AP Calculus BC Formulas, Definitions, Concepts & Theorems to Know

[Pages:5]AP Calculus BC Formulas, Definitions, Concepts & Theorems to Know

Def. of e:

e

lim

n

1

1 n

n

and

ea

lim

n

1

a n

n

Absolute Value: x if x 0

x x if x 0

Definition of Derivative:

f x lim f x h f x

h0

h

Alternative form of Def of Derivative:

f a lim f x f a

xa x a

Definition of Continuity:

f is continuous at c if and only if

lim f x lim f x f c

xc

xc

Differentiability: A function f is not differentiable at x a if 1) f is not continuous at x a 2) f has a cusp at x a 3) f has a vertical tangent at x a

Euler's Method:

Used to approximate a value of a function,

given dy dx and x0, y0

Use

y y0

dy dx

(

x

x0

)

repeatedly.

Average Rate of Change of f x on a,b

is the slope:

f

b

b

f a

a

b

1

a

b

a

f

x dx

Instantaneous Rate of Change of f x with respect to x is f x .

Intermediate Value Theorem (IVT):

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

Mean Value Theorem (MVT):

If f is continuous on a,b and differentiable on a,b then there exists a number c in a,b such

that f c f b f a . (Think: The slope at

ba x c is the same as the slope from a to b .)

Trig Identities to Know: sin 2x 2sin x cos x cos2x cos2 x sin2 x cos2 x 1 1 cos 2x 22 sin2 x 1 1 cos 2x 22

Definition of a Definite Integral:

b a

f

x dx

lim

n

n i1

f

a

(b

a)i n

b

n

a

Also know Riemann Sums ?

Left, Right, Midpoint, Trapezoidal

Average Value of a function f x on a,b :

f AVE

1 ba

b f x dx

a

Curve Length of f x on a,b :

L

b a

1 f x2dx

Logistic Differential Equation:

dP kP L P

dt

;

P

t

1

L ce

Lkt

,

c

L

P0 P 0

d dx

f

g x

f

g

x

gx

(chain rule)

d uv uv uv

dx (product rule)

d dx

u v

uv uv v2

(quotient rule)

d xn nxn1

dx (power rule)

f 1 a

1

f f 1 a

(derivative of an inverse)

Derivatives

d sin u cosu u

dx

d cosu sin u u

dx

d tan u sec2 u u

dx

d secu secu tan u u

dx

d cot u csc2 u u

dx

d cscu cscu cot u u

dx

d ln u u

dx

u

d dx

logb

u

u u

1 ln b

d eu ueu

dx

d au uau ln a

dx

d arcsin u u

dx

1 u2

d dx

arctan

u

1

u u2

d arcsecu u

dx

u u2 1

d arccosu u

dx

1 u2

d dx

arccot

u

1

u u

2

d arccscu u

dx

u u2 1

undu un1 c

n 1 (power rule for integrals)

1 u

du

ln

u

C

eudu eu C

audu au C ln a

ln udu u ln u u C

Integrals

cosudu sin u C sin udu cosu C sec2 udu tan u C secu tan udu secu C csc2 udu cot u C cscu cot udu cscu C tan udu ln cosu C secudu ln secu tan u C cot udu ln sin u C cscudu ln csc u cot u C

du arcsin u C

a2 u2

a

du

u2 a2

1 a

arctan

u a

C

du 1 arcsec u C

u u2 a2 a

a

Techniques of Integration:

u-substitution; Partial Fractions; Completing the Square; Integration By-Parts: udv uv vdu

Max/Min, Concavity, Inflection Point

Critical Number at x c if:

Absolute Maxima:

f c 0 or f c is undefined

The absolute max on a closed interval a,b is

First Derivative Test: Let c be a critical number.

If f x changes from to at x c

f a , f b , or a relative maximum. The absolute min on a closed interval a,b is

f a , f b , or a relative minimum.

then f has a relative max of f c . If f x changes from to at x c

Test for Concavity:

If f x 0 for all x in I , then the graph of f

then f has a relative min of f c

Second Derivative Test:

If f c 0 and f c 0

is concave up on I .

If f x 0 for all x in I , then the graph of f

is concave down on I .

then f has a relative min of f c . If f c 0 and f c 0

Inflection Point:

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

then f has a relative max of f c .

f changes sign at x c

Fundamental Theorems

First Fundamental Theorem of Calculus:

Second Fundamental Theorem of Calculus:

b

a

f x dx

f

b

f

a

(The accumulated change in f from a to b )

d dx

gx

a

f

t dt

f

gx gx

or

d dx

gx

h x

f

t dt

f

gx gx

f

h x h x

Area & Volume Functions in the form y f x or x f y

Area Between Curves

b

A a topcurve bottomcurve dx

d

A c rightcurve leftcurve dy

Volume ? Cylindrical Shell Method

V

2

b

a

radius

height

dx

V

2

d

c

radius height dy

Note that the Shell Method is NOT tested on the AP Exam.

