typed by Sean Bird, Covenant Christian High School

updated May 2014

AP CALCULUS

Stuff you MUST know Cold * means topic only on BC

Curve sketching and analysis

y = f(x) must be continuous at each:

critical point:

dy

dx

= 0 or undefined

local minimum:

dy

dx

goes (–,0,+) or (–,und,+) or

2

2

d y

dx

>0

local maximum:

dy

dx

goes (+,0,–) or (+,und,–) or

2

2

d y

dx

<0

point of inflection: concavity changes

2

2

d y

dx

goes from (+,0,–), (–,0,+),

Differentiation Rules

Chain Rule

( ) ‘( )

d du dy dy du

f u f u

dx dx dx du

R

x

O

d

Product Rule

( ) ‘ ‘

d du dv

uv v u OR u

dx

v

dx dx

uv

Quotient Rule

2 2

‘ ‘du dv

dx dxv u u v uv

O

d u

R

dx v vv

Approx. Methods for Integration

Trapezoidal Rule

1

0 12

1

( ) [ ( ) 2 ( ) …

2 ( ) ( )]

b

b a

na

n n

f x dx f x f x

f x f x

With data, do each trap separately

using ½ (f(x1)+f(x2)). It is an over

approximation if ( ) (concave up)

Riemann Sums are rectangles.

Left Riemann sums under approximate

when f(x) is increasing ( ( ) )…

Theorem of the Mean Value

i.e. AVERAGE VALUE

Basic Derivatives

1n nd

x nx

dx

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

1

ln

d du

u

dx u dx

u ud du

e e

dx dx

where u is a function of x,

and a is a constant.

“PLUS A CONSTANT” 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

( )

( )

( )

b

a

f x dx

f c

b a

This value f(c) is the “average value” of

the function on the interval [a, b].

The Fundamental Theorem of

Calculus

( ) ( ) ( )

where ‘( ) ( )

b

a

f x dx F b F a

F x f x

Corollary to FTC

( )

( )

( ( )) ‘( ) ( ( ))

( )

‘( )

b x

a x

f b x b x f a x

f t dt

d

d

a x

x

Solids of Revolution and friends

Disk Method

2

( )

x b

x a

V R x dx

Washer Method

2 2

( ) ( )

b

a

V R x r x dx

General volume equation (not rotated)

( )

b

a

V Area x dx

*Arc Length

2

1 ‘( )

b

a

L f x dx

2 2

‘( ) ‘( )

b

a

x t y t dt

Intermediate Value Theorem

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 .

More Derivatives

1

2 2

1

sin

d u du

dx a dxa u

1

2

1

cos

1

d

x

dx x

1

2 2

tan

d u a du

dx a dxa u

1

2

1

cot

1

d

x

dx x

1

2 2

sec

d u a du

dx a dxu u a

1

2

1

csc

1

d

x

dx x x

ln

u x u xd du

a a a

dx dx

1

log

ln

a

d

x

dx x a

Mean Value Theorem

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

( ) ( )

‘( )

f b f a

f c

b a

.

Distance, Velocity, and Acceleration

velocity = d

dt

(position)

acceleration = d

dt

(velocity)

*velocity vector = ,

dx dy

dt dt

speed = 2 2

( ‘) ( ‘)v x y *

displacement =

f

o

t

t

vdt

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

Rolle’s Theorem

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

‘( ) 0f c .

Derivative Formula for Inverses

df

dx df

dx

x f a

x a

1

1

( )

(+,und,–), or (–,und,+)

(and look out for endpoints)

BC TOPICS and important TRIG identities and values

l’Hôpital’s Rule

If

( ) 0

or

( ) 0

f a

g b

,

then

( ) ‘( )

lim lim

( ) ‘( )x a x a

f x f x

g x g x

Slope of a Parametric equation

Given a x(t) and a y(t) the slope is

dy

dt

dx

dt

dy

dx

Values of Trigonometric

Functions for Common Angles

θ sin θ cos θ tan θ

0° 0 1 0

6

1

2

3

2

3

3

4

2

2

2

2

1

3

3

2

1

2

3

2

1 0 “ ”

π 0 1 0

Know both the inverse trig and the trig

values. E.g. tan(π/4)=1 & tan-1

(1)= π/4

Euler’s Method

If given that ( , )dy

dx f x y and that

the solution passes through (xo, yo),

1 1 1

( )

( ) ( ) ( , )

o o

n n n n

y x y

y x y x f x y x

In other words:

new oldx x x

old old

new old

,x y

dy

y y x

dx

Polar Curve

For a polar curve r(θ), the

AREA inside a “leaf” is

2

1

2

1

2 r d

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

0.

The SLOPE of r(θ) at a given θ is

/

/

sin

cos

d

d

d

d

dy dy d

dx dx d

r

r

Integration by Parts

udv uv vdu

Ratio Test

The series

0

k

k

a

converges if

1

lim 1k

k

k

a

a

If the limit equal 1, you know nothing.

Trig Identities

Double Argument

sin2 2sin cosx x x

2 2 2

cos2 cos sin 1 2sinx x x x

2 1

cos 1 cos2

2

x x

Integral of Log

Use IBP and let u = ln x (Recall

u=LIPET)

ln lnxdx x x x C

Taylor Series

If the function f is “smooth” at x =

a, then it can be approximated by

the nth

degree polynomial

2

( )

( ) ( ) ‘( )( )

”( )

( )

2!

( )

( ) .

!

n

n

f x f a f a x a

f a

x a

f a

x a

n

Lagrange Error Bound

If ( )nP x is the nth

degree Taylor polynomial

of f(x) about c and ( 1)

( )n

f t M

for all t

between x and c, then

1

( ) ( )

1 !

n

n

M

f x P x x c

n

2 1

sin 1 cos2

2

x x

Pythagorean

2 2

sin cos 1x x

(others are easily derivable by

dividing by sin2

x or cos2

x)

2 2

2 2

1 tan sec

cot 1 csc

x x

x x

Reciprocal

1

sec cos sec 1

cos

x or x x

x

1

csc sin csc 1

sin

x or x x

x

Odd-Even

sin(–x) = – sin x (odd)

cos(–x) = cos x (even)

Some more handy INTEGRALS:

tan ln sec

ln cos

sec ln sec tan

xdx x C

x C

xdx x x C

Maclaurin Series

A Taylor Series about x = 0 is

called Maclaurin.

2 3

1

2! 3!

x x x

e x

2 4

cos 1

2! 4!

x x

x

3 5

sin

3! 5!

x x

x x

2 31

1

1

x x x

x

2 3 4

ln( 1)

2 3 4

x x x

x x

Alternating Series Error Bound

If

1

1

N

n

N n

k

S a

is the Nth

partial sum of a

convergent alternating series, then

1N NS S a

Geometric Series

2 3 1 1

1

n n

n

a ar ar ar ar ar

diverges if |r|≥1; converges to

1

a

r

if |r|<1

This is available at http://covenantchristian.org/bird/Smart/Calc1/StuffMUSTknowColdNew.htm