AB Calculus
Formulas & Theorems
Page 1
I.
TRIGONOMETRIC FORMULAS
1.
sin 2   cos 2  1
4.
sin    sin 
2.
1  tan 2   sec 2 
5.
cos   cos 
3.
1  cot 2   csc 2 
 6.
7.
sin A  B   sin A cos B  cos Asin B
8.
sin A  B   sin A cos B  cos Asin B
 9.
cos A  B   cos A cos B  sin A sin B
 10.
cos A  B   cos A cos B  s in A s in B
 11.
sin 2  2 sin  cos 




12.
cos 2  cos 2   sin 2   2 cos 2  1 1  2 sin 2 
13.
tan  
sin 
1

cos  cot 
14.
cot  
cos 
1

sin  tan 
15.
sec  
1
cos 
16.
csc  
1
sin 
18.
sin 2    cos 


17.
cos 2    sin 


19.
sin A sin B sin C


a
b
c
20.
a 2  b 2  c 2  2bc cos A
b 2  a 2  c 2  2ac cos B

c 2  a 2  b 2  2ab cos C





tan   tan 

AB Calculus
Formulas & Theorems
Page 2
II.
1.
d
x n   nx n 1

dx
2.
d
dx
3.
d  f  g f  f g
 
dx g 
g2
4.
d
f g x   f g x g x 
dx
5.
d
sin x   cos x
dx
7.
d
tan x   sec
dx





9.

11.

13.


 fg  
f g  g f 
2
x
d
sec x   sec x tan x
dx
d
e x  e x

dx
d
1
ln x  

dx
x
14.
d
1
Arc sin x  

dx
1  x2
15.
d
1
Arc tan x  

dx
1  x2


DIFFERENTIATION FORMULAS
6.
d
cos x    sin x
dx
8.
d
cot x    csc 2 x

dx
10.
d
csc x    csc x cot x
dx
12.
d
a x   a x ln a

dx




AB Calculus
Formulas & Theorems
Page 3
III. INTEGRATION FORMULAS
1.
a
dx  ax  C
2.

x n 1
x dx 
 C, n  1
n 1
3.

1
dx  ln x  C
x


4.
n
 ex
5.
dx  e x  C
ax
C
 a dx 
ln a
x
 6.
 ln x
dx  x ln x  x  C
 7.
 sin
x dx   cos x  C
 9.
 tan
x dx  ln sec x  C or  ln cos x  C

10.
 cot

x dx  ln sin x  C or  ln csc x  C
 11.
 sec
x dx  ln sec x  tan x  C
 12.
 csc
x dx  ln csc x  cot x  C
 13.
 sec 2
 15.
 sec
x tan x dx  sec x  C
 16.
 csc
x cot x dx   csc x  C
 17.
 tan 2

18.

19.




 8.
x dx  tan x  C
x dx  tan x  x  C
x 
dx
1

Arc
tan
  C
a2  x2 a
a 
x 
dx
 Arc sin   C
a 
a 2 x 2
14.

 cos
x dx  sin x  C
 csc 2
x dx   cot x  C
AB Calculus
Formulas & Theorems
Page 4
IV. DEFINITIONS, FORMULAS, & THEOREMS
1.
Even and Odd Functions
A function
y  f x  is even if f x   f x  for every x in the
function’s domain. Every even function is symmetric about the y-axis.
y  f x  is odd if f x    f x  for every x in the


 even function is symmetric
function’s
domain. Every
about the origin.
A function
2.
Periodicity

A function f x  is periodic with period p, p  0  if f x  p   f x 
for every value of x .


Note:
The
 period of the function
y A sin Bx  C or y  A cos Bx  C is
2

and the amplitude is A . The period of y  A tan Bx  C  is
.
B
B


3.

Limits and Continuity

The limit


f x  exists if and only if both corresponding
 one-sided
lim
x a
limits exist and are equal – that is,
f x   L  lim f x   L  lim f x 
lim


x a

x a
A function

y  f x  is continuous at x  a if:
f a  is defined
i.

ii.
 iii.

