 # Calculus Summary Calculator

## Functions

A function is an expression with a variable. y = x2 + 6x + 9 can also be written ƒ(x) = x2 + 6x + 9
The ƒ(x) piece allows you to evaluate the function at a point. In other words, you input a value, and you get a value back in return → ƒ(x) = c
ƒ(0) = 9 shown here
ƒ(1) = 16 shown here
ƒ(-1) = 4 shown here

## Derivatives

A derivative is a measure of how a function changes when it's input changes denoted ƒ'(x)
 Δƒ(x) Δx
 =
 ƒ(x + h) - ƒ(x) h

Power Rule for ƒ(x) = axn = anxn - 1
Power Rule Example: Given ƒ(x) = 3x2 + 6x + 8, we calculate ƒ'(x) at x = 1 denoted ƒ'(1) = 12 shown here

Product Rule of Derivatives for (f · g) = f · g' + f' · g = (u · v) = uv' + vu'

Quotient Rule for Derivatives. If you have a fraction for a function, your derivative using the quotient rule is shown below:
 ƒ(x)  = g(x) h(x)

 ƒ'(x)  = h(x)g'(x) - h'(x)g(x) [h(x)]2

## Integrals

Integral are the opposite of derivatives
abƒ(x)dx = F(b) - F(a)

01 12x3dx is shown here giving us an answer of 45
01 6x2 + 4x - 1dx is shown here giving us an answer of 3

## Cartesian (Rectangular) and Polar Coordinates:

Translating Cartesian Coordinates of (x,y) to Polar Coordinates of (r,θ) where r = √x2 + y2 and θ = tan-1(y/x)
Translating Cartesian Coordinates of (3,4) to Polar Coordinates of (5,53.13°) is shown here
Translating Polar to Cartesian Transformation is (r,θ) → (x,y) = (rcosθ,rsinθ)
Translating Polar Coordinates of (r,θ) = (20,30°) = (17.3205,10) shown here

## Arithmetic and Geometric Series

The explicit formula for an arithmetic series is an = a1 + (n - 1)d
The explicit formula for the artithmetic series 1,5,9,13,17 is an = 1 + 4(n - 1) shown here

The explicit formula for a geometric series is an = a1r(n - 1) where r is the common ratio between each term shown below:
 r  = an an - 1

The explicit formula for the geometric series 2,4,8,16,32 is an = (2)2(n - 1) shown here