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52 Cards in this Set
 Front
 Back
The Taylor Polynomial of order n based at a real number a for functions

Rn (look up)


When does a function have an inverse?

When it's 11 » if x=x2 then F(x) = f(x2)


How do you find the inverse?

Switch x and y » solve for y


How do you prove a function is one to one?

f(x) is strictly monotomic if it is increasing or decreasing on an interval I. If f(x) is monotomic it is 11.


What is the relationship between a function and its inverse?

the inverse is the unique function with domain equal to the range of f that satisfies the equation:
f(f¯¹(x))=x 

inverse(b)=

1/f(a)


Domain and Range of ln(x)

Domain:(0, )
Range: all Real numbers 

Know graph of ln(x)

¡


Derivative of ln(x)

1/x


The definition of ln(x) or L(x)

L(x)= çdt/t (integral from 1 to x) x>0


Logrithmic Differentiation

g(x)=g(x)[g1(x)/g1(x) + g2(x)/g2(x)+. . .g'n(x)/gn(x)]


relationship between e^x and ln(x)

ln(e^x)=x


defininition of the number e

e is where L(x)=1


Domain and Range of e^x

Domain: all Real numbers
Range:(0, ) 

know graph of e^x

__/


derivate/integral of e^x

dy/dx e^x = e^x
ç e^x = e^x dy/dx 

Definition of exponential function to the base "a"

a function of the form F(x)=p^x


Domain and Range of a^x

Domain:all real numbers
Range:(0,) 

dy/dx of a^x

a^x (ln x) du/dx


integral of a^x

1/ln a (a^x) + C


dy/dx of x^n where n is a variable

set f(x)=x^n
use log diff 

the logarithm of x to the base p

log x = ln x/ln p
p 

log base p of p raised to the t =

t


dy/dx log x
a 
1/xlnp


logarithm to the base e

ln=log e


restrictions of domain for
sine tan sec 
sin [¨ö¬á, ¨ö¬á]
tan (¨ö¬á, ¨ö¬á) sec [0,¨ö¬á)U(¨ö¬á,¬á] 

domain and range of inverse sin of x

D: [1,1]
R: [¬á¨ö, ¬á¨ö] 

domain and range of inverse tangent of x

D: (¡Ä, ¡Ä)
R: (¨ö¬á, ¨ö¬á) 

domain and range of inverse secant of x

D:[0, ¨ö¬á) U (¨ö¬á, ¬á]
R:(¡Ä, 1] U [1, ¡Ä) 

Relation between [r, ө] and (x,y)

x=rcosө
y=rsinө 

polar coordinates for a circle with a radius a centered at the origin

r=a


area of 2 parametric equations

A= ¡ò¨ö([p2ө]©÷[p1ө]©÷)dө


Length of a curve (parametric)

L=¡ò¡î([x¡¯(t)]©÷+[y¡¯(t)]©÷)


Length of a curve (cartesian)

L= ¡ò¡î(1+[f¡¯(x)]©÷)dx


Length of a curve (polar coordinates)

L= ¡ò¡î([p(Ө)]©÷+[p¡¯(Ө)©÷]dӨ


Surface Area (parametric)

SA=¡ò2¬áy(t)¡î([x¡¯(t)]©÷+[y¡¯(t)]©÷)dt


Surface Area (cartesean)

SA=¡ò2¬áf(x)¡î(1+[f'(x)]©÷)dx


Greatest upper bound

highest number a set approaches
GUB of (4,1]=1 

The limit as n goes to infinity of x^n is

¡Þ x>1
1 x=1 0 1<x<1 DNE x¡Ü1 

Indeterminate Forms

0/0, /, 0(), , 0^0, ^0, 1^


Infinate Series

Given a sequence of {a sub n} an expression of the form a1 + a2 +a3. . .+an is called an infinate series


Series vs. Sequence

sequence: adding terms to form new terms
series: a f(x) defined on the set of positive integers 

sequence of partial sums

Sn=a0 + a1 + a2 + a3. . .+an=


geometric series

sum going from k=0 to infinity of x^k


When does a^k converge? diverge?

lxl < 1, then converges to 1/1x
lxl > or = 1, then diverges 

Formula for the Partial Sum of a Geometric Series

1/1x


pseries

1/k^p


pseries converge? diverge?

converge if p>1


Basic Divergence Test

If Ak does not got to O, then the sum/series of parital sums diverges


harmonic series

1/k


Integral Test

the sum of f(k) converges if the integral from 1 to infinity of f(x) converges


Basic Comparison Test

nonnegative terms
Ak<Bk for all terms sufficiently large a)if Bk convergesAk converges b)if Bk divergesAk diverges 