AP Calc BC

r²=

X²+Y²

Slope of polar curves: Dy/Dx=

(dy/dtheta)/(dx/dtheta)

The area of the region between the origin and the curve

A=int AtoB ½ r²dtheta

Area between polar curves

A=int AtoB ½ (r2²-r1²)dtheta

x=?

rcostheta

y=?

rsintheta

Displacement from t=a to t=b

(int AtoB v1(t) dt, int AtoB v2(t)dt)

Total distance traveled?

int AtoB sqrt((v1(t))²+(v2(t))²)dt

Speed of a vector

magnitude of V(t)= sqrt(V1²+v2²)

Arc length of a parametrized curve

int T1toT2 sqrt( (dy/dt)²+(dx/dt)²)dt

d²y/dx²=

(dy’/dt)/(dx/dt)

Remainder estimation theory

|Rn(x)|<=M((r^n+1*|x-a|^n+1)/(n+1)!)

1/(1-x)

series from 0 to infinity, x^n (|x|<1)

1/(1+x)

series from 0 to inf, -1^n x^n (|x|<1)

e^x

series from 0 to inf, (x^n/n!) (all real x)

Sin x

series from 0 to inf, -1^n ((x^(2n+1))/(2n+1)!) for all real x

cos x

series from 0 to inf, -1^n (x^(2n+1)/2n!, all real x

ln(1+x)

series from 1 to inf, -1^n-1 x^n/n (-1<x<=1)

tan^-1(x)

series from 0 to inf, -1^n (x^(2n+1)/2n+1, |x|<=1

what happens when L’hopitale reaches #/0 or 0/#

if numerator function approaches 0 then the limit is 0, if the denominator approaches 0 then the limit is infinity

fluid pressure

p(force per unit area) P=wh h=depth, w=weight density

force on one side

integral from 0 to H, (density)(one side of base)(height)

Hooke’s law

force=constant K(x)
Work(work)=int 0toH (Kx)dx

Length of a smooth curve

Int from AtoB sqrt(1+(dy/dx)²)dx

integrating cylindrical shells

int AtoB 2pi (shellradius)(shell height)dx
radius is the opposite axis we are revolving around
height is (upper curve-lower curve)

Area of a circle

A=piR²
A=pi(outer²-inner²)dx

Volume/washers

int AtoB pi(outer²-inner²) dx

half life

Ln(2)/k

average value over an interval

1/(b-a) int AtoB

revolved around a line y=k

upper and lower radius become k-r

areas between curves

A=int AtoB (f(x)-g(x)) dx
intersection points give limit of integration
right-left

sin(2x)

2sin(x)cos(x)

cos(2x)

cos²(x)-sin²(x)

2cos²(x)-1

1-2sin²(x)

sec²(x)

1+tan²(x)

csc²(x)

1+cot²(x)