Stuff you MUST know Cold for the AP Calculus Examin the morning of Wednesday, May 7, 2008.

AP Physics & CalculusCovenant Christian High School7525 West 21st StreetIndianapolis, IN 46214Phone: 317/390.0202 x104Email: seanbird@covenantchristian.org Website: http://cs3.covenantchristian.org/bird Psalm 111:2

Sean Bird

Updated April 24, 2009

Curve sketching and analysisy = f(x) must be continuous at each: critical point: = 0 or undefined. And don’t forget endpoints

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

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

point of inflection: concavity changes

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

dydx

dydx

2

2

d ydx

2

2

d ydx

2

2

d ydx

dydx

Basic Derivatives

1n nd x nxdx

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

1lnd duudx u dx

u ud due edx dx

Basic Integrals

1n nd x nxdx

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

1lnd duudx u dx

u ud due edx dx

Some more handy integrals 1

1n naax dx x C

n

tan ln sec

ln cos

sec ln sec tan

x dx x C

x C

x dx x x C

More Derivatives 1

2

1sin1

d duudx dxu

1

2

1cos1

d xdx x

12

1tan1

d xdx x

12

1cot1

d xdx x

1

2

1sec1

d xdx x x

1

2

1csc1

d xdx x x

lnx xd a a adx

1loglna

d xdx x a

Recall “change of base”lnloglna

xxa

Differentiation Rules Chain Rule

( ) '( )d du dy dy duf u f udx dx dx du

Rx

Od

Product Rule

( ) ' 'd du dvuv v u OR udx

vdx dx

uv

Quotient Rule

2 2

' 'du dvdx dxv u

Od udx v v

u v uvv

R

The Fundamental Theorem of Calculus

Corollary to FTC

( )

( )

( ( )) '( ) ( ( ))

( )

'( )

b x

a x

f b x b x f a x

f t dtdd

a xx

( ) ( ) ( )

where '( ) ( )

b

af x dx F b F a

F x f x

Intermediate Value Theorem

. Mean Value Theorem

( ) ( )'( ) f b f af cb 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 cb 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

Approximation Methods for IntegrationTrapezoidal Rule

10 12

1

( ) [ ( ) 2 ( ) ...

2 ( ) ( )]

b

a

n n

b anf x dx f x f x

f x f x

Simpson’s Rule

10 1 23

2 1

( )

[ ( ) 4 ( ) 2 ( ) ...2 ( ) 4 ( ) ( )]

b

a

n n n

f x dx

f x f x f xf x f x

xf x

Simpson only works forEven sub intervals (odd data points) 1/3 (1 + 4 + 2 + 4 + 1 )

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 *bc topic

Distance, Velocity, and Acceleration

velocity =

ddt

(position) ddt

(velocity)

,dx dydt 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

xt

acceleration =

*velocity vector =

*bc topic

Values of Trigonometric Functions for Common Angles

12

32

33

22

22

32

3

0–10π,180°

∞01 ,90°

,60°

4/33/54/553°

1 ,45°

3/44/53/537°

,30°

0100°

tan θcos θsin θθ

126

4

3

2

π/3 = 60° π/6 = 30°

sine

cosine

Trig IdentitiesDouble Argument

sisi n c2n 2 osx xx

2 2 2cos2 cos sin 1 2sinx x x x

Trig IdentitiesDouble Argument

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

2 1sin 1 cos22

x x

2 1cos 1 cos22

x 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

dydtdxdt

dydx

//

sincos

dd

dd

dy dy ddx 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

12

A 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

( ) 0 or( ) 0

f ag b

( ) '( )lim lim( ) '( )x a x a

f x f xg x g x

Integration by Parts

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

udv uv vdu

( ) ' 'd du dvuv v u OR udx

vdx dx

uv We know the product rule

( )( )

( )

d uv v du u dvu 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 31 11

x x xx

2 3 4

ln( 1)2 3 4x x xx x

Taylor Series

2

( )

( ) ( ) '( )( )''( ) ( )2!

( ) ( ) .!

nn

f x f a f a x af a x a

f a x an