Calculus BC Formulas

8/8/2019 Calculus BC Formulas

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A. P. CALCULUS BC FORMULA BOOKLET

GRAPHING CALCULATORS

Each student will be expected to bring to the examination a graphing calculator on which the student

can:

1. produce the graph of a function within an arbitrary viewing window;

2. find the zeros of a function;

3. compute the derivative of a function numerically, and

4. compute definite integrals numerically.

Pay special attention to calculator syntax; i.e., placement of parentheses, commas, variables, and order of

operations. Important functions include graph, root, solve, nDeriv, andfnInt.

CALCULATORS should be in RADIAN MODE.

CONTINUITY: The function f(x) is said to be continuous at x = c if

1) f(c) is a finite number;

2) limx c

f(x) exists;

3) limx c

f(x) = f(c) .

DIFFERENTIABILITY: The function is continuous at x = c .

DIFFERENTIABILITY IMPLIES CONTINUITY,

BUTCONTINUITY DOES NOT IMPLY DIFFERENTIABILITY.

LIMITS: ZEROS IN NUMERATOR/DENOMINATOR OF A FRACTION

("c" is a constant.)

Zero (Root)0

c= 0

Vertical Asymptote

c

0=

= D.N.E.

( )Point of Exclusion (Removable Discontiuity)

0

0 = undefined( )

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DERIVATIVE OF A FUNCTION: f ' x( ) = limh0

f x + h( ) f x( )h

or f'

a( ) = limxa

f x( ) f a( )x a

DIFFERENTIATION RULES:

(Where "u" and "v" are differentiable functions of x, and "a" is a constant.)

d

dxau = a

du

dx

d

dxu + v( )=

du

dx+

dv

dx

d

dxu

n= n u

n1

du

dx

d

dxa = 0

d

dxuv( ) = u

dv

dx+ v

du

dxd

dx

u

v

=

v du

dx u

dv

dx

v2

CHAIN RULE:dy

dx=

dy

du

du

dx

d

dxsinu = cos u

du

dx

d

dxcosu = sinu

du

dx

d

dxtan u = sec2 u

du

dx

d

dxcotu = csc2 u

du

dx

ddx

sec u = secu tanu dudx

ddx

csc u = cscu cotu dudx

d

dxln u =

1

u

du

dx

d

dxe

u = eudu

dx

d

dxa

u = a u ln adu

dx

d

dxsin

1u =

du

dx

1 u2d

dxcos

1u =

du

dx

1 u 2

ddx

tan 1 u =

du

dx1+ u 2

ddx

cot 1 u =

du

dx1+ u 2

d

dxsec

1u =

du

dx

u u2 1

d

dxcsc

1u =

du

dx

u u2 1

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Decreasing

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VELOCITY: ACCELERATION:

V=ds

dt

a =dv

dt=

dv

ds

ds

dt= v

dv

ds

a =

d2s

dt2

i) If v > 0 and a > 0, the speed is increasing.

ii) If v > 0 and a < 0, the speed is decreasing.

iii) If v < 0 and a > 0, the speed is decreasing. ( Note: speed = v t( ) )iv) If v < 0 and a < 0, the speed is increasing.

DISTANCE: Ifv = f t( ) , the distance traveled by a body between t= a and t= b is given by

f(t)a

b

dt

(Be careful. Does the object change directions between a and b?)

EQUATION OF A TANGENT LINE:

y y1 = f ' (x1 ) x x1( )

EQUATION OF A NORMAL LINE:

y y1 = 1

f ' (x1)

x x1( )

TANGENTS (function must exist at xi )

Vertical tangents: f ' xi( )does not existHorizontal tangents: f ' xi( )= 0

LINEAR APPROXIMATION The linear approximation to f x( ) near x = xo is given byy = yo + f ' xo( ) x xo( ) for x sufficiently close to xo .

EULER'S METHOD ("Numerical Solutions to a Differential Equation")

Iterative use of the Linear Approximation with a given step value.

y1 = y0 + f ' xo( ) x1 xo( )

y2 = y1 + f ' x1( ) x2 x1( )y3 = y2 + f ' x2( ) x3 x2( )etc.

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INVERSES: To find the inverse of y = f(x), solve for x in terms of y, then interchange x and y.

f[ f1

x( )] = x and f1[f x( )] = x

f1( )' d( ) = 1

f' c( )or

dx

dy=

1dy

dx

MEAN-VALUE THEOREM (SPECIAL CASE -- ROLLE'S THEOREM): If the function f(x) is

continuous at each point on the closed interval a < x < b and has a derivative at each point on the open

interval a < x < b, then there is at least one number c, a < c < b, such that

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

b a

a bc

MEAN VALUE THEOREM

a b

y'=0

c

ROLLE'S THEOREM

"Where average velocityf(b) f(a)

b a

meets instantaneous velocity f ' ( c)( )."

