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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 1-1 » 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 1-1.
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:
Domain and Range of ln(x)
Domain:(0, ‡)
Range: all Real numbers
Know graph of ln(x)
Derivative of ln(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)[g’1(x)/g1(x) + g’2(x)/g2(x)+. . .g'n(x)/gn(x)]
relationship between e^x and ln(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
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
log base p of p raised to the t =
dy/dx log x
logarithm to the base e
ln=log e
restrictions of domain for
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)
polar coordinates for a circle with a radius a centered at the origin
area of 2 parametric equations
A= ¡ò¨ö([p2ө]©÷-[p1ө]©÷)dө
Length of a curve (parametric)
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)
Surface Area (cartesean)
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 1/1-x
lxl > or = 1, then diverges
Formula for the Partial Sum of a Geometric Series
p-series 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
Integral Test
the sum of f(k) converges if the integral from 1 to infinity of f(x) converges
Basic Comparison Test
non-negative terms
Ak<Bk for all terms sufficiently large
a)if Bk converges--Ak converges
b)if Bk diverges--Ak diverges