AP CALCULUS AB & BC FORMULA LIST

AP CALCULUS AB & BC FORMULA LIST

Definition of e:

e

lim

n

1

1 n

n

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x if x 0 Absolute value: x x if x 0

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Definition of the derivative:

f (x) lim f x h f x

h0

h

f a lim f x f a

xa x a

(Alternative form)

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Definition of continuity: f is continuous at c iff

1) f (c) is defined;

2) lim f (x) exists; xc

3) lim f (x) f (c). xc

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f b f a

Average rate of change of f (x) on [a, b] = ba

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Rolle's Theorem: If f is continuous on [a, b] and differentiable on (a, b) and if f (a) = f (b), then there is at least one number c on (a, b) such that f (c) 0.

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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 (c)

f

b

f

a

.

ba

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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.

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sin 2x 2sin x cos x

cos2 x sin2 x

cos

2x

1

2 sin 2

x

2 cos2 x 1

cos2 x 1 cos 2x 2

sin2 x 1 cos 2x 2

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Definition of a definite integral:

b a

f

x dx

lim

x0

n i 1

f

xi

xi

d c 0

dx

d uv uv vu

dx

d dx

f

g x

f

g xgx

d [sin u] cos u du

dx

dx

d [tan u] sec2 u du

dx

dx

d [sec u] secu tan u du

dx

dx

d [ln u] 1 du

dx

u dx

d [eu ] eu du

dx

dx

d [arcsin u] 1 du

dx

1 u2 dx

d dx

[arctan

u]

1

1 u

2

du dx

d [arcsecu] 1 du

dx

u u2 1 dx

d dx

xn

nxn1

d dx

u v

vu uv v2

d [cos u] sin u du

dx

dx

d [cot u] csc2 u du

dx

dx

d [csc u] cscu cot u du

dx

dx

d dx

[loga

u]

1 u ln a

du dx

d [au ] au ln a du

dx

dx

d [arccos u] 1 du

dx

1 u2 dx

d dx

[arc

cot

u]

1

1 u

2

du dx

d [arc csc u] 1 du

dx

u u2 1 dx

f 1 a

1

f f 1 a

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cosu du sin u C sec2 u du tan u C

sin u du cosu C csc2 u du cot u C

secu tan u du secu C

1 u

du

ln

u

C

cscu cot u du cscu C

 tan u du ln cosu C

cot u du ln sin u C

secu du ln secu tan u C

cscu du ln cscu cot u C

eudu eu C

du arcsin u C

a2 u2

a

audu au C

ln a

du

u2 a2

1 arctan u

a

a

C

du 1 arc sec u C

u u2 a2 a

a

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Definition of a Critical Number:

Let f be defined at c. If f c 0 or if f is undefined at c, then c is a critical number of f.

______________________________________________________________________________ First Derivative Test: Let c be a critical number of a function f that is continuous on an open interval I containing

c. If f is differentiable on the interval, except possibly at c, then f c can be classified as

follows.

1) If f x changes from negative to positive at c, then f c is a relative minimum of f.

2) If f x changes from positive to negative at c, then f c is a relative maximum of f.

______________________________________________________________________________ Second Derivative Test: Let f be a function such that the second derivative of f exists on an open interval containing c.

1) If f c 0 and f c 0 , then f c is a relative minimum.

2) If f c 0 and f c 0 , then f c is a relative maximum.

______________________________________________________________________________ Definition of Concavity: Let f be differentiable on an open interval I. The graph of f is concave upward on I if f is

increasing on the interval and concave downward on I if f is decreasing on the interval.

______________________________________________________________________________ Test for Concavity: Let f be a function whose second derivative exists on an open interval I.

1) If f x 0 for all x in I, then the graph of f is concave upward in I.

2) If f x 0 for all x in I, then the graph of f is concave downward in I.

______________________________________________________________________________ Definition of an Inflection Point:

A function f has an inflection point at c, f c

1) if f c 0 or f c does not exist and

2) if f changes sign from positive to negative or negative to positive at x c

OR if f x changes from increasing to decreasing or decreasing to increasing at x = c.

First Fundamental Theorem of Calculus:

b

a

f

x dx

f

b

f

a

Second Fundamental Theorem of Calculus:

d dx

x

a

f

t dt

f

x

Chain Rule Version:

d dx

gx

a

f

t dt

f

g x g x

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Average value of f (x) on [a, b]:

f AVE

1 ba

b

f (x)dx

a

Volume around a horizontal axis by discs: V b[r(x)]2 dx a

Volume around a horizontal axis by washers: V b ([R(x)]2 [r(x)]2 )dx a

b

Volume by cross sections taken perpendicular to the x-axis: V a A(x)dx

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If an object moves along a straight line with position function s t , then its

Velocity is vt st

Speed = v t

Acceleration is at vt st

Displacement (change in position) from

x

a

to

x

b

is

Displacement

=

b a

v

t

dt

Total Distance traveled from

x a to x b

is

Total Distance =

b

a

v

t

dt

or Total Distance =

c

a

v

t

dt

b

c

v

t

dt

,

where

v t changes sign at

x c.

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CALCULUS BC ONLY

Differential equation for logistic growth: dP kP L P, where L lim P t

dt

t

Integration by parts: u dv uv v du

Length of arc for functions: s b 1[ f (x)]2 dx a

_____________________________________________________________________________ If an object moves along a curve, its

Position vector = xt , y t

Velocity vector = xt , yt

Acceleration vector = xt , yt

Speed (or magnitude of velocity vector) =

v(t)

dx dt

2

dy dt

2

Distance traveled from

t a to t b (or length of arc) is s

b a

dx dt

2

dy dt

2

dt

In polar curves, x r cos and y r sin

Slope of polar curve: dy r cos rsin dx r sin rcos

Area inside a polar curve: A 1 b r2d

2a

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Definition of a Taylor polynomial:

If f has n derivatives at c, then the polynomial

Pn x

f

c

f cxc

f c x c2

2!

f c x c3 ...

3!

f

n c x cn

n!

is called the nth Taylor polynomial for f at c.

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Lagrange Error Bound for a Taylor Polynomial (or Taylor's Theorem Remainder):

If f is differentiable through order n 1 in an interval I containing c, then for each x in I,

there exists z between x and c such that

f

x

f

c

f cx c

f c x c2

2!

...

f n c x cn

n!

Rn x,

where

Rn x

f n1 z n 1!

x

c

n1

.

Rn x

gives a bound for the size of the error

found by the nth degree Taylor polynomial.

The remainder represents the difference between the function and the polynomial. That is,

Rn f x Pn x .

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Alternating Series Remainder: If a series has terms that alternate, decrease in absolute value, and have a limit of 0 (so that the

series converges by the Alternating Series Test), then the absolute value of the remainder Rn

involved in approximating the sum S by Sn is less than the first neglected term. That is,

Rn S Sn an1 .

______________________________________________________________________________ Maclaurin series that you must know:

ex 1 x x2 x3 xn

2! 3!

n0 n!

cos x 1 x2 x4 x6

1n

x2n

2! 4! 6! n0

(2n)!

sin x x x3 x5 x7

1 n

x2n1

3! 5! 7! n0

(2n 1)!

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