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Level 621

Partial Fraction Decomposition


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Type 1
Distinct Linear Factors
Type 3
Distinct Quadratic Factors
Type 4
Quadratic Factors Repeated
x is constant A
Put one degree lower in the numerator
Use long division to get into proper form
Degree in numerator must be smaller than the degree in the numerator. If not you must?
Linear factors
Trick to set factor to 0 then solve when linear factor only
∫constant/linear
C * ln|linear| + C
= f(x) + p(x)/q(x)
any rational function P(x)/Q(x) can be written as...
?/(x-3)(x-2)
Ex: ?/x^2 - 5x + 6
Ex: ?/(x+1)^2 (x-7)
= A/(x-7) + B/(x+1) + C/(x+1)^2
Ex: ?/(x^2+4)(x-1)
= A/(x-1) + Bx+C/(x^2+4)
Ex: ?/(x^2+4)^2 (x+5)
= A/(x+5) + Bx+C/(x^2+4) + Dx+E/(x^2+4)^2
Partial fractions
Review this in notes!
Substitution undoes?
Chain rule
Integration by parts: formula to memorize?
(Integral of)udv = uv - (integral of) v du
The product rule
Integration by parts undoes
LIPET
Integration by part, order to choose u?
See notes
Integration by parts?
Integration by parts: defiant integrals (tips?)
Find indefiant integral then substitute integral
The integral of tan ?
-ln |(cosx) | +c
Vshell= (integral) 2(pie)rhdx
Volume formula for cylindrical shell?
Volume formula for cylindrical shells: radius
Radius is the distance from the axis of revolution to the strip (never involves function)
Volume formula for cylindrical shells: height
The "tallness" or "length" of the strip. (Will include function)
Cylindrical shells
Rectangle strip must be parallel to the axis of revolution
Indeterminate forms of limits
L'Hopital's Rule: helps us determine
(o/0) , (∞/∞)
When do you Use L'Hopitals Rule directly ?
∞•0, ∞ -∞
When do you Try to rearrange to get (o/0, ∞/∞) and then use L'Hopitals rule
L'Hopitals rule in action does what
You take the derivative of the top and bottom (if there is a top and bottom) see notes!
f(x)
f of x
X->a
Lim f(x) =Lim e^ln(f(x))
The INSTEAD and to replace the instead
when using limits for L'Hopitals rule question don't forget
Improper integrals
Have to written with limits
diverges
P-series ∑1/nᴾ....p<1
P> 1: the integral converges
When does it converge when does it diverge
Make a direct comparison!
For improper integrals that you don't knew the integral to?
Euler tangent
See notes
(dx/dt) = ky
Exponential growth formula and solution cube formula for exponential growth
Logistic growth
The grower rate is promotional to the amount present and the difference between the carrying capacity (A) and the amount present
Formula for logistic growth (memorize)
(dy/dx) = Ky (A-y)
Y = (A)/(1+Be^-AKt)
Solution curve for logistic growth (memorize)
Point of inflection:
Solution curve for logistic growth: half way to the carrying capacity what is happening to the graph?
Y= A/2
Solution curve for logistic growth: half way to the carrying capacity: What is the y value? What is the x value
(A/(1+B)) (sub in 0 for time to equation)
Solution curve for logistic growth: what is the inital amount
You start above the carrying capacity
You will not have a point of inflection on a logistic growth graph if?
But fitting it in the logistic growth equation
What is one way you can answer the "is this logistic growth?" Question?
Limits
When an using comparison for improper fraction, remember you can use?
3.Use natural logs
I ^∞, 0^ 0, ∞^0
Ln (0)
Does not exists
S f(x)/g(x) dx
format for a partial fraction
if the top is bigger
first factor, then use synthetic or long division
if the bottom is bigger
first try u sub, if that doesnt work use AB technique
For AB remember...
each power gets its own fraction, squared factor Ax+B Bx+c etc
a^u/lna +C
∫a^u du =
Make the denominators the same.
How do you add or subtract fractions?
1/4 + 1/3 = ?
3/12 + 4/12 = 7/12
How do you multiply fractions?
'Just do it' - multiply both numerators together and both denominators.
2/3 * 1/4 = ?
=2/12 which simplifies to 1/6
(6x-2)/(x-3)(x+1)
A/(x-3) and B/(x+1)
A=4
You have found that 6x-2=A(x+1)+B(x-3). Compare coefficients to find A and B.
(4x+1)/(x+1)(x-2)
1/(x+1) + 3/(x-2) =?
A/(x+4) + B/(x-1) + C/(x-1)^2
When (9x-14)/(x+4)(x-1)^2 is split into partial fractions, what are the denominators?
substituting x=-3
Substitute appropriate values of x to find values of A and B for this equation: 4x+6=A(x-3)+B(x+3)
A/(ax+b) + B/(ax+b)^2 + C/(cx+d)
Give the algebraic structure for the partial fraction of (px+q)/(ax+b)^2 (cx+d)