Otherwise,
 and
f x  exists,
lim
x a
f x   f a 
lim
x a
f
is discontinuous at


x a

x  a.
AB Calculus
4.
Formulas & Theorems
Page 5
Important Limits

lim
0

lim
0
sin 
1

cos  1
0

 
 1 
1    e
lim
n 
n  
 
1  n n  e
n
1
lim
n 0
5.  Intermediate-Value Theorem
 A function y  f x  that is continuous on a closed interval a,b  takes
on every value between
f a  and f b .
Note:

If f is continuous on
equation
6.
f x 
0
 in sign, then the
a,b  and f a and f b  differ

has at least one solution in the open interval a,b .
Limits of Rational Functions at Inifinity



lim
x 
o
 
lim
x 
o
 
lim
x 
o




f x 

 0 if the degree of f x  < degree of g x 
gx 
x2  2x
0
Example:
lim
3
x  x  3


f x 
f
is infinite if the degree of x  > degree of g x 
g x 
x3  2x

 
Example:
lim
2
x  x  8


f x 
is finite if the degree of f x  = degree of g x 
g x 
2x2  3x  2
2



Example:
lim
2
5
x  10 x  5 x



AB Calculus
7.
Horizontal and Vertical Asymptotes


8.
Formulas & Theorems
Page 6
A line
y b
either
f x   b
lim
x 
9.

f x   b .
lim
x 
A line x  a is a vertical asymptote of the graph
 of

either
lim f x   or lim f x   .
x a
y  f x  if
y  f x  if
x a

Average
and Instantaneous Rate of Change


or


y  f x , then the
average rate of change of y with respect to x over the interval x 0 , x 1  is
f x 1   f x 0  y 1  y 0 y


.
x 1 x 0
x 1  x 0 x




If x 0 , y 0  is a point on the graph of y  f x , then the instantaneous
rate of change of y with respect to x at x 0 is f x 0 .
x0 , y0
If 

is a horizontal asymptote of the graph of
 and x 1 , y 1 are points on the graph of
Definition of the Derivative

f x   lim
h 0

f x  h   
f x  
f x   f a 

or f a   lim
h
x a
x a
The latter definition of the derivative is the instantaneous rate of change
of f x with respect to x at x  a .



Geometrically, the derivative of a function at a point is the slope of the
tangent line to the graph of the function at that point.

10.


Increasing/Decreasing Test




f x   0 on an interval, then f is increasing on that interval.
If f x   0 on an interval, then f is increasing on that interval.
If


AB Calculus
11.
Formulas & Theorems
Page 7
Rolle’s Theorem
If
f
a,b  and differentiable on a,b  such that
is continuous on
f a   f b , then there is at least one number c in the open interval
a,b  such that f c  0 .



12.

Mean Value Theorem
a,b  and differentiable on a,b , then there is at
f b   f a 
 f c, or,
least one number c in a,b  such that
b a
equivalently, f b   f a   f cb  a .


If



f
is continuous
on


Note:

 slope of the secant line is the same as
Geometrically this states that the
the 
slope of the tangent line at some point in the interval.
13.
Critical Number
A critical number of a function
such that either
14.
Local Extremum
If
of
15.





c in the domain of f



f hasa local maximum
or minimum at c , then c is a critical number

f.
Suppose that

is a number
f c  0 or f c does not exist.
The First Derivative Test

f


c is a critical number of a continuous function f .
If f  changes from positive to negative at c , then f has a local
maximum at c .
If f  changes from negative to positive at c , then f has a local

minimum at c .
If f  does not change sign at c , 
then f 
has no local maximum or

local minimum at c .






AB Calculus
16.
Formulas & Theorems
Page 8
Concavity Test


f x   0 for all x in I , then the graph of f is concave upward
on I .
If f x   0 for all x in I , then the graph of f is concave
downward
 on I.
If



17. Point of Inflection




A point P 
on a curve y  f x  is called an inflection point if f is
continuous there and the curve changes from concave upward to
concave downward or from concave downward to concave upward at P .
To locate the points of inflection of y  f x , find the points where



f x   0 or where f x  fails to exist. These are the only 
candidates
f x  may have a point of inflection, Then test these points to
make sure that f x   0
on one side and f x   0 on the other.
where

18.


The Second Derivative Test
Suppose
 f  is continuous near
c
.
f c  0 and f c  0 , then f has a local minimum at c .

