AP CALCULUS BC Stuff you MUST Know Cold

[Pages:2]l'Hopital's Rule

If

f (a) 0 =

or =

,

g(a) 0

f (x)

f '( x )

then lim

= lim

x a g ( x ) x a g '( x )

Average Rate of Change (slope of the secant line)

If the points (a, f(a)) and (b, f(b)) are on the graph of f(x) the average rate of change of f(x) on the interval [a,b] is

f (b) - f (a)

b-a

Definition of Derivative (slope of the tangent line)

f (x + h) - f (x)

f

'( x)

=

lim

h0

h

Derivatives

( ) d xn = nxn-1

dx

d (sin x) = cosx

dx

d (cos x) = -sin x

dx

d ( tan x) = sec2x

dx

d (cot x) = -csc2x

dx

d (sec x) = tanx sec x

dx

d (csc x) = -cotx csc x

dx

d

(ln u )

=

1 du

dx

u

( ) d eu = eudu

dx

d dx

(loga

x)

=

1 x ln

a

d (au ) = ax (ln a) du

dx

AP CALCULUS BC Stuff you MUST Know Cold

Properties of Log and Ln 1. ln1 = 0 2. ln ea = a 3.eln x = x 4. ln xn = n ln x

5. ln (ab) = ln a + ln b 6.ln (a b ) = ln a - ln b

Differentiation Rules

Chain Rule

d

[

f

(u)]

=

f

du '(u)

dx

dx

Product Rule

d

( uv )

dv =u

du +v

dx

dx dx

Quotient Rule

du dv

d dx

u v

=

v

-u dx

v2

dx

Mean Value & Rolle's Theorem

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

f (b) - f (a) that f '(c) =

b-a

if f(a) = f(b), then f '(c) = 0.

Curve sketching and analysis

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

critical point:

dy dx

=0

or undefined.

dy

local minimum: goes (-,0,+) or

dx

d2y

(-,und,+) or

>0

dx2

dy

local maximum: goes (+,0,-) or

dx

d2y (+,und,-) or dx2 < 0

Absolute Max/Min.: Compare local

extreme values to values

at endpoints.

pt of inflection : concavity changes.

d2y

goes (+,0,-),(-,0,+),

dx2

(+,und,-), or (-,und,+)

"PLUS A CONSTANT"

The Fundamental Theorem of

Calculus

b

f (x)dx = F (b) - F (a)

a

where F '(x) = f (x)

2nd Fundamental Theorem of Calculus

d g (x)

f (x)dx = f (g(x)) g '(x) dx #

Average Value

If the function f(x) is continuous on

[a, b] and the first derivative exist

on the interval (a, b), then there

exists a number x = c on (a, b) such

that

f (c) =

1

b

f (x)dx

b-a a

f (c) is the average value

Euler's Method dy

If given that = f (x, y) and dx

that the solution passes through (x0, y0), then

x new = x old + x dy

ynew = yold + dx (xold , yold ) x

Logistics Curves

L

P(t) =

,

1 + Ce-(Lk )t

where L is carrying capacity

Maximum growth rate occurs when

P = ? L

dP = kP(L - P) or

dt

dP

P

= (Lk)P(1 - )

dt

L

Integrals

kf (u)du = k f (u)du

du = u + C

undu = un+1 + C, n -1 n +1

1 u

du

=

ln

|

u

|

+C

eudu = eu + C

au du

=

1 ln a

au

+

C

cos udu = sin u + C

sin udu = - cos u + C

tan udu = - ln | cos u | +C

cot udu = ln | sin u | +C

sec udu = ln | sec u + tan u | +C

csc udu = - ln | csc u + cot u | +C

du

1

|u|

u u2 - a2 = a arc sec a + C

du

u

a2 - u2 = arcsin a + C

du 1

u

a2 + u2 = a arctan a + C

Integration by Parts

udv = uv - vdu

Arc Length

For a function, f(x)

b

L = 1 + [ f '(x)]2 dx

a

For a polar graph, r()

2

L = [r()]2 + [r '()]2 d

1

Lagrange Error Bound

If Pn(x) is the nth degree Taylor polynomial of f(x) about c, then

max f (n+1) (z)

f (x) - Pn(x)

( n +1) !

n+1

x-c

for all z between x and c.

Distance, velocity and Acceleration

Velocity = d ( position)

dt

Acceleration = d (velocity )

dt dx dy

Velocity Vector = , dt dt

Speed = |v(t)| = ( x ')2 + ( y ')2 .

Distance Traveled =

final

final

time

time

v(t) dt = ( x ')2 + ( y ')2 dt

initial time

initial time

b

x(b) = x(a) + x '(t)dt

a

b

y(b) = y(a) + y '(t)dt

a

Polar Curves

For a polar curve r(), the

2

[ ] Area inside a "leaf" is

1 2

r() 2 d

1

where 1 and 2 are the "first" two

times that r = 0.

The slope of r() at a given is

[ ] dy

=

dy d

=

d d

r() sin

[ ] dx

dx d

d d

r() cos

Ratio Test

(use for interval of convergence)

The series an converges if

n=0

lim an+1 < 1 a n

n

CHECK ENDPOINTS

Alternating Series Error Bound

If SN =

N (-1)n

n =1

an

is

the

Nth

partial sum of a convergent

alternating series, then

S - SN aN +1

Volume

Solids of Revolution

b

Disk Method: V = [ R(x)]2 dx

a

Washer Method:

b

( ) V = [R(x)]2 - [r(x)]2 dx

a

b

Shell Method: V = 2 r(x)h(x)dx

a

Volume of Known Cross Sections

Perpendicular to

x-axis:

b

V = A(x)dx

a

y-axis:

d

V = A( y)dy

c

Taylor Series

If the function f is "smooth"at x = c, then it can be approximated by the nth degree polynomial f (x) f (c) + f '(c)(x - c)

+ f "(c) (x - c)2 + ... 2!

+ f '''(c) (x - c)3 + ... 3!

+ f (n) (c) (x - c)n n!

Elementary Functions

Centered at x = 0

ex = 1 + x + x2 + x3 + ... 2! 3!

x2 x4 x6 cos x = 1 - + - + ...

2! 4! 6! x3 x5 x7 sin x = x - + - + ... 3! 5! 7! 1 = 1 + x + x2 + x3 + ... 1- x

x2 x3 x4 ln(x + 1) = x - + - + ...

234

Most Common Series

1

(-1)n

diverges

converges

A(r)n converges to

A

if |r| ................
................

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