ABSOLUTE-VALUE THEOREM:

f(x)= x =x, if x 0

x, if x < 0

GREATEST-INTEGER THEOREM:

g(x) = [x] is the greatest integer not greater than x.

e.g. g(5.2) = 5, g(-1.5) = -2, g(1) = 1

DIRECT VARIATION: y = kx ("y " is directly proportional to "x ")

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INVERSE VARIATION: y =k

xor xy = k ("y " is inversely proportional to "x ")

REFLECTIONS:

The graph ofy = f x( ) is the reflection ofy = f x( ) in the x-axis;eg. y = x

2; y = x2

whereas the graph ofy = f x( ) is the reflection of the graph ofy = f x( ) in the y-axis.eg. y = x ; y = x

ODD/EVEN FUNCTIONS:

EVEN: f x( ) = f x( )ODD: f x( ) = f x( )

e. g. Even function: y = x2

or y = cosx

Odd function: y = x3 or y = sinx

SYMMETRY:

w.r.t. x-axis .... equivalent equations when y replaced by -y

w.r.t. y-axis .... equivalent equations when x replaced by -x

w.r.t. origin .... equivalent equations when x replaced by -x

and y replaced by -y

RELATIONSHIPS between the graphs of and the graphs ofy = f x( ) and the graphs ofy = kf x( ), y = f kx( ), y k= f x h( ), y = f x( ), and y = f x( ).

LOGARITHMIC FUNCTIONS:

y = loga x iff ay= x

y = ln x iff ey= x

PROPERTIES:

ln ab( )= ln a + ln b

lna

b

= ln a ln b

ln ar = r ln a

loga x =lnx

ln a

ln1 = 0lne =1

ln ex = x

e ln x = x

x

a

xb=

x

a +b

x

ab=

x

ab

xa

xb = x

a bx

o=1

xa( )b = xab x a = 1

xa

NATURAL LOGARITHM: lnx =dt

t1

x

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EQUATIONS FOR EXPONENTIAL GROWTH AND DECAY: Equations of the form y ' = ky aresolved as.

A = Aoekt

or A = Aoert

LAWS OF LOGISTIC GROWTH : Equations of the form y ' = ky A y( ).

y =A

1+ Bekt

NB. A=the Maximum Capacity and the POI x,A

2

is the moment of maximum growth.

SLOPE FIELDS

Tips associating the slope field to a particular Differential Equation:

1) Horizantal Dashes dy

dx= 0

2) Dashes\ dy

dx< 0

3) Dashes / dy

dx> 0

4) All Dashes in any column // to each other dy

dxhas no y

5) All Dashes in any row // to each other dy

dxhas no x

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INTEGRATION FORMULAS:

f(x) dx = F(x) + C, whereF' (x) = f(x)

d

dx

f(t) dt= f(x)a

x

[First Fundamental Theorem]

Remember the Chain Rule!!!:d

dxf(t) dt= f(u)

a

u

Du

f(x) dx = F(b) F(a), where F' (x)= f(x)a

b

[Second Fundamental Theorem]

un

du =un+1

n +1+ C, n 1

du

u = ln u + C

eu

du = eu+ C a

udu =

au

ln a+ C, a > 0, a 1( )

sinu du = cos u + C cosu d u = sinu + C

sec2u du = tan u + C csc

2u d u = cotu + C

sec u tan u du = secu + C csc u cot u du = cscu + C

secu du = ln sec u + tanu + C

sin2u du =

1

2u

1

4sin 2u + C cos

2u du =

1

2u +

1

4sin 2u + C

du

a2

u2= sin

1 u

a

+ C

du

a2

+ u2 =

1

a

tan 1u

a

+ C

Integration by Parts: u dv = uv v du

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Integral Boundary Rules

f x( )a

a

dx = 0

f x( )a

b

dx = f x( )b

a

dx

f x( )a

b

dx + f x( )b

c

dx = f x( )ac

dx

If f x( ) g x( ) on a,b[ ], then f x( )a

b

dx g x( )ab

dx

AVERAGE (MEAN) VALUE: If the function y = f x( ) is continuous on the interval a < x < b, then theaverage or mean value ofy with respect to x over the interval [a,b] is

yav( )x

=1

b af(x) dx

a

b

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AREA APPROXIMATIONS

RIEMANN SUMS

A = limn +

f ci( )i=1

n

x

A = f x( )a

b

dx

Left-Hand Rectangles Midpoint Rectangles Right-Hand Rectangles

TRAPEZOIDAL RULE:

f(x) dxa

b

b a

2nf(x0 )+ 2 f(x1 ) + 2f(x2 )+....+ 2f(xn 1 )+ f(xn )[ ]