If f c  0 and f c  0 , then f has a local maximum at c .


19. Extreme Value Theorem




on a closed interval a,b , then f x 
attains an
If f is continuous


If

f c and an absolute minimum value f d  at
some numbers c and d in a,b .


absolute maximum value

20.
The Closed Interval Method




To find the absolute
 maximum and absolute minimum values of a
continuous function f on a closed interval a,b :
a,b 

Find the values of


Find the values of
at the endpoints of the interval
The largest of the values is the absolute maximum value; the
 of the values is the 
smallest
absolute minimum value.


f
f
at the critical numbers of


f
in
AB Calculus
21.
Formulas & Theorems
Page 9
Differentiability Implies Continuity
If a function is differentiable at a point x  a , it is continuous at that
point. The converse is false, that is, continuity does not imply
differentiability.
22.
Local Linearity and Linear Approximation

The linear approximation of
y  f x 0   f x 0 x  x 0 .
f x  near x  x 0 is given by
To estimate the slope of a graph at a point, draw a tangent line to the

graph at that point.On a graphing
calculator, another way is to “zoom
in” around the point in question until the graph “looks” straight. This
method almost always works. If we “zoom in” and the graph “looks”
straight at a point, say x  a , then the function is locally linear at that
point.

y  x has a sharp corner at x  0 . This corner cannot be
smoothed out 
by “zooming in” repeatedly. Consequently, the derivative
The graph of
y  x does not exist at x  0 , hence, is not locally linear at x  0 .


Newton’s
Method
of
23.



24.

Let f be a differentiable
function and suppose r is a 
real zero of f . If
x n is an approximation to r , then the next approximation x n 1 is given
f x n 
provided f x n   0 . Successive approximations
f x n 


can be found using
 this method.

by x n 1  x n 
L’Hôpital’s Rule

If
f x 
lim
x a g x 
f x 
lim
x a g x 



is of the form
 lim
x a
f x 
.
gx 
 
0

or
, and if
0


lim
x a
f x 
exists, then
gx 
AB Calculus
25.
26.
Formulas & Theorems
Page 10
Related Rates

Read the problem carefully and draw a diagram if possible.

Introduce notation. Assign symbols to all quantities that are
functions of time.

Express the given information and the required rate in terms of
derivatives.

Write an equation that relates the various quantities of the
problem. If necessary, use the geometry of the situation to
eliminate one of the variables by substitution.

Use Chain Rule to differentiate both side of the equation with
respect to t .

Substitute the given information into the resulting equation and
solve for the unknown rate.

Common Geometric Formulas
Triangle
Circle
Sector of a Circle
A  21 bh
A  r 2
A  21 r 2 
A  21 ab sin 
C  2 r
s  r




Sphere
Cylinder
Cone
V  43 r 3
V  r 2 h
V  13 r 2 h
A  4 r 2
A  2rh  2 r 2
A  r r 2  h 2


AB Calculus
27.

Properties of the Exponential Function

The exponential function y  e

The domain is the set of all real numbers,

The range is the set of all positive numbers, y  0

d
e x  e x

dx

y  e x is continuous, increasing, and concave up for all x

e
lim
x 
 


28.
Formulas & Theorems
Page 11

x
x
is the inverse function of y  ln x
  x  


  and


e
lim
x 
x
0

e ln x  x , for x  0
ln e x  x for all x


Properties of the Natural Logarithmic Function

 
The domain of y  ln x is the set of all positive numbers, x  0

The range of y  ln x is the set of all real numbers,   y  

y  ln x is continuous, increasing, and concave down everywhere


in its domain


ln ab   ln a  ln b

a 
ln   ln a  ln b
b 

ln a r  r ln a
 








ln x  0 if 0  x 1 and ln x  0 if x 1
ln x    and lim ln x   
lim
x 
x 0


ln x 
log a x 
ln a


AB Calculus
29.
Formulas & Theorems
Page 12
Comparing Rates of Change
The exponential function y  e grows very rapidly as x   while the
natural logarithmic function y  ln x grows very slowly as x  .
x
Exponential functions like y  2 or y  e grow more rapidly as x
power of x. The function y 
than any positive
 ln x grows slower as


x   than any nonconstant polynomial.
x
We say, that as x :





