Untitled Flashcards Set

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