Calculus Summary Calculator

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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 = anx^{n - 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
_a∫^bƒ(x)dx = F(b) - F(a)

₀∫¹ 12x³dx is shown here giving us an answer of 45
₀∫¹ 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 a_n = a₁ + (n - 1)d
The explicit formula for the artithmetic series 1,5,9,13,17 is a_n = 1 + 4(n - 1) shown here

The explicit formula for a geometric series is a_n = 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 a_n = (2)2^{(n - 1)} shown here