AP Calculus Study Sheet
Limits
Graphically
Limit from the left of f ( x ) as x  a
lim f ( x )
x4
Notation:
lim f ( x )
Limit from the right of f ( x ) as x  a
lim f ( x )
lim f ( x)
x4
x 4
x 0
Notation:
Theorems:
lim x a f ( x)  F and lim x a g ( x)  G
Limit Evaluation Techniques
Factor and cancel
lim x a ( f ( x)  g ( x)) 
lim
lim x a ( f ( x)  g ( x)) 
x 2
x 2  4 x  12
x 2  2x
Rationalize Numerator/Denominator
lim
x 9
3 x
x 2  81
lim x a ( f ( x)  g ( x)) 
Rational Expressions (LCD)
lim x a ( f ( x)) n 
1
1

lim x  h x
x 0
h
Limits at Infinity
lim x a
f ( x)

g ( x)
3x 2  5 x  7
x  8  2 x  6 x 2
lim
L’Hopital’s Rule
sin x

x
1  cos x
lim x 0

x
lim x 0
f ( x) 0
f ( x)  
 or lim

x

a
f ( x) 0
f ( x)  
f ( x) f ' ( x)
then lim

, a is a number,  or - 
x a f ( x)
g ' ( x)
If
lim
x a
Definition of Continuity:
Vertical Asymptotes
A function is continuous at the point x=a if and only if:
1.
Horizontal Asymptotes:
2.
3.
Derivatives
Definition of Derivative
Alternate Form of Def. of Derivative
d
 f ( x)  
dx
d
 f ( x)  at x  a
dx
Notatation of the Derivative
If y=f(x), then all of the notations for a
derivative are:
Interpretation of the Derivative
If y = f(x)
1. m=f’(x) is ____________
2. f’(a) is also called the _______________
Equation of the Tangent line is given by:
Average Rate of Change:
Equation of the Normal line is given by:
Derivative properties and formulas
Where u is a function of x and a is a constant
d
c  f (x) =
dx
d n
(x ) 
dx
d
(sin u ) 
dx
d
(cos u ) 
dx
d
(tan u ) 
dx
d
(cot u ) 
dx
d
(sec u ) 
dx
d
(csc u ) 
dx
d
 f ( x )  g ( x ) =
dx
d
(ln u ) 
dx
d u
(e ) 
dx
d
(sin 1 u ) 
dx
d
(cos 1 u ) 
dx
d
(tan 1 u ) 
dx
d
(cot 1 u ) 
dx
d u
(a ) 
dx
d
(log a u ) 
dx
Product Rule
Quotient Rule
Chain Rule
d
 f ( x ) g ( x )
dx
d  f ( x) 
dx  g ( x) 
d
 f ( g ( x))
dx
Higher order derivatives
The second derivative is denoted as _____ or ______ and is found by:
The third derivative is denoted as _____ or _____ and is found by :
Implicit Differentiation
Find
dy
for 2 x  3xy  2 y  cos y  3
dx
Curve analysis:
Critical points:
Absolute Extrema Steps:
f(x) is Increasing:
Relative maximum:
f(x) is Decreasing:
Relative minimum:
f(x) is concave up:
f(x) is concave down:
Second Derivative Test:
f(x) has a point of inflection:
Intermediate Value Theorem
Mean Value Theorem
Conditions:
1.
2.
Formula:
Related Rates:
Steps:
EX1: A 15 foot ladder is resting along a wall.
The bottom is initially 10 ft away from the wall
and is being pushed towards the wall at ¼
ft/sec. How fast is the top moving after 12 sec?
EX2: Two people are 50 feet apart when one
starts walking north. The angle between 
them changes at a rate of 0.01 rad/min. What
rate is the distance between them changing
when   0.5 rad?
Optimization
Steps:
EX: A farmer wants to enclose a rectangular field with 500 feet of fence. One side will be along a river.
Determine the dimensions that will maximize the area.
Integration Properties and Formulas
Where k , a and n are constants:
 kf ( x)dx =
 sec x tan xdx =
e
  f ( x)  g ( x)dx =
 csc x cot xdx =
 tan xdx =
 kdx =
b

n
dx =
( n  1)
b
dx =
____
 f ( x)dx
f ( x)dx 
a
x
x
____
 cot xdx =
c

f ( x)dx   f ( x)dx 
a
b
 sec xdx =
a
 sin xdx =
 f ( x)dx 
 csc xdx =
a
b
________   f ( x)dx  ________
 cos xdx =
a
If f ( x)  g ( x) , then
b
 sec
 csc
2
xdx =

2
xdx =
dx
x =

b
f ( x)dx ______  g ( x)dx
a
a
a
u
du 
du
a  u2
2
du
u
a
=
u2  a2
2
du
=
 u2
=
Riemann Sums
Left:
Right:
Midpoint
Trapezoidal Approximation
Fundamental Theorem of Calculus
Part 1:
b
 f ' ( x)dx 
a
Part 2:
d x
f  t  dt 
dx  a
Or
If F (x ) is the antiderivative of f (x )
b
 f ( x)dx 
a
d a
f  t  dt 
dx  x
d u x 
f  t  dt 
dx  a
Average Value of a Function:
Integration Techniques
U-Substitution
Steps for Indefinite Integrals:
Steps for Indefinite Integrals:
Differential Equations
Steps for solving.
Slope Fields
What does it look like?
How do I find the slopes?
General solution:
Particular Solution:
Distance, Velocity, and Acceleration
s(t) is the position function,  x(t ), y (t )  is the position in parametric
velocity =
acceleration =
velocity vector =
acceleration vector =
speed (rectangular and parametric) =
displacement =
distance (rectangular and parametric) =
average velocity =
Population Density
Linear
Circular
Area
Net Area is given by:
Area above x-axis
Area below x-axis
Area between curves
A=
A=
A=
Volume of known cross sections
Formula:
Area formulas:
Squares
Equilateral Triangles
Isosceles Right Triangles
Rectangles
Semicircles
Semiellipses
Volume of Solids of Revolution
Discs Formula:
Washers Formula:
Radius:
Outer Radius:
Outer radius:
Inner Radius:
Inner Radius:
Eulers Method
What is it?
Steps:
1.
2.
3.
Logistic Equations
General Equations
Carrying Capacity
Fastest Growth:
Partial Fractions:
EX:
x
2
dx

 4 x  12
Arc Length
Formula:
Integration by Parts
Formula:
Improper Integrals



a

f ( x)dx 
 f ( x)dx 
f ( x)dx 

a

Parametrics and Vectors
dy

dx
d2y

dx 2

dx
dt
dy
y t 
dt
x t 

position vector at any time t:
Horizontal Tangent lines:
velocity vector at any time t:
Vertical Tangent lines:
acceleration vector at any time:
Speed of the particle or the magnitude (length) of the velocity vector=
Length of the arc from t  a to t  b
Distance traveled by the particle from t  a to t  b
Polar Equations
Rectangular coordinates are in the form  x, y  .
Area of Polar Curve:
Polar coordinates are in the form  r ,   .
x  r cos 
r  x2  y 2
y  r sin 
tan  
y
x
dy

dx
Series
Geometric Series test
p-Series test
Integral Test
nth Term Test
Direct Comparison test
Limit Comparison test
Ratio Test
Root Test
Alternating Series Test
Telescoping Series Test
Taylor Polynomials
Formula:
Power Series Interval of Convergence
Steps:
Taylor Series:
Formula:
Special Maclaurin Series:
ex
sin( x )
cos(x)
Alternating Series Remainder
Geometric Power Series:
Lagrange Remainder: