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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
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Rn (look up)
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When does a function have an inverse?
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When it's 1-1 » if x=x2 then F(x) = f(x2)
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How do you find the inverse?
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Switch x and y » solve for y
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How do you prove a function is one to one?
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f(x) is strictly monotomic if it is increasing or decreasing on an interval I. If f(x) is monotomic it is 1-1.
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What is the relationship between a function and its inverse?
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the inverse is the unique function with domain equal to the range of f that satisfies the equation:
f(f¯¹(x))=x |
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inverse(b)=
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1/f(a)
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Domain and Range of ln(x)
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Domain:(0, )
Range: all Real numbers |
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Know graph of ln(x)
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¡
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Derivative of ln(x)
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1/x
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The definition of ln(x) or L(x)
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L(x)= çdt/t (integral from 1 to x) x>0
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Logrithmic Differentiation
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g(x)=g(x)[g1(x)/g1(x) + g2(x)/g2(x)+. . .g'n(x)/gn(x)]
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relationship between e^x and ln(x)
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ln(e^x)=x
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defininition of the number e
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e is where L(x)=1
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Domain and Range of e^x
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Domain: all Real numbers
Range:(0, ) |
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know graph of e^x
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__/
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derivate/integral of e^x
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dy/dx e^x = e^x
ç e^x = e^x dy/dx |
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Definition of exponential function to the base "a"
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a function of the form F(x)=p^x
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Domain and Range of a^x
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Domain:all real numbers
Range:(0,) |
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dy/dx of a^x
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a^x (ln x) du/dx
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integral of a^x
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1/ln a (a^x) + C
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dy/dx of x^n where n is a variable
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set f(x)=x^n
use log diff |
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the logarithm of x to the base p
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log x = ln x/ln p
p |
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log base p of p raised to the t =
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t
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dy/dx log x
a |
1/xlnp
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logarithm to the base e
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ln=log e
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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] |
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domain and range of inverse sin of x
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D: [-1,1]
R: [¬á¨ö, -¬á¨ö] |
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domain and range of inverse tangent of x
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D: (-¡Ä, ¡Ä)
R: (-¨ö¬á, ¨ö¬á) |
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domain and range of inverse secant of x
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D:[0, ¨ö¬á) U (¨ö¬á, ¬á]
R:(-¡Ä, -1] U [1, ¡Ä) |
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Relation between [r, ө] and (x,y)
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x=rcosө
y=rsinө |
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polar coordinates for a circle with a radius a centered at the origin
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r=a
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area of 2 parametric equations
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A= ¡ò¨ö([p2ө]©÷-[p1ө]©÷)dө
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Length of a curve (parametric)
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L=¡ò¡î([x¡¯(t)]©÷+[y¡¯(t)]©÷)
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Length of a curve (cartesian)
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L= ¡ò¡î(1+[f¡¯(x)]©÷)dx
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Length of a curve (polar coordinates)
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L= ¡ò¡î([p(Ө)]©÷+[p¡¯(Ө)©÷]dӨ
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Surface Area (parametric)
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SA=¡ò2¬áy(t)¡î([x¡¯(t)]©÷+[y¡¯(t)]©÷)dt
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Surface Area (cartesean)
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SA=¡ò2¬áf(x)¡î(1+[f'(x)]©÷)dx
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Greatest upper bound
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highest number a set approaches
GUB of (-4,-1]=-1 |
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The limit as n goes to infinity of x^n is
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¡Þ x>1
1 x=1 0 -1<x<1 DNE x¡Ü1 |
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Indeterminate Forms
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0/0, /, 0(), -, 0^0, ^0, 1^
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Infinate Series
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Given a sequence of {a sub n} an expression of the form a1 + a2 +a3. . .+an is called an infinate series
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Series vs. Sequence
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sequence: adding terms to form new terms
series: a f(x) defined on the set of positive integers |
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sequence of partial sums
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Sn=a0 + a1 + a2 + a3. . .+an=
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geometric series
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sum going from k=0 to infinity of x^k
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When does a^k converge? diverge?
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lxl < 1, then converges to 1/1-x
lxl > or = 1, then diverges |
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Formula for the Partial Sum of a Geometric Series
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1/1-x
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p-series
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1/k^p
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p-series converge? diverge?
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converge if p>1
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Basic Divergence Test
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If Ak does not got to O, then the sum/series of parital sums diverges
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harmonic series
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1/k
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Integral Test
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the sum of f(k) converges if the integral from 1 to infinity of f(x) converges
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Basic Comparison Test
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non-negative terms
Ak<Bk for all terms sufficiently large a)if Bk converges--Ak converges b)if Bk diverges--Ak diverges |