User:Saul/calculus

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Revision as of 01:00, 6 October 2019 by Saul (talk | contribs) (Approximating Roots)

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) = 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)

Uses

Finding x-Intercept

Differentiation can be used to find the x-intercepts of a line.
The following formulae is used to find this:
xn+1 = xn - ƒ(xn) / ƒ(xn)
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.

Approximating Roots and Powers

Differentiation can be used to approximate roots.

For example approximating the answer to:
3√(120)

First round to the nearest known value:
3√(125) = 5

Differentiate the original equation:
ƒ(x) = 3√x = x1/3
ƒ(x) = x-2/3 / 3

Substitute the approximation:
ƒ(120) = 120-2/3 / 3
120-2/3 / 3 = 0.014

Place this into the equation of gradient:
(y - 5) / (x - 125) = 0.014

Prearrange to the formulae of a line:
y = 0.014x + 3.29

Substitute for the number to approximate with x:
y = 0.014(120) + 3.29
y = 4.934

Therefore:
3√(120) ≈ 4.934
3√(120) = 4.932 (calculated)

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)

(xn + xm) dx = xn dx + xm dx

( 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

Note: ƒ2(x) = ƒ(x) * ƒ(x)