AP Calculus Formula List - Math Tutoring with Misha

[Pages:6]AP CALCULUS FORMULA LIST

Definition of e:

e

=

lim

n

1

+

1 n

n

____________________________________________________________________________________

Absolute Value:

x x =

if

x0

-x if x < 0

____________________________________________________________________________________

Definition of the Derivative:

f '( x) = lim f ( x + x) - f ( x)

x

x

f '( x) = lim f ( x + h) - f ( x)

h

h

f '(a) = lim f (a + h) - f (a)

h

h

derivative at x = a

f '( x) = lim f ( x) - f (a)

xa

x-a

alternate form

____________________________________________________________________________________

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

____________________________________________________________________________________

Average Rate of Change of f ( x) on [a, b] = f (b) - f (b)

b- a ____________________________________________________________________________________

Rolle's Theorem:

If f is continuous on [a, b] and differentiable on (a, b) and if f (a) = f (b), then there exists a number c on (a, b) such that f '(c) = 0.

____________________________________________________________________________________

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)

.

b-a

Note : Rolle's Theorem is a special case of The Mean Value Theorem

If f (a) = f (b) then f '(c) = f (a)- f (b) = f (a)- f (a) = 0.

b-a

b-a

Source: adapted from notes by Nancy Stephenson, presented by Joe Milliet at TCU AP Calculus Institute, July 2005

AP Calculus Formula List

Math by Mr. Mueller

Page 1 of 6

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.

____________________________________________________________________________________

Definition of a Critical Number:

Let f be defined at c. If f '(c) = 0 or f ' is is undefined at c, then c is a critical number of f .

____________________________________________________________________________________

First Derivative Test:

Let c be a critical number of the function f that is continuous on an open interval I containing c.

If f is differentiable on I , 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 f '(c) = 0 and the second derivative exists

on an open interval containing c.

(1) If f "(c) > 0, then f (c) is a relative minimum. (2) If 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 on I. (2) If f "( x) < 0 for all x in I, then the graph of f is concave downward on I.

____________________________________________________________________________________

Definition of an Inflection Point:

A function f has an inflection point at (c, f (c)) if

(1) f "(c) = 0 or f "(c) does not exist, and if (2) f changes concavity at x = c.

____________________________________________________________________________________

Exponential Growth:

dy = ky dt

y(t) = Cekt

Source: adapted from notes by Nancy Stephenson, presented by Joe Milliet at TCU AP Calculus Institute, July 2005

AP Calculus Formula List

Math by Mr. Mueller

Page 2 of 6

( ) ( ) Derivative of an Inverse Function:

f -1

'(x) =

f'

1

f -1 ( x)

____________________________________________________________________________________

First Fundamental Theorem of Calculus:

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

____________________________________________________________________________________

Second Fundamental Theorem of Calculus:

d x f (t ) dt = f ( x)

dx a

d g(x) f (t ) dt = f ( g ( x)) g '( x) Chain Rule Version

dx a ____________________________________________________________________________________

The Average Value of a Function:

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

f AVE

=

1 b-a

b f ( x) dx

a

____________________________________________________________________________________

Volume of Revolution:

Volume around a horizontal axis by discs:

V =

b a

r

(

x

)

2

dx

Volume around a horizontal axis by washers:

{ } V = b R ( x)2 - r ( x)2 dx a

____________________________________________________________________________________

Volume of Known Cross-Section:

Cross-sections perpendicular to x-axis:

V = b A( x) dx a

____________________________________________________________________________________

Position, Velocity, Acceleration:

If an object is moving along a straight line with position function s (t ), then

Velocity is:

v(t) = s'(t)

Speed is:

v(t)

Acceleration is:

a(t) = v'(t) = s"(t)

Displacement from x = a to x = b is:

Note : Displacement is a change in position.

Total Distance traveled from x = a to x = b is:

Displacement = bv (t ) dt a

Total Distance = b v (t ) dt a

Source: adapted from notes by Nancy Stephenson, presented by Joe Milliet at TCU AP Calculus Institute, July 2005

AP Calculus Formula List

Math by Mr. Mueller

Page 3 of 6

TRIGONOMETRIC IDENTITIES

Pythagorean Identities:

sin2 x + cos2 x = 1

tan2 x +1 = sec2 x

1+ cot2 x = csc2 x

____________________________________________________________________________________

Sum & Difference Identities

sin ( A ? B) = sin Acos B ? cos Asin B

cos ( A ? B) = cos Acos B sin Asin B

tan ( A ? B) = tan A ? tan B

1 tan A tan B ____________________________________________________________________________________

Double Angle Identities

sin 2x = 2sin x cos x cos2 x - sin2 x

cos 2x = 1- 2 sin2 x 2 cos2 x -1

cos2 x = 1+ cos 2x 2

sin2 x = 1- cos 2x 2

tan

2

x

=

1

2 -

tan tan

x

2

x

____________________________________________________________________________________

Half Angle Identities

sin x = ? 1- cos x

2

2

cos x = ? 1+ cos x

2

2

tan x = ? 1- cos x = sin x = 1- cos x 2 1+ cos x 1+ cos x sin x

Source: adapted from notes by Nancy Stephenson, presented by Joe Milliet at TCU AP Calculus Institute, July 2005

AP Calculus Formula List

Math by Mr. Mueller

Page 4 of 6

CALCULUS BC ONLY

Integration by Parts: u dv = uv - v du

____________________________________________________________________________________

Arc Length of a Function:

For a function f ( x) with a continuous derivative on [a, b] :

Arc Length is :

s =

b a

1+ f '( x)2 dx

____________________________________________________________________________________

Area of a Surface of Revolution:

For a function f ( x) with a continuous derivative on [a, b]:

Surface Area is :

S = 2 b r ( x) a

1+ f '( x)2 dx

____________________________________________________________________________________

Parametric Equations and the Motion of an Object:

Position Vector = ( x (t ), y (t )) Velocity Vector = ( x '(t ), y '(t )) Acceleration Vector = ( x"(t ), y"(t ))

Speed (or, magnitude of the velocity vector):

v(t) =

dx dt

2

+

dy dt

2

Distance traveled from t = a to t = b is:

s =

b a

dx dt

2

+

dy dt

2

dt

Note : The distance traveled by an object along a parametric curve is the same as

the arc length of a parametric curve.

Slope (1'st derivative) of curve C at ( x (t ), y (t )) is:

dy = dy dt dx dx dt

Second derivative of curve C at ( x (t ), y (t )) is:

d dy

d2y dx2

=

d dx

dy dx

=

dt dx dx dt

____________________________________________________________________________________

L'H?pital's Rule:

f (x)

f '(x)

lim

xc

g

(x)

=

lim

xc

g

'(x)

Source: adapted from notes by Nancy Stephenson, presented by Joe Milliet at TCU AP Calculus Institute, July 2005

AP Calculus Formula List

Math by Mr. Mueller

Page 5 of 6

Polar Coordinates and Graphs:

For r = f ( ) :

x = r cos , y = r sin , r2 = x2 + y2 , tan = y x

Slope of a polar curve:

dy dx

=

dy dx

d d

=

f ( ) cos + - f ( )sin +

f '( )sin f '( )cos

=

r cos + r 'sin -r sin + r 'cos

Area inside a polar curve:

A = 1

2

f

(

2

) d

=

1 2

r 2d

Arc length (*):

s=

f ( )2 + f '( )2 d =

r2

+

dr d

2

d

Surface of Revolution (*) :

about the polar axis:

S = 2 f ( )sin

f ( )2 + f '( )2 d

about = : 2

S = 2 f ( ) cos

f ( )2 + f '( )2 d

____________________________________________________________________________________

Euler's Method:

Approximating the particular solution to: y ' = dy = F ( x, y)

dx

xn = xn-1 + h

( ) yn = yn-1 + h F xn-1, yn-1

given: h = x, ( x0, y0 )

____________________________________________________________________________________

Logistic Growth:

dP dt

=

kP 1-

P L

P

(

t

)

=

1

+

L Ce-

kt

k is the proportionality constant where : L is the Carrying Capacity C is the integration constant

____________________________________________________________________________________

The n'th Taylor Polynomial for f at c :

Pn ( x) =

f

(c)+

f

'(c)( x - c) +

f

"(c) ( x - c)2

2!

+

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

n!

The n'th MacLaurin Polynomial for f is the Taylor Polynomial for f when c = 0.

Pn ( x) =

f

(0) +

f

'(0) x +

f

"(0) x2

2!

+

f

"(0) x3

3!

+

+ f (n) (0) xn

n!

____________________________________________________________________________________

ex = 1+ x + x2 + x3 + = xn

2! 3!

n=0 n!

cos x = 1- x2 + x4 - x6 +

2! 4! 6!

=

n=0

( -1)n

x2n

(2n)!

sin x = x - x3 + x5 - x7 +

3! 5! 7!

=

n=0

( -1)n

x 2 n +1

(2n +1)!

Source: adapted from notes by Nancy Stephenson, presented by Joe Milliet at TCU AP Calculus Institute, July 2005

AP Calculus Formula List

Math by Mr. Mueller

Page 6 of 6

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