AP Calculus - Sequences and series equations + Exam hard equations

nth term test tests for

divergence

nth term test is divergent if…

the limit of the function as n goes to infinity is NOT 0

geometric test only converges if

-1<r<1

integral test qualifications

positive, continuos, and decreasing

integral test is converging or diverging if

the integral from n to infinity is converging or diverging (respectively)

P series test is converging if

p >1

P series test is diverging if

p<=1

direct comparison test converges when (blank) function converges

bigger

direct comparison test diverges when (blank) function diverges

smaller

criteria for limit comparison test

the limit of the original function being divided by the function it is being compared to equals a number that is not 0 or infinity.

alternating series will converge

f(n+1) < f(n) and the limit as n goes to infinity for f(n) is 0

absolute convergense test

if the absolute values of a function converges, then the unabsolute value will converge as well.

ratio test converges

if the limit of the absolute value of a(n+1)/a(n) is < 1

ratio test diverges

if the limit of the absolute value of a(n+1)/a(n) is > 1

the ratio test is iconclusive

if the limit of the absolute value of a(n+1)/a(n) =1

(d/dx) log base b of x

1/(xlnb)

(d/dx) b to the power of x

b to the x times lnb

(d/dx) tanx

sec squared x

(d/dx) secx

secxtanx

(d/dx) arcsinx

1/(squareroot 1-x²)

(d/dx) arccosx

-1/(squareroot 1-x²)

(d/dx) arctanx

1/(1+x²)

integral of udv =

uv - integral of vdu

1/(cx+d)(hx+k) =

A/(cx+d) + B/(hx+k)

Lagrange error

error <= abs. val of (f^(n+1)(c )(b-a)^(n+1))/(n+1)!)

logistic dP/dt =

(k/P)P(M-P)

logistic P =

M/1+Ce^-kt

carrying capacity

M

Taylor series

f^n(a)/n!(x-a)^n starting at n =0

Euler method

(x,y)|dy/dx| change in x | (dy/dx) times change in x = change in y

AROC

F(b)-f(a)/b-a

MVT

if f(x) is continuos and defferentiable on (a,b), there must be some point c were F^1(c ) = AROC for a and b.

IVT

a function f(x) that is continuos on [a,b] takes on every y value between f(a) and f(b)

EVT

if f(x) is continuos on [a,b] then f(x) must have both an absolute min and an absolute max on the interval [a,b]

Arc length (cartesian)

integral from a to b of sqrt. (1+(dy/dx)²)

arc length (parametric) and also total distance traveled

integral from t1 to t2 sqrt. ((dx/dt)² + (dy/dt)²)

speed

sqrt. ((dx/dt)² + (dy/dt)²)

polar area

½ (intergral from theta1 to theta 2 r²)

area of a trapezoid

1/2h(b1+b2)