The Calculus AB Review Sheet
Derivative Formulas
Derivative Notation:
For a function f(x), the derivative would be f ' ( x)
Leibniz's Notation:
For the derivative of y in terms of x, we write
dy
dx
For the second derivative using Leibniz's Notation:
d2y
dx 2
Product Rule:
y  f ( x) g( x)
y  x 2 sin x
dy
 f ' ( x) g( x)  g' ( x) f ( x)
dx
dy
 2 x sin x  cos x ( x 2 )
dx
dy
 2 x sin x  x 2 cos x
dx
Quotient Rule:
f ( x)
y
g( x)
dy
f ' ( x) g( x)  g' ( x) f ( x)

dx
( g ( x )) 2
y
dy
dx
dy
dx
sin x
x3
cos x ( x 3 )  3x 2 sin x

x6
x cos x  3 sin x

x4
Chain Rule:
y  ( f ( x )) n
y  ( x 2  1) 3
dy
 n( f ( x )) n 1 ( f ' ( x ))
dx
dy
 3( x 2  1) 2  2 x
dx
dy
 6 x ( x 2  1) 2
dx
Natural Log
Power Rule:
y  ln( f ( x ))
y  ln( x 2  1)
dy
1

 f ' ( x)
dx
f ( x)
dy
1
 2
 2x
dx
x 1
dy
2x
 2
dx
x 1
y  xa
y  2x5
dy
 ax a 1
dx
dy
 10 x 4
dx
Constant with a Variable Power:
y  a f ( x)
y  2x
dy
 a f ( x )  ln a  f ' ( x )
dx
dy
 2 x ln 1
dx
y  3x
2
2
dy
 3 x  ln 3  2 x
dx
Variable with a Variable power
y  f ( x) g ( x )
Take ln of both sides!
y  x sin x
ln y  ln x sin x
ln y  sin x ln x
1 dy
1
 cos x ln x  sin x
y dx
x
dy
1
 x sin x [cos x ln x  sin x ]
dx
x
Implicit Differentiation:
Is done when the equation has mixed variables:
x2  x2 y3  y4  5
derivative  2 x  [ 2 xy 3  3 y 2
dy 2
dy
x ]  4y3
0
dx
dx
dy
dy
 4y3
 2 x  2 xy 3
dx
dx
dy
2 x  2 xy 3


dx
3y 2 x 2  4 y 3
 3y 2 x 2
Trigonometric Functions:
d
sin x  cos x
dx
d
tan x  sec 2 x
dx
d
cos x   sin x
dx
d
sec x  sec x tan x
dx
Inverse Trigonometric Functions:
y  arcsin x
dy
1

1
dx
1 x2
y  arctan x
dy
1

1
dx 1  x 2
y  arcsin x 4
dy
1

 4x 3
8
dx
1 x
y  arctan x
dy
1

 3x 2
6
dx 1  x
Integral Formulas
Basic Integral
 5 dx
 5x  C
,where C is an arbitrary constant

 x  C
Variable with a Constant Power
x

a
dx
x a 1
C
a 1
 x dx
3

x4
C
4
Constant with a Variable Power
a
x
dx
ax

C
ln a
 5 dx
x
5x

C
ln 5
3

Fractions
2x
dx
32 x
C
2 ln 3
if the top is the derivative of the bottom

1
dx
x4
x
4
1
 x dx
-unless-
dx
 ln| x|C
x 3

3
x3
 x 4  1 dx
1
 ln| x 4  1| C
4
Substitution
When integrating a product in which the terms are somehow related, we
usually let u= the part in the parenthesis, the part under the radical, the
denominator, the exponent, or the angle of the trigonometric function
x
x 2  1  dx; u  x 2  1
 cos 2 x dx;
du  2 x  dx
1
2 x ( x 2  1) 1/ 2  dx
2
1
  u 1/ 2 du
2
1 2
  u 3/ 2  C
2 3
1
 ( x 2  1) 3/ 2  C
3
u  2x
du  2  dx
1
2 cos 2 x  dx
2
1
  cos u du
2
1
 sin u  C
2
1
 sin 2 x  C
2


