing and Decreasing Functions A function y = f(x) is said to be increasing/decreasing on an interval if its deriva-
tive is positive/negative on the interval.

Maximum, Minimum, and Inflection Points
The curve y = f(x) has a local (relative) minimum at a point where x = c if the first derivative changes signs from negative
to positive at c.

The curve y = f(x) has a local maximum at a point where x = c if the first deivative changes signs from positive to negative.

The curve y = f(x) is said to be concave upward on an interval if the second derivative is positive on that interval. Note that
this would mean that the first derivative is increasing on that interval.

The curve y = f(x) is siad to be concave downward on an interval if the second derivative is negative on that interval. Note
that this would mean that the first derivative is decreasing on that interval.

The point where the concavity of y = f(x) changes is called a point of inflection.

The curve y = f(x) has a global (absolute) minimum value at x = c on [a, b] if f(c) is less than all y values on the interval.