AP Calculus AB
Formulas & Justifications
1
Limits at Infinity:
To find lim ( )
x
f x
→±∞
think
Top Heavy ⇒limit is ±∞
Bottom Heavy ⇒limit is 0
Equal ⇒limit is ratio of coefficients
2
Limits with Infinity (at vertical asymptotes):
When finding a one-sided limit at a vertical asymptote,
the answer is either ±∞.
3
Justifying that a function is continuous at a point:
f is continuous at c iff:
1. f c( ) is defined
2. lim ( )
x c
f x
→
exists
3. f c( ) = lim ( )
x c
f x
→
4
Definition of the Derivative:
0 0
( ) ( ) ( ) ( ) '( ) lim lim
h x
f x h f x f x x f x f x
→ ∆ → h x
+ − + ∆ −
= =
∆
( ) ( ) '( ) limx a
f x f a f a
→ x a
−
=
−
(Alternate form for a derivative at a given value.)
5
Justifying that a derivative exists at a point, c :
Show algebraically that lim '( ) lim '( )
x c x c
f x f x
→ → − +
= .
6
Average Rate of Change of f on [a, b]:
a.r.c. =
f b f a ( ) ( )
b a
−
−
(algebra slope of y
x
∆
∆
) (slope of secant line)
7
Instantaneous Rate of Change of f at a :
f a '( ) (derivative at the given value) (slope of tangent line)
AB Calculus Formulas & Justifications - 2 -
8
Power Rule:
d n n 1
x nx
dx
−
=
9
Common Derivatives to Remember:
2
d 1 1
dx x x
−
=
1
2
d
x
dx x
=
10
Trig Function Derivatives:
[ ] sin cos d
x x
dx
=
[ ] cos sin d
x x
dx
= −
[ ] 2
tan sec
d
x x
dx
=
[ ] 2
cot csc
d
x x
dx
= −
[ ] sec sec tan
d
x x x
dx
= [ ] csc csc cot
d
x x x
dx
= −
11
Derivatives of Inverse Trig Functions:
2
1
[arcsin ]
1
d
x
dx x
=
−
2
1
[arccos ]
1
d
x
dx x
−
=
−
2
1
[arctan ]
1
d
x
dx x
=
+
2
1
[ cot ]
1
d
arc x
dx x
−
=
+
2
1
[ sec ]
1
d
arc x
dx x x
=
−
2
1
[ csc ]
1
d
arc x
dx x x
−
=
−
12
Derivatives of Exponential and Logarithmic Functions:
1
[ln ] , 0 d
x x
dx x
= >
1
[log ]
ln a
d
x
dx x a
=
[ ] d x x
e e
dx
=
[ ] ln d x x
a a a
dx
=
13
Justifications for horizontal tangent lines:
f x( ) has horizontal tangents when 0
dy
dx
= .
AB Calculus Formulas & Justifications - 3 -
14
Chain Rule:
dy dy du
dx du dx
= •
[ ] ( ( )) '( ( )) '( ) d
f g x f g x g x
dx
= •
15
Product Rule:
[ ] ( ) ( ) ( ) '( ) ( ) '( ) d
f x g x f x g x g x f x
dx
= +
16
Quotient Rule:
[ ]2
( ) ( ) '( ) ( ) '( )
( ) ( )
d f x g x f x f x g x
dx g x g x
−
=
17
Derivatives of Inverse Functions:
The derivative of an inverse function is the reciprocal of the derivative of the
original function at the “matching” point.
If (a, b) is on f x( ) , then (b, a) is on 1
f x( ) −
and 1 1
( )'( )
'( )
f b
f a
−
= .
18
Justifications for horizontal tangent lines:
f x( ) has vertical tangents when dy
dx
is undefined.
19
Justifications for Particle Motion:
Particle is moving right/up because v t( ) 0 > (positive).
Particle is moving left/down because v t( ) 0 < (negative).
Particle is speeding up (|velocity| is getting bigger) because v t( ) and a t( ) have
same sign.
Particle is slowing down (|velocity| is getting smaller) because v t( ) and a t( )
have different signs.
20
Intermediate Value Theorem:
If f is continuous on [a, b] and k is any number between f a( ) and f b( ), then
there is at least one number c between a and b such that f c k ( ) = .
21
Extreme Value Theorem:
If f is continuous on the closed interval [a, b], then f has both a minimum
and a maximum on the closed interval [a, b].
AB Calculus Formulas & Justifications - 4 -
22
Justification for an Absolute Extrema.
1. Find critical numbers.
2. Identify endpoints.
3. Find f ( ) critical numbers and f ( ) endpoints .
4. Determine absolute max/min values by comparing the y-values. State in a
sentence.
23
Mean Value Theorem:
If f is continuous on [a, b] and differentiable on (a, b) then there exists a
number c on (a, b) such that ( ) ( ) '( ) f b f a f c
b a
−
=
−
.
(Calculus slope = Algebra Slope)
24
Rolle’s Theorem:
If f is continuous on [a, b] and differentiable on (a, b) and if f a f b ( ) ( ) = ,
then there exists a number c on (a, b) such that f c '( ) 0 = .
25
Justification for a Critical Number:
x c = is a critical number because f x '( ) 0 = or f x '( ) is undefined.
26
Justification for Increasing/Decreasing Intervals:
Inc: f x( ) is increasing on [____, ____] b/c f x '( ) 0 > .
Dec: f x( ) is decreasing on [____, ____] b/c f x '( ) 0 < .
