Difference between revisions of "User:Saul/calculus"

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(Vertical Line Test)
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You can visually test if a graphed curve is a function if you can draw a vertical line at any given point and have that line intersect '''no more than once'''.<br>
 
You can visually test if a graphed curve is a function if you can draw a vertical line at any given point and have that line intersect '''no more than once'''.<br>
 
This makes sense if you look back in the function's definition ''associates to every element of a first set exactly one element of the second set'' and hence if it intersects more than once the given x has more than one solution.
 
This makes sense if you look back in the function's definition ''associates to every element of a first set exactly one element of the second set'' and hence if it intersects more than once the given x has more than one solution.
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=== Piecewise Functions ===
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A '''piecewise function''' is a function described in pieces by applying a different formulae on different parts of it's domain.<br>
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This can be thought of as applying a condition on the inputs.<br>
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An example of this is the '''absolute''' function: '''|x|'''<br>
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This function outputs '''x''' if '''x &#8805; 0''' otherwise outputs '''-x''' if '''x &#60; 0'''<br>
  
 
== Differentiation ==
 
== Differentiation ==

Revision as of 06:25, 3 March 2020

Functions

Wikipedia provides a good definition for functions:

A function is a relation between sets that associates to every element of a first set exactly one element of the second set.

Notations Used

Some quick definitions of the notations used:
{0, 1, 2} defines a set containing the items 0, 1 and 2.
Parentheses - () defines an exclusive interval and square brackets - [] define an inclusive interval, these can be mixed for example:

(0, 10) defines an interval containing the numbers 1 to 9
[0, 10] defines an interval containing the numbers 