Stuff you MUST know Cold for the AP Calculus Exam

Curve sketching and analysisy = f(x) must be continuous at each: critical point: = 0 or undefined.

local minimum: goes (–,0,+) or (–,und,+) or > 0 at stationary pt

local maximum: goes (+,0,–) or (+,und,–) or < 0 at stationary pt

point of inflection: concavity changes

goes from (+,0,–), (–,0,+),(+,und,–), or (–,und,+)

dy

dx

dy

dx

2

2

d y

dx2

2

d y

dx

2

2

d y

dx

dy

dx

Basic Derivatives

1n ndx nx

dx

sin cosd

x xdx

cos sind

x xdx

2tan secd

x xdx

2cot cscd

x xdx

sec sec tand

x x xdx

csc csc cotd

x x xdx

1ln

d duu

dx u dx

u ud due e

dx dx

x xe dx e

Basic Integralssin cosx dx xcos sinx dx x 2sec tanx dx x

sec tan secx x dx x

1lndx x

x

ln

xx a

a dxa

Some more handy integrals 1

1n na

ax dx x Cn

tan ln sec

ln cos

sec ln sec tan

x dx x C

x C

x dx x x C

More Derivatives 1

2

1sin

1

d duu

dx dxu

1

2

1cos

1

dx

dx x

12

1tan

1

dx

dx x

12

1cot

1

dx

dx x

1

2

1sec

1

dx

dx x x

1

2

1csc

1

dx

dx x x

lnx xda a a

dx

1log

lna

dx

dx x a

Recall “change of base”ln

loglna

xx

a

Differentiation Rules Chain Rule

( ) '( )d du dy dy du

f u f udx dx dx du

Rx

Od

Product Rule

( ) ' 'd du dv

uv v u OR udx

vdx dx

uv

Quotient Rule

2 2

' 'du dvdx dxv u

Od u

dx v v

u v uv

vR

The Fundamental Theorem of Calculus

Other part ofthe FTC

( )

( )

( ( )) '( ) ( ( ))

( )

'( )

b x

a x

f b x b x f a x

f t dtd

da x

x

( ) ( ) ( )

where '( ) ( )

b

af x dx F b F a

F x f x

Intermediate Value Theorem

. Mean Value Theorem

( ) ( )'( )

f b f af c

b a

.

If the function f(x) is continuous on [a, b], AND the first derivative exists on the interval (a, b), then there is at least one number x = c in (a, b) such that

If the function f(x) is continuous on [a, b], and y is a number between f(a) and f(b), then there exists at least one number x = c in the open interval (a, b) such that f(c) = y.

( ) ( )'( )

f b f af c

b a

If the function f(x) is continuous on [a, b], AND the first derivative exists on the interval (a, b), then there is at least one number x = c in (a, b) such that

If the function f(x) is continuous on [a, b], AND the first derivative exists on the interval (a, b), AND f(a) = f(b),then there is at least one number x = c in (a, b) such that f '(c) = 0.

Mean Value Theorem & Rolle’s Theorem

If the function f(x) is continuous on [a, b], then f has both an absolute maximum and an absolute minimum on [a,b]

The absolute extremes occur eitherat the critical pointsor at the endpoints.

Extreme Value Theorem

Approximation Methods for Integration

Trapezoidal Rule

10 12

1

( ) [ ( ) 2 ( ) ...

2 ( ) ( )]

b

a

n n

b anf x dx f x f x

f x f x

Riemann Sum 1

n

kk

f c x

LRAM when ck is a LEFT endpointRRAM when ck is a RIGHT endpointMRAM when ck is a MIDPOINT

Theorem of the Mean Valuei.e. AVERAGE VALUE If the function f(x) is continuous on [a, b] and the

first derivative exists on the interval (a, b), then there exists a number x = c on (a, b) such that

This value f(c) is the “average value” of the function on the interval [a, b].

( )( )

( )

b

af x dx

f cb a

Solids of Revolution and friends Disk Method

2( )

x b

x aV R x dx

2 2( ) ( )

b

aV R x r x dx

( )b

aV Area x dx

Washer Method

General volume equation (not rotated)

21 '( )

b

aL f x dx

Arc Length

2 2b

aL x t y t dt

2

2b

a

drL r d

d

Distance, Velocity, and Acceleration

velocity =

d

dt

(position) d

dt

(velocity)

,dx dy

dt dt

speed = 2 2( ') ( ')v x y

displacement = f

o

t

tv dt

final time

initial time

2 2

distance =

( ') ( ')

f

o

t

t

v dt

x y dt

average velocity = final position initial position

total time

x

t

acceleration =

velocity vector =

Values of Trigonometric Functions for Common Angles

1

23

2

3

3

2

2

2

2

3

2

3

0–10π,180°

∞01 ,90°

,60°

1 ,45°

,30°

0100°

tan θcos θsin θθ

1

26

4

3

2

π/3 = 60° π/6 = 30°

sine

cosine

Trig IdentitiesDouble Argument

sin 2 2sin cosx x x2 2 2cos2 cos sin 1 2sinx x x x

2 1sin 1 cos2

2x x

2 1cos 1 cos2

2x x

Pythagorean2 2sin cos 1x x

sine

cosine

Slope – Parametric & PolarParametric equation Given a x(t) and a y(t) the slope is

Polar Slope of r(θ) at a given θ is

dydt

dxdt

dy

dx

/

/

sin

cos

dd

dd

dy dy d

dx dx d

r

r

What is y equal to in terms of r and θ ? x?

Polar Curve

For a polar curve r(θ), the AREA inside a “leaf” is

(Because instead of infinitesimally small rectangles, use triangles)

where θ1 and θ2 are the “first” two times that r = 0.

2

1

212 r d

1

2A bh

r d

r

21 1

2 2dA rd r r d

and

We know arc length l = r θ

l’Hôpital’s Rule

If

then

( ) 0or

( ) 0

f a

g b

( ) '( )lim lim

( ) '( )x a x a

f x f x

g x g x

Other Indeterminate forms:

0

0 01 0

Write as a ratio

Use Logs

Integration by Parts

Antiderivative product rule(Use u = LIPET)e.g.

udv uv vdu

( ) ' 'd du dv

uv v u OR udx

vdx dx

uv We know the product rule

( )

( )

( )

d uv v du u dv

u dv d uv v du

u dv d uv v du

ln x dxlnx x x C

Let u = ln x dv = dx du = dx v = x1

x

LIPET

Logarithm

Inverse

Polynomial ExponentialTrig

Maclaurin SeriesA Taylor Series about x = 0 is called

Maclaurin.

If the function f is “smooth” at x = a, then it can be approximated by the nth degree polynomial

2 3

12! 3!

x x xe x

2 4

cos 12! 4!

x xx

3 5

sin3! 5!

x xx x

2 311

1x x x

x

2 3 4

ln( 1)2 3 4

x x xx x

Taylor Series

2

( )

( ) ( ) '( )( )

''( )( )

2!

( )( ) .

!

nn

f x f a f a x a

f ax a

f ax a

n