27
Justification for a Relative Max/Min Using 1st Derivative Test:
Local Max: f x '( ) changes from + to -.
Local Min: f x '( ) changes from - to +.
28
Justification for Relative Max/Min Using 2nd Derivative Test:
Local Max: f c '( ) 0 = (or und) and f x ''( ) 0 < .
Local Min: f c '( ) 0 = (or und) and f x ''( ) 0 > .
AB Calculus Formulas & Justifications - 5 -
29
Justification for a Point of Inflection:
Using 2nd derivative: f x ''( ) 0 = (or dne) AND f x ''( ) changes sign.
Using 1st derivative: f x ''( ) 0 = (or dne) AND slope of f x '( ) changes sign.
30
Justification for Concave Up/Concave Down:
Concave Up: f x( ) is concave up on (____, ____) because f x ''( ) 0 > .
Concave Down: f x( ) is concave down on (____, ____) because f x ''( ) 0 < .
31
Justifications for linear approximation estimates:
A linear approximation (tangent line) is an overestimate if the curve is
concave down.
A linear approximation (tangent line) is an underestimate if the curve is
concave up.
32
Integration Rules:
1 1
1
n n x dx x C
n
+
= + ∫ +
cos sin xdx x C = + ∫
( ) ( ) 1
cos sin kx dx kx C
k
= + ∫
sin cos xdx x C = − + ∫
( ) ( ) 1
sin cos kx dx kx C
k
= − + ∫
2
sec tan xdx x C = + ∫
2
csc cot xdx x C = − + ∫
sec tan sec x xdx x C = + ∫
csc cot csc x xdx x C = − + ∫
1
dx x C ln
x
= + ∫
tan ln cos xdx x C = − + ∫
x x e dx e C = + ∫
kx kx 1
e dx e C
k
= + ∫
2 2
1 1
arctan
x
dx C
x a a a
= + + ∫
AB Calculus Formulas & Justifications - 6 -
33
Justifications for Reimann Sums:
Left-Riemann Sums:
The sum is an overestimate if the curve is decreasing.
The sum is an underestimate if the curve is increasing.
Right-Riemann Sums:
The sum is an overestimate if the curve is increasing.
The sum is an underestimate if the curve is decreasing.
34
First Fundamental Theorem of Calculus:
'( ) ( ) ( )
b
a
f x dx f b f a = − ∫
(Finds the signed area between a curve and the x-axis)
35
Properties of Integrals:
( ) ( ) ( ) ( )
b b b
a a a
f x g x dx f x dx g x dx + = + ∫ ∫ ∫
( ) ( ) ( ) ( )
b b b
a a a
f x g x dx f x dx g x dx − = − ∫ ∫ ∫
( ) ( )
b b
a a
cf x dx c f x dx =
∫ ∫
( ) ( )
b a
a b
f x dx f x dx = − ∫ ∫
( ) 0
a
a
f x dx =
∫
36
Average Value of a Function:
1
( )
b
avg
a
f f x dx
b a
=
−
∫
37
Second Fundamental Theorem of Calculus:
( ) ( )
x
a
d
f t dt f x
dx
= ∫
( )
( ) ( ( )) '( )
g x
a
d
f t dt f g x g x
dx
= • ∫
AB Calculus Formulas & Justifications - 7 -
38
“Net Change” Theorem:
( )
b
a
f x dx ∫
represents the “net change” in the function f from time a to b.
39
Finding Total Amount:
( ) ( ) '( )
b
a
f b f a f x dx = + ∫
(want = have + integral)
40
Steps for Solving Differential Equations:
“Find a solution (or solve) the separable differentiable equation...”
1. Separate the variables
2. Integrate each side
3. Make sure to put C on side with independent variable (normally x)
4. Plug in initial condition and solve for C (if given)
5. Solve for the dependent variable (normally y)
41
Exponential Growth and Decay:
“The rate of change of a quantity is directly proportional to that quantity”
Gives the differential equation: dy ky
dt
=
Which can be solved to yield: kt y Ce =
AB Calculus Formulas & Justifications - 8 -
42
Particle Motion Formulas:
Velocity: v t s t ( ) '( ) =
Acceleration: a t v t s t ( ) '( ) ''( ) = =
Speed: speed = v t( )
Average Velocity: (given s t( )) s b s a ( ) ( )
b a
−
−
(given v t( )) 1
( )
b
a
v t dt
b a −
∫
Average Acceleration: (given v t( )) v b v a ( ) ( )
b a
−
−
(given a t( )) 1
( )
b
a
a t dt
b a −
∫
Displacement: ( )
b
a
v t dt ∫
Total Distance: ( )
b
a
v t dt ∫
Position at b: ( ) ( ) ( )
b
a
s b s a v t dt = + ∫
43
Areas in a Plane:
Perpendicular to x-axis: [ ] ( ) ( )
b
a
f x g x dx − ∫
f x( ) is top curve, g x( ) is bottom curve, a and b are x-coordinates of
point of intersection
Perpendicular to y-axis: [ ] ( ) ( )
b
a
f y g y dy − ∫
f y( ) is right curve, g y( ) is left curve, a and b are y-coordinates of
point of intersection
44
Steps to Finding Volume:
Volume = Area ∫
1. decide on whether it’s a dx or dy
2. find a formula for the area in terms of x or y
3. find the limits (making sure they match x or y)
4. integrate and evaluate
AB Calculus Formulas & Justifications