g x 
lim
x 
For example,

f x 
  or if
g x 
If

ex

lim
3
x  x
x4
4


x grows faster than ln x as x   since lim
x  ln x


 same rate as x 2 as x   since

x 2  2 x grows at the

x2  2x
1 

lim
x 2 
x 




To find some of these limits as x  , you may use the graphing
 

f x  grows faster than g x  as x  , then


g x  grows slower than f x  asx  .
f x  and g x  grow at the same rate as 
x   if


f x 
 L  0 (where
lim
 L is finite and nonzero).
x  g x 



0.
lim
x  f x 


f x  grows faster than g x  as x   if



x
e x grows faster than x 3 as x   since
calculator. Make sure that an appropriate viewing window is used.


AB Calculus
30.
Formulas & Theorems
Page 13
Inverse Functions
f gx   x for every x in
the domain of g , and g f x   x , for every x in the domain of f
, then f and g are inverse functions of each other.





A function f has an inverse function
if and only if no horizontal

line
intersects
once.
its graph more than



If f and g are two functions such that

If f is either increasing or decreasing in an interval, then f has
aninverse function over that interval.

If f is differentiable at every point on an interval I , and




f x   0
g  f 1 x  is differentiable at every point of the interior
1
of the interval f I  and g  f x  
.
f x


on I , then
 of the Definite Integral
Properties
31.
Let


f x  and g x  becontinuous on a,b .

 a c  f x  dx

  a f x  dx  0
a
b
 c  a f x  dx , c is a nonzero constant
b

 
b f x  dx
b
   a f x dx
 
a f x  dx

a
b
a f x  dx
c

c f x  dx , where
b
f is continuous on an

interval containing the numbers a , b , and c , regardless of the order a , b ,

f x  is an odd function, then a f x  dx  0
 

 
a
a
If f x  is an even function, then a f x  dx  2 0 f x  dx
and c .


a
If

If
f x   0 on a,b , then


If
gx   f x  on a,b 
, then





 a f x  dx
b
0
 a gx  dx
b

 a f x  dx
b
AB Calculus
32.
Formulas & Theorems
Page 14
Velocity, Speed, and Acceleration

The velocity of an object tells how fast it is going and in which
direction. Velocity is an instantaneous rate of change of the
position – that is, v t  x  t .



The speed of an object is the absolute value of the velocity,
v t  . It
tells how fast its is going disregarding its direction.


The speed of a particle increases (speeds up) when the velocity and
acceleration have the same signs. The speed decreases
(slows

down) when the velocity and the acceleration have opposite signs.

The acceleration is the instantaneous rate of change of velocity – it
is the derivative of the velocity – that is, a t  v  t . Negative


acceleration (deceleration) means that the velocity is decreasing.
The acceleration fives the rate at which the velocity is changing.
Therefore, if

x t  is the position of a 
moving object and t is time, then:


velocity = v t  x  t 
dx
dt
dv d 2 x
 2

  acceleration = a t   x t   v t  
dt
dt

v t    a t  dt



x t    v t  dt
 Note:

The average velocity of a particle over the time interval from t 0 to another
time t , is Average Velocity =
x t   x t 0 
t  t0
, where
the particle at time t .



x t  is the position of


AB Calculus
33.
Formulas & Theorems
Page 15
Definition of Definite Integral as the Limit of a Sum
Suppose that function
b a
.
n
Choose one number in each subinterval, that is, x 1 in the first, x 2 in the


th
th
second, … , x i in the i , … , and x n in the n . Then
n

b

f
x
x

f x  dx .




i
lim
a
n  i1





Summation
Formulas 
Divide the interval into
34.

f x  is continuous on the closed interval a,b .
n equal subintervals, of length x 
n
1  n
i1
n
 c  nc
i1
n

i 
i1
n

 i2
i1

2
n n 1 2n 1 
6
n n 1 

 i  
i1
 2 

2
n
3


n n 1 
35.
Trapezoidal Rule
a,b  where a,b 
subintervals x 0 , x 1 , x 1 , x 2 , … ,
If a function f is continuous on the closed interval
has been partitioned into
n
b a
, then
x n
1 , x n , each of length
n


b
b a
f x 2   ...  2 f x n 1   f x n .
 a f x  dx 
 f x 0   2 fx 1   2 
2n


 is the average of the left-hand and right-hand
The Trapezoidal Rule
Riemann sums.
AB Calculus
36.
Formulas & Theorems
Page 16
Fundamental Theorem of Calculus


 
 a f x  dx
b
d
dx
d
dx
 Fb   Fa , where Fx   f x 
0 f t  dt
x
g x 
a
 f x 
f t  dt  f g x
gx 
37. Average Value of a Function

