Math 12
Terms
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- 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:
f(f¯¹(x))=x - inverseÂ’(b)=
- 1/fÂ’(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
- gÂ’(x)=g(x)[gÂ’1(x)/g1(x) + gÂ’2(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 1/1-x
lxl > or = 1, then diverges - Formula for the Partial Sum of a Geometric Series
- 1/1-x
- p-series
- 1/k^p
- 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
- 1/k
- 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