User:Saul/calculus

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Revision as of 22:42, 5 October 2019 by Saul (talk | contribs) (Solid of Revolution)

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 Rules

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

ƒ(x) = xn
ƒ(x) = nxn-1

ƒ(x) = axn
ƒ(x) = anxn-1

ƒ(x) = a
ƒ(x) = 0

ƒ(x) = x
ƒ(x) = 1


ƒ(x) = fg
ƒ(x) = fg + gf

ƒ(x) = f/g
ƒ(x) = (fg - gf) / 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)

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 = kx dx

xn dx = xn + 1 / (n + 1)

( ƒ(x) ± g(x) )dx = ƒ(x)dx ± g(x)dx

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

abƒ(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 ab ƒ2(x) dx