AP Calc BC Memorized Formulas

Limit Definition of Derivative

lim as x approaches a = f(x)-f(a)/(x-a)

FTC 1

On the integral of a to b of f(x)dx =F(b)-F(a) where F’(x)=f(x)

lim as b approaches -inf of arctanb

-pi/2

lim as b approaches inf (1-3/b) to the b

=e^-3

IVT terms and conditions

If a function is continuous on a closed interval [a, b], and y is between f(a) and f(b) then it takes on every value between f(a) and f(b) at least once on that open interval

MVT

If a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) such that the derivative at that point equals the average rate of change of the function over [a, b].

Derivative of an inverse

1 state what you need to find f^-1 (a) 2) f(b)=a 3) f’(b) 4) f^-1 (a) =1/f’(b)

Exponential growth/decay differential and solution

dy/dt=ky y(t)=ce^kt

area of an equilateral triangle

The area of an equilateral triangle can be calculated using the formula A = (sqrt(3)/4) s², where s is the length of a side.

Area of a trapezoid

The area of a trapezoid can be calculated using the formula A = (1/2)(b1 + b2)h, where b1 and b2 are the lengths of the two bases and h is the height.

Average Value Formula

The average value of a function f on the interval [a, b] is given by 1/b-a int on a to b abc value of f’(x)dx

Average rate of change formula

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

Newton’s Cooling Law Differential and Solution

dT/dt=K(T-Tsub s) T=T sub s + (T0-Ts) e^kt

EVT

states that a continuous function on a closed interval [a, b] must attain both a maximum and a minimum value within that interval.

Logistic Growth Differential and Solution

dP/dt = kP(1 - P/M), P(t) = M/(1 + Ae^{-kt}) A=M-Po/Po

Volume of disc

pi int a to b [R(x)]² dx

Volume of washer

V=pi int from a to b [R(x)] -[r(x)]² dx

Euler’s method

x, y, m, m(change in x), y new

Integration By Parts

uv- int vdu using LIATE

Parametric Derivatve

dy/dt / dx/dt =dy/dt

Parametric second derivative

d/dx (dy/dx0 over dx/dt

Speed

abs value of speed is the square root of (x’)² + y’(t)²) and is also arc length

Arc lengths formulas

int of a to b square root 1+ f’(x) ² dx or of the square root [r²]+[dr/d theta]² d theta

TD traveled

t intital to t final integral of abs value of v dt or int of t0 to tf square root x’(t)² + y’(t)² dt

Polar Form of Area Equation

Area = 1/2 int of theta1 to theta2 r² dtheta

Sum of a Geometric Series

a sub (n+1) over (1-abs value of r)

Maclaurin Series of 1/(1+x)

summation n=0 to inf (-1)^n x^n =1+x+x²/2!+x³/3!

Macluarin Series for cos(x)

summation n=0 to fin (-1)^n x²n= 1 - x²/2! + x⁴/4! - x⁶/6! + …

Maclaurin Sries of e^x

summation n=0 to inf (x^n/n!) = 1 + x + x²/2! + x³/3! + …

Maclaurin Series for sin(x)

summatin n=0 to inf (-1)^n x^(2n+1)/(2n+1)! = x - x³/3! + x⁵/5! - …

Lagrange Error

If P sub n (x) is the nth degree Taylor polynomial of f(x) anout c and abs value of f to the (n+1) (t) is less than or equal to M for alll t between x and c, |f(x)-P sub n (x)| less than or equal to (M/(n+1)!) |x-c|^n+1

Alt series error

|Sinf-Sn| less than or equal to |an+1| = abs value first negleted term.

If S sub n = summation of (-1)^n a sub n is the nth partial sum of a convergent alt series if then the first sentence.