AREA FORMULAS

Function: A = f(x) g(x)[ ]a

b

dx or A = f(y) g(y)[ ]cd

dy

Polar: A =1

2r ( )[ ]

2

d

Parametric: Eliminate the parameter:

i) isolate the parameter in one equation, and

ii) substitute into the other equation

and then use the Function formula

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ARC LENGTH:

Function: L = 1+dy

dx

2

a

b

dx

Polar: L = r2 + drd

2

d

Parametric: L =dx

dt

2

+

dy

dt

2

a

b

dt

PARAMETRIC, POLAR AND VECTOR FORMS

Parametric: dydx

=

dy

dtdxdt

a function in t( ) and d2

ydx

2 =

dy'

dtdxdt

Vertical Tangent:dx

dt= 0

Horizontal Tangent:dy

dt= 0

Area: Eliminate the parameter and use the Function formula

Arc length: L = dx

dt

2

+dy

dt

2

a

b

dt

Polar: Area: A =1

2r ( )[ ]

2

d

Arc Length: L = r2 +dr

d

2

d

Parametric Polar: x ( )= r cos and y ( ) = r sin

Vector: Velocity v = x' t( ) i + y' t( ) j

Speed= v = x' t( )[ ]2

+ y' t( )[ ]2

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HORIZONTAL ASYMPTOTES (Maximum Capacity) and LIMITS AT INFINITY

xLim f x( ) or

xLim f x( )

= (

L'Hpital's Rule: If limxx 0

f(x)

g(x)

is indeterminate of the form

0

0or

, and if

limxx 0

f ' (x)

g ' (x)

exists, then lim

xx 0

f(x)

g(x)

= lim

xx 0

f ' (x)

g ' (x)

.

IMPROPER INTEGRALS

1. Boundary at infinity: f x( )dxa

= limb

F b( ) F a( )[ ]

2. Boundary is a Veritical Asymptote:

f x( )dxa

b

= limcb

f x( )dxa

c

or = limc a+

f x( )dxc

b

3. Region includes a Vertical Asymptote at x=c: f x( )dxa

b

= f x( )dxa

c

+ f x( )dxcb

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TAYLOR POLYNOMIALS

f x( )= f a( )+ f ' a( ) x a( )+f' ' a( ) x a( )2

2!+

f'' ' a( ) x a( )3

3!+ +

fn a( ) x a( )n

n!+ R

x( )

where R x( )=fn+1 c

( )x

c

( )

n+1

n+ 1( )! for some c x, a( )

McLauren series=Taylor Series where a=0

SERIES OF KNOWN FUNCTIONS

y = sin x =

**y = cos x = 1 x2

2!+

x4

4!

x6

6!+ +

1( )n x2n

2n( ) !+ =

1( )n x2n

2n( ) !0

y = ex =

y =1

1 x= 1+ x + x

2+ x

3+ + xn + = xn

0

on 1< x < 1

**y =1

1+ x= 1 x + x

2 x3 + + x( )n+ = 1( )

nx

n

0

on 1< x < 1

**y =1

1+ x2= 1 x

2+ x

4 x6 + + x( )2n

+ = 1( )nx

2n

0

on 1< x < 1

**y = tan 1 x = x x3

3+

x5

5

x7

7+ +

1( )n x2n+1

2n+1+ =

1( )n x2n +1

2n +10

on 1 x 1

**y = ln 1+ x( ) = x x2

2+

x3

3

x4

4+ +

1( )n xn

n+ =

1( )n xn

n1

on 1< x 1**These can be derived from the unmarked series.

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If a n1

IS an Alternating Series:

Alternating Series Definition: 1( )n+1

an1

or a n cosn0

Liebnitz Alternating Series Test: 1( )n+1

an1

converges if

1. all an are positive,

2. an an +1,

and 3.nLim an = 0

Absolute Convergence vs Conditional Convergence (only applies to Alternating Series)

a n is absolutely convergent if an converges.

a n is conditionally convergent if a n converges but an diverges.

