User:Saul/calculus

Differentiation

We will use the function notation ƒ(x) which just applies some action to x like this function will double x: ƒ(x) = 2x
Differentiation of a function is taking a function (usually a curve) and finding the gradient at the single instant of x.
For example the function ƒ(x) = x² will represent a 'U' shaped curve, the gradient at point x will be: ƒ^′(x) = 2x
A derivative of a function will be notated with ƒ^′, second derivatives are marked ƒ^′′ and so on.

Various Notations

The following are various notations for derivatives.
Note that: y = ƒ(x) = ƒ

dy / dx
dƒ / dx
dƒ(x) / dx
y^′
[y]^′
ƒ^′
[ƒ]^′

Notation for second derivative:
ƒ^′′
d²y / dx²

Various Rules

Note: n represents a real number, and a represents a constant.

ƒ(x) = xⁿ
ƒ^′(x) = nx^n-1

ƒ(x) = axⁿ
ƒ^′(x) = anx^n-1

ƒ(x) = a
ƒ^′(x) = 0

ƒ(x) = x
ƒ^′(x) = 1


ƒ(x) = fg
ƒ^′(x) = f^′g + g^′f

ƒ(x) = f/g
ƒ^′(x) = (f^′g - g^′f) / g²


ƒ(x) = sin(x)
ƒ^′(x) = cos(x)

ƒ(x) = cos(x)
ƒ^′(x) = -sin(x)

ƒ(x) = tan(x)
ƒ^′(x) = 1 / cos²(x) = sec²(x)

Uses

Finding x-Intercept

Differentiation can be used to find the x-intercepts of a line.
The following formulae is used to find this:
x_n+1 = x_n - ƒ(x_n) / ƒ^′(x_n)
Applying this function until ƒ(x) ≈ 0
The initial x is just a guess.
Note: if this results in divergence (x does not converge at 0 or x → ∞) then x₁ was a bad guess.

Integration

Integration is the reverse of differentiation, sometimes known as anti-derivative.
Integration is usually noted like so:
∫ƒ^′(x)dx = ƒ(x) + c
Where dx shows that it will be the integral relative to x (there may be other variables in the equation).
c represents a unknown constant - this must exist because when a function is derived (differentiation) some information is lost so c is to make up for that loss.

Various Rules

k represents a constant.
∫k dx = k + c

∫kx dx = k∫x dx

∫xⁿ dx = x^{n + 1} / (n + 1)

∫(xⁿ + x^m) dx = ∫xⁿ dx + ∫x^m dx

∫( ƒ(x) ± g(x) )dx = ∫ƒ(x)dx ± ∫g(x)dx

∫(1 / x) dx = Ln(x) + c
Note: Ln(x) = Log_e(x)

Uses

Area

A use of integration is to find the area between a line and the x axis between two points on a graph.
The area between the two points a and b (where x = 0) can be found like so:

First find the derivative:

_a∫^bƒ^′(x) dx = ƒ(x)

Say that the derivative is F, then find the difference of the values for F for each limit like so:

F(b) - F(a) = Area

Note that the area will be negative if it is under the x axis and positive if above.

Also if the line crosses the x axis more than twice then there will be more than two values where x = 0 and at least one positive area and a negative area, these will at least partially cancel each other out if only the left most and right most x values are taken, so to effectively calculate the area the absolute areas between every set of points (where x = 0) muse be calculated.

Solid of Revolution

A cool use for integration is to calculate a solid of revolution - the volume of a shape that is made from the area of two points spun around a axis, for example a triangle could create a cone.
The formulae for this is as follows:
V = Π _a∫^b ƒ²(x) dx

Note: ƒ²(x) = ƒ(x) * ƒ(x)