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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)