38.
The average value of
f on a,b  is
1 b
 f  x  dx .
ba a
Area Between Curves

If f and g are continuous
functions such that
then the area between the curves is

Note:

b


left top
right
 bottom
 dx .
General form for an approximating rectangle perpendicular to the
y–axis is area
39.
 a  f x   g x  dx .
General form for an approximating rectangle perpendicular to the
x–axis is area A 

f x   gx  on a,b ,
A
bottom right
top
 left  dy

Slope Fields
At every point

x , y , a differential equation of the form
dy
 f x , y 
dx
gives the slope of the member of the family of solutions that contains
that point. A slope field is a graphical representation of this family of
curves. At each point in the plane, a short segment is drawn whose
slope
is equal to the value of the derivative at that
 point. These segments
are tangent to the solution’s graph at the point.
The slope field allows you to sketch the graph of the solution curve even
though you do not have its equation. This is done by starting at any
point (usually the point given by the initial condition), and moving from
one point to the next in the direction indicated by the segments of the
slope field.
Some calculators have built in operations for drawing slope fields; for
calculators without this feature there are programs for drawing them.
AB Calculus
40.
Formulas & Theorems
Page 17
Solving Differential Equations by Separating the Variables
There are many techniques for solving differential equations. Any
differential equation you may be asked to solve on the AB Calculus Exam
can be solved by separating the variables. Rewrite the equation as an
equivalent equation with all the y and dy terms on one side and all the
x and dx terms on the other. Antidifferentiate both sides to obtain an
equation without dy or dx , but with one constant of integration. Use
the initial condition to evaluate this constant.



41.  Volumes of Solids of Revolution


Let f be nonnegative and continuous on
a,b , and let
R be the region
y  f x , below by the x–axis, and the sides by the
lines x  a and x  b .


 about the x–axis, it generates a

When this region R is revolved
solid,
cross-sections, whose volume
 having circular
2
b

V   a  f x  dx .
bounded above by



When this region R is revolved about the y–axis, it generates a
solid, having circular cross-sections, whose volume
V 
a 2xf x  dx .
b


42.
Note:
Any closed region R can be rotated about either the x–axis or the y–axis.
If you choose your approximating rectangle perpendicular to the axis of
rotation, then the volume can be viewed as a sum of disks or a sum of
washers. If you choose your approximating rectangle parallel to the axis
of rotation,
 then the volume can be viewed as a sum of shells.
Volumes of Solids with Known Cross-Sections

For cross-sections of area
volume

V 
 a A x  dx .
b
For cross-sections
of area
volume
V 
c A y  dy .
d


A x , taken perpendicular to the x–axis,
A y , taken perpendicular to the y–axis,
AB Calculus
43.
Integration by Parts
If
u  f x  and v  gx  and if f x  and g x  are continuous, then
u


Formulas & Theorems
Page 18
dv  uv 
v
du .
Note:
The goal
of the procedure
 is to choose

to solve than the original problem.
u and v so that  v du is easier
Suggestion:
 L is the logarithmic
 L.I.A.T.E.,

When “choosing” u , remember
where
function, I is an inverse trigonometric function, A is an algebraic
function, T is a trigonometric function, and E is the exponential function.
Just choose u as the first expression in L.I.A.T.E. (and dv will be the
remaining
 part of the integrand).
For example:
  When integrating  x ln x dx , choose u 
ln x , since L comes first
in L.I.A.T.E., and dv  x dx .

When integrating  xe x dx , choose


x
in L.I.A.T.E., and dv  e dx .
u  x , since A comes before E


When integrating  xArc tan x dx , choose u  Arc tan x , since I
and dv  x dx .
 A in L.I.A.T.E.,
comes before