RADIUS OF CONVERGENCE

R is the radius of convergence whennLim

an +1 x a( )n+1

an x a( )n < 1 leads to x a < R

INTERVAL OF CONVERGENCE

Solve x a < R (from the Radius of Convergence) and test convergence at the endpoints

SPECIAL LIMITS (for comparison)

limx0

sinx

x= 1 lim

x01 cos x

x= 0 lim

x0

ex 1

x=1

limn

ln n

n= 0 lim

nnn = 1 lim

nx

1n= 1

limn

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KNOWN SERIES (for comparison)

Geometric Series: arn

1

--converges toa

1 rfor r1--diverges for p1

Harmonic Series:1

n1

= 1+1

2+

1

3+

1

4+ Diverges

Alternating Harmonic Series:1( )n+1

n1

= 11

2+

1

3

1

4+ Converges conditionally

Telescoping Series: Any series that can be simplified by Partial Fractions such that consecutive

terms add to 0, leaving only the first and last terms e.g.,1

n n+ 1( )1

It will generally converge, by the integral Test and partial fractions.

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absolute convergence

absolute minimum

absolute maximum

acceleration

acceleration vector

algebraic functionalternating series

amplitude

antiderivative

antidifferentiation

arc length

arccosine

arcsine

arctangent

asymptote

average rate of changeaxis of rotation

axis of symmetry

base (exponential and log)

bounded above

bounded below

bounded

cartioid

Cartesian Coordinate System

Chain Rule

circlecircular functions

closed interval [a,b]

coefficient

Comparison Test

complex number

components of a vector

composition f g

concave down

concave up

conditional convergenceconic section

constant function

constant of integration

continuity at a point

continuity on an interval

continuous function

convergent improper integral

convergent sequence

convergent series

coordinate axes

cosecant function

cosine function

cotangent functioncritical point

critical value

cross-sectional area

decay model

decreasing function

decreasing on an interval

definite integral

degree

delta notation

derivativedifference quotient

differentiability

differential

differential equation

differentiation

discontinuity

disk method

distance (from velocity)

distance formula

divergent improper integraldivergent sequence

divergent series

domain

dummy variable of integration

dy/dx (leitniz notation)

e

ellipse

end behavior

endpoint extrema

essential discontinuityEuler's Method

even function

exponential function

exponential growth and decay

exponential laws

extremum

factorial

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First Derivative Test

Frequency of a periodic function

function

Fundemental Theorem of Calculus

geometric sequence

geometric series

graphgrowth models

growth rate

half-life

harmonic series

hyperbola

imaginary number

implicit differentiation

improper integral

increasing function

increasing on an intervalincrement

indefinite integral

indterminate form

infinite limit

inflection point

initial condition

initial value problem

inscribed rectangle

instantaneous rate of change

instantaneous velocityinteger

integrable function

integrand

integration by partial fractions

integration by parts

integration by substitution

Intermediate Value Theorem

interval

interval of convergence

inverse functionirrational number

Lagrange Error Bound

Law of Cosines

Law of Sines

left-hand limit

left-hand sum

Leibniz, Gottfried

L'Hopital's Rule

limit

limt at infinity

limit of integration

linear approximation

linear function

local extremalocal linearity

local linearization

logarithmic function

logarithmic laws

logistic equation

logistic growth

lower bound

Maclaurin series

maximum

mean valueMean Value Theorem

midpopint formula

minimum

monotonic

motion

natural log

Newton, Isaac

non-removable discontinuity

normal line

numerical derivativenumerical integration

odd function

one-to-one function

open interval (a,b)

optimization

order of a derivative

origin

parabola

parallel curves

parameterparametric curve

partial fractions

partial sum of a series

partition of an interval

percentage error

period

periodic function

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perpendicular curves

piece-wise defined functions

polar coordinates

polynomial

position function

position vector

power seriesprime notation f'(x)

Product Rule

proportionality

p-series

quadrant

quadratic formula

Quotient Rule

radian

radius of a circle

radius of convergencerange

rate of change

rational function

Ratio Test

real number

rectangular coordinates

region (in a plane)

related rates

relative error

relative maximumrelative minimum

removable discontinuity

Rhiemann sum

right-hand limit

right-hand sum

root of an equation

roundoff error

scalar

secant function

secant linesecond derivative

Second Derivative Test

separable differential equation

sequence

series

set

sigma notation

sine function

slope

slope field

solid (in 3-space)

solid of revolution

speed

spheresubset

symmetry

tangent function

tangent line

tangent vector

Taylor polynomial

Taylor series

term of a sequence or series

transcendental function

Trapezoidal Ruletruncation error for power series

trigonometric functions

unit circle

unit vector

upper bound

u-substitution

vector

vertex

viewing window

volume by slicingx-axis

x-intercept

y-axis

y-intercept

zero of a function

Calculus BC Formulas

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