Integration by Parts (Not an AB topic)
When taking an integral of a product, substitute for u the term whose derivative would
eventually reach 0 and the other term for dv.
The general form:
Example:
 xe
ux
x
dx
dv  e x dx
du  1 dx v  e x
 x  e x   e x dx
 xe x  e x  C
uv 
 vdu (pronounced "of dove")
Example 2:
2
 x cos x
u  x2
dv  cos x dx
du  2 x dx v  sin x
x 2 sin x   2 x sin x
u  2 x dv  sin x dx
du  2 dx v   cos x
x 2 sin x  [2 x cos x    2 cos x
 x 2 sin x  2 x cos x  2 sin x  C
Inverse Trig Functions (Second Semester)
Formulas:
1

a2  x2
x
 arcsin  C
a
a

2
1
 x2
1
x
arctan  C
a
a
Examples:
1

; a  3; v  x
9  x2
 arcsin
x
C
3
1
 16  x

2
; a  4; v  x
1
x
arctan  C
4
4
More examples:
1

1
3
; a  2; v  3 x
4  9x 2

3
4  9x 2
1
3x
 arcsin
C
3
2
1
a  4; v  3 x
 16
1
3
2
3 9 x  16
1 1
3x
  arctan  C
3 4
4
1
3x
 arctan  C
12
4
 9x

2
Trig Functions
 sin x dx   cos x  C
 sec
2
x dx  tan x  C
sin 2 x
 cos 2 x dx  2  C  sec x tan x  sec x  C
 tan x dx   ln cos x  C  cot x dx  ln sin x  C
Properties of Logarithms (Second Semester)
Form
logarithmic form
<=>
exponential form
y = loga x
<=>
ay = x
Log properties
y = log x3
=>
y = 3 log x
log x + log y = log xy
log x – log y = log (x/y)
Change of Base Law
This is a useful formula to know.
y  log a x 
log x
log a
 or 
ln x
ln a
Properties of Derivatives
1st Derivative shows: maximum and minimum values, increasing and decreasing intervals, slope of
the tangent line to the curve, and velocity
2nd Derivative shows: inflection points, concavity, and acceleration
- Example on the next page -
Example:
y  2 x 3  3x 2  36 x  2
dy
 6 x 2  6 x  36
dx
0  6( x 2  x  6)
0  6( x  3)( x  2)
Find everything about this function
1st derivative finds max, min, increasing, decreasing
x  3,2
max
(-2, 46)
increasing
(   , -2] [3,  )
min
(3, -79)
decreasing
(-2, 3)
d2y
 12 x  6
dx 2
0  6(2 x  1)
1
x
2
2nd derivative finds concavity and inflection points
inflection pt
( ½ , -16 ½)
concave up
(   , ½)
concave down
(1/2,  )
Miscellaneous
Newton’s Method
Newton’s Method is used to approximate a zero of a function
c
f (c )
f ' (c )
where c is the 1st approximation
Example:
If Newton’s Method is used to approximate the real root of
x3 + x – 1 = 0, then a first approximation of x1 = 1 would lead to a
third approximation of x3:
f (1) 3

or .750  x 2
f ' (1) 4
3
f( )
3
4  59 or .686  x

3
3
4
86
f '( )
4
1
f ( x)  x 3  x  1
f ' ( x)  3x 2  1
Separating Variables
Used when you are given the derivative and you need to take the integral. We separate
variables when the derivative is a mixture of variables
Example:
dy
 9 y 4 and if y = 1 when x = 0, what is the value of y when
dx
1
x= ?
3
If
dy
dy
 9 y 4  4  9 dx
dx
y

dy
y 3

9
dx

 9x  C
3
y4

Continuity/Differentiable Problems
f(x) is continuous if and only if both halves of the function have the same answer at the
breaking point.
f(x) is differentiable if and only if the derivative of both halves of the function have the same
answer at the breaking point
Example:
 2 x  6( plug in 3)
 x2 , x  3
f ' ( x) 
f ( x) 
 6 x  9, x  3
-
66
At 3, both halves = 9, therefore, f(x) is continuous
At 3, both halves of the derivative = 6, therefore, f(x) is differentiable
Useful Information
-
We designate position as x(t) or s(t)
The derivative of position x’(t) is v(t), or velocity
The derivative of velocity, v’(t), equals acceleration, a(t).
We often talk about position, velocity, and acceleration when we’re discussing particles
moving along the x-axis.
A particle is at rest when v(t) = 0.
A particle is moving to the right when v(t) > 0 and to the left when v(t) < 0
-
To find the average velocity of a particle:
b
1
v (t ) dt
ba a

Average Value
Use this formula when asked to find the average of something
1
ba
b
 f ( x)dx
a
Mean Value Theorem
NOT the same average value.
According to the Mean value Theorem, there is a number, c, between a and b, such that the
slope of the tangent line at c is the same as the slope between the points (a, f(a)) and (b, f(b)).
f ' (c) 
f (b)  f (a)
ba
Growth Formulas
Double Life Formula: y  y 0 (2) t / d
Half Life Formula: y  y 0 (1 / 2) t / h
Growth Formula: y  y 0 e kt
y = ending amount y0 = initial amount t = time
k = growth constant d = double life time h = half life time
Useful Trig. Stuff
Double Angle Formulas:
sin 2 x  2 sin x cos x
cos 2 x  cos 2 x  sin 2 x
Identities:
sin 2 x  cos 2 x  1
1
 sec
cos
sin 
 tan 
cos
1  tan 2 x  sec 2 x
1  cot 2 x  csc 2 x
1
 csc
sin 
cos
 cot 
sin 
Integration Properties
Area
b
 [ f ( x)  g ( x)]dx
f(x) is the equation on top
a
Volume
f(x) always denotes the equation on top
About the x-axis:
About the y-axis:
b
b
  [ f ( x)] dx

2 x[ f ( x)]dx
2
a
a
b
b
  [( f ( x)) 2  ( g ( x)) 2 ]dx

2 x[ f ( x)  g ( x)]dx
a
a
about line y = -1
about the line x = -1
b
b
  [ f ( x)  1] 2 dx

2 ( x  1)[ f ( x)]dx
a
a
Examples:
f ( x)  x 2 [0,2]
x-axis:
y-axis:
2
2
2
  ( x ) dx    x dx
2 2
0

2

2 x[ x ]dx  2 x 3 dx
4
0
2
0
0
about y = -1
In this formula f(x) or y is the radius of the shaded region. When we rotate about the
line y = -1, we have to increase the radius by 1. That is why we add 1 to the radius
2
2
  [ x  1] dx    ( x 4  2 x 2  1)dx
2
0
about x = -1
2
0
In this formula, x is the radius of the shaded region. When we rotate about the line x =
-1, we have the increased radius by 1.
2

2

2 ( x  1)[ x 2 ]dx  2 ( x 3  x 2 )dx
0
0
Trapeziodal Rule
Used to approximate area under a curve using trapezoids.
Area 
ba
 f ( x0 )  2 f ( x1 )  2 f ( x2 )  ....  2 f ( xn1 )  f ( xn )
2n
where n is the number of subdivisions
Example:
f(x) = x2+1. Approximate the area under the curve from [0,2] using trapezoidal rule
with 4 subdivisions
a0
b2
n4
A
20
 f (0)  2 f (.5)  2 f (1)  2 f (1.5)  f (2)
8
1
1  2(5 / 4)  2(2)  2(13 / 4)  5
4
1
76
 (76 / 4) 
 4.750
4
16

Riemann Sums
Used to approximate area under the curve using rectangles.
a) Inscribed rectangles: all of the rectangles are below the curve
Example: f(x) = x2 + 1 from [0,2] using 4 subdivisions
(Find the area of each rectangle and add together)
I=.5(1) II=.5(5/4) III=.5(2) IV=.5(13/4)
Total Area = 3.750
b) Circumscribed Rectangles: all rectangles reach above the curve
Example: f(x) = x2 + 1 from [0,2] using 4 subdivisions
I=.5(5/4) II=.5(2)
Total Area = 5.750
III=.5(13/4)
IV=.5(5)
Reading a Graph
When Given the Graph of f’(x)
Make a number line because you are more familiar with number line.
This is the graph of f’(x).
Make a number line.
- Where f’(x) = 0 (x-int)
is where there are
possible max and mins.
- Signs are based on if the
graph is above or below
the x-axis (determines
increasing and
decreasing)
min
x = -4, 3
max
x=0
increasing
(-4,0) (3,6]
decreasing
[-6,-4] (0,3)
To read the f’(x) and figure out inflection points and concavity, you read f’(x) the same way
you look at f(x) (the original equation) to figure out max, min, increasing and decreasing.
For the graph on the previous page:
Signs are
determined by if
f’(x) is increasing
(+) and decreasing
(-)
inflection pt
x = -2, 2, 4, 5
concave up
(-6,-2) (2,4) (5,6)
concave down
(-2,2) (4,5)
Original Source:
http://www.geocities.com/Area51/Stargate/5847/index.html