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AP Calculus

Exam Study Guide

Derivatives
Constant Rule

d
[c ]  0
dx

Constant Multiple Rule

d
[cu ]  cu '
dx

d
d
[ f  g ]  f ' g '
[ x n ]  nx n 1
Sum and Difference Rules
dx
dx
d
d  f 
gf ' fg '
[ fg ]  gf ' fg '
Product Rule
Quotient Rule
g
dx
dx  
g2
1
d
1 '
[ f (u )]  f '(u )  u ' Inverse  f  ( x)  f ' ( f 1 ( x))
Chain Rule
dx
Power Rule

d
[sin x]  cos x
dx

d
[cos x]   sin x
dx
d
[eu ]  e u  u '
dx
d
[arc sin u ] 
dx

u'
1 u2

d
[sec x]  sec x tan x
dx

d
[tan x]  sec 2 x
dx

d
d
[cot x]   csc 2 x
[csc x]   csc x cot x
dx
dx
d
1
d
[ln u ]   u '
[a u ]  (ln a)a u  u '
dx
u
dx
u'
d
u ' d
[arctan u ] 
[arccos u ] 
2 dx
1 u2
dx
1 u

Definition

Notation

If y= f (x) then the derivative is defined to be
f ( x  h)  f ( x) or
f '( x)  lim

If y= f (x) then all of the following are equivalent notations
for the derivative
...


f '(a)  y ' x a 

df
dx


x a

dy
dx

 Df (a)
x a

Interpretation of the Derivative
If y= f (x) then,
1
...
This formula can also be written as y  y1  m( x  x1 ) where m  f '( x1 )

2
...
This is sometimes referred to as
the slope of the curve f at x  a
...

 x 
Differentiability
A function is not differentiable at a point x=a (meaning the derivative does not exist at x=a) if the function
1
...
) 2
...
) has a vertical tangent has a vertical tangent at x=a (slope of the tangent
is undefined)
...
If the function is
continuous, it may be differentiable but that is not certain
...
) Slope

m

y2  y1 y

x2  x1 x

Facts about motion…
1
...
) direction change v(t )  0
3
...
If v(t) and a(t) have the same sign, speed is

2
...
For vertical lines,
slope is undefined and the equation is

x  a
...
) Parallel lines have the same slope
...


m1  

increasing
...
) Moving right or up – v(t) >0 Moving left or down – v(t)<0
(also state v(t)=0)
5
...

6
...
Units for acceleration
- distance/ time2
Derivative

Integral

Position

1

...
If f '(c) = 0 or f ' is undefined at c, then c is a critical
number of f
...
If f is differentiable on I , except possibly at c , then f (c) can be classified as follows
...
) If f '(c) changes from negative to positive at c , then f (c) is a relative minimum of f
...
) If f '(c) changes from positive to negative at c , then f (c) is a relative maximum of f
...

1 ) If f "(c)  0 , then f (c) is a relative minimum
...
) Concave Up – tangent line underneath graph
...
) Inflection point – f”must change sign
...




1 b
Average value of a function=
f ( x)dx
b  a a



Net Change in a function=

b



b

 ( f ( x)  g ( x))dx
a

a

 0 dx  0  C  k

  f ( x)  g ( x) dx  

b

 r
a

b

x

2

Total Distance Traveled:

v(t )dt

dx  kx  C

f ( x) dx   g ( x) dx

n 1

a

v(t ) dt

 kf ( x) dx  k  f ( x) dx
 f ( x)dx   f ( x)dx
a

b

a

 cot udu  ln sin u  C



 sec
 csc

2

x dx  tan x  C

2

x dx   cot x  C

 e du  e
u

u

C

1
u
du  arcsin    C
2
a u
a
2

Important Area Formulas

Approximating Integrals

Circle
Square

1
h(b1  b2 )
2
A   r2
A  (si de)2

Rectangle

A   length  width 

Triangle

1
A  bh
2
3
2
A
 side 
4

Equilateral Triangle



b

1
 u du  ln u  C Does not apply for exponents other than -1

1
1
u
du  arctan    C
2
u
a
a

Trapezoid

 r 2 )dx

(rectangle perpendicular to the axis of
rotation)

tan udu   ln cos u  C  ln sec u  C

a

2

a

 x dx  n  1  C , n  1
 cos x dx  sin x  C
 sec x tan x dx  sec x  C
 csc x cot x dx   csc x  C
 sin x dx   cos x  C



dx

  (R

b

n

2

(rectangle perpendicular to the axis of
rotation)
Washer

V   (area of cross section)dx

a

Second Fundamental Theorem of Calculus:

Disk

b



f ( x)  F (b)  F (a) (Definite Integral)

Version)
Volumes of solids of revolution

b

Volumes of solids with perpendicular cross sections—
no rotation

b

a

d g ( x)
f (t )  f ( g ( x)) g '( x) (Chain Rule
dx a

f ( x)  g ( x) is top-bottom or right-left
...
) Riemann Sums (Left / Right/ Midpoint)
2
...


Limits, Continuity and other Miscellaneous Info
Types of Limits:
Limit at Infinity : We say lim f ( x)  L if we can make
x 

f (x) as close to L as we want by taking x large enough
and positive
...

x 

Infinite Limit : We say lim f ( x)   if we
x a

can make f (x) arbitrarily large (and positive) by taking
x sufficiently close to a (on either side of a) without
letting x=a
...

Right hand limit : lim f ( x)  L This has the same

x a

definition as the limit except it requires x>a
...
This has the same

x a

Relationship between the limit and one-sided
limits

lim f ( x)  L  lim f ( x)  lim f ( x)  L


x a

x a

x a

lim f ( x)  lim f ( x)  lim f ( x)


x a 

x a

x a

Evaluation Techniques for Limits
i
...
) Rationalize Numerator/Denominator
iii
...
) Piecewise Function – 2 one-sided
limits
Limits at Infinity
To compute lim
x 

p ( x)
, consider dominance (which
q ( x)

one is increasing faster? Use degree for
polynomials
...
) BOT – Big on Top - there is no limit
...
) BOB – Big on Bottom - the limit is zero
3
...


definition as the limit except it requires x ...

i
...
) Find f(a) and f(b)
2
...
) Conclude “Therefore, by IVT, a  c  b
”

x a

iii

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Title: Calculus Exam Study Guide
Description: This is a great study guide for AP calculus students. I got a 5 on the exam with some review of this sheet. It includes formulas and basic explanations.

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