Volume ? General Volume Formula

Volume

b

a

A( x)

dx,

where

A

x

area

Volume

d

c

A(

y)

dy,

where

A

y

area

Volume ? Disc/Washer Method

V

b

a

R

x

2

r x2

dx

V

d

c

R

y2

r

y2

dy

Volume by Cross-Sections

x-axis:

V

b

a

A x dx

y-axis:

V

b

a

A

y dy

Asemicircle

8

s2

;

ARtIsosc

1 s2 2

(leg as

s)

AEquil

3 s2 4

;

ARtIsosc

1 s2 4

(hypot as

s)

Horizontal/Vertical Motion

Position Function: s t

Velocity Function: v t st

Acceleration Function: a t vt st

Displacement (change in position) over a,b

b

a

v

t

dt

s

b

s

a

Total Distance Traveled over

a,b

b

a

v

t

dt

Speed = v t

Speed Increases if v t and a t have same sign

Speed Decreases if v t and a t different signs

Motion Along a Curve (Parametrics & Vectors)

Position Vector x t, y t

|Displacement| x b x a2 y b y a2

Velocity Vector xt, yt

Speed (or Magnitude/Length of Velocity Vector)

Acceleration Vector xt, yt

v t xt2 yt2

dy

Slope=

dy dx

dt dx

dt

d dy

d2y dx2

d dx

dy dx

dt

dx dx

dt

Speed Increases if d speed 0

dt

Distance Traveled (or Length of Curve)

b v tdt b

a

a

xt2 yt2dt

Polar Curves

x r cos , y r sin Slope of polar curve: dy dy d rsin r cos

dx dx d rcos r sin

Area inside a polar curve: A 1 r2d

2

Series

Convergence/Divergence of Series

Lagrange Error Bound (aka Taylor's Theorem)

10 Tests: nth Term Test, Telescoping Series Test, Geometric Series Test, p-Series Test, Integral Test, Direct Comparison Test, Limit Comparison Test,

f

x Pn x

Rn x

max

f n1 z x cn1 n 1!

Alternating Series Test, Ratio Test, and Root Test.

Alternating Series Error Bound If a series is alternating in sign and decreasing in magnitude, and to zero, then

Power Series of ex, sin x, cos x , centered at x 0 ex 1 x1 x2 x3 ...

1! 2! 3!

error first disregarded term

Taylor Series

P x f c f c x c1 f c x c2

1!

2!

f c x c3

f n cx cn

...

...

3!

n!

is called the nth degree Taylor Series for f x ,

sin x x1 x3 x5 x7 ... 1! 3! 5! 7! x2 x4 x6

cos x 1 ... 2! 4! 6!

ex, sin x, and cos x converge for all real x-values

centered at x c .

Hyperbolic Trig Functions are not part of the curriculum for AP Calculus BC. They are used in Differential Equations and other math courses, as well as in some of the sciences. Here is a crash-course on hyperbolic functions. You'll notice many similarities to basic trig functions.

Definitions of Hyperbolic Trig Functions:

sinh x ex ex csch x 1

2

sinh x

cosh x ex ex sech x 1

2

cosh x

tanh x sinh x cosh x

coth x 1 tanh x

Derivatives of hyperbolic trig functions: d sinh u cosh u u dx d cosh u sinh u u dx d tanh u sech2 u u dx d coth u csch2 u u dx d sech u sech u tanh u u dx d csch u csch u coth u u dx

Inverses of hyperbolic trig functions:

sinh1 x ln x x2 1 D ,

cosh1 x ln x x2 1 D 1,

tanh1 x 1 ln 1 x 2 1 x

D 1,1

coth1 x 1 ln x 1 D : ,1 1,

2 x 1

sech1 x ln 1 1 x2 x

D : (0,1]

csch1

x

ln

1 x

1 x2 x

D : ,0

0,

Identities involving hyperbolic trig functions: cosh2 x sinh2 x 1 tanh2 x sech2 x 1 coth2 x csch2 x 1 sinh 2x 2sinh x cosh x cosh 2x cosh2 x sinh2 x

sinh2 x 1 1 cosh 2x

22

cosh2 x 1 1 cosh 2x

22

Integrals involving hyperbolic trig functions:

cosh udu sinh u C sinh udu cosh u C sech2 udu tanh u C csch2 udu coth u C sech u tanh udu sech u C csch u coth udu csch u C

Derivatives of inverse hyperbolic trig functions:

d sinh1 u u

dx

u2 1

d cosh1 u u

dx

u2 1

d dx

tanh1

u

1

u u2

d sech1 u u

dx

u 1 u2

d csch1 u u

dx

u 1 u2

d dx

coth

1

u

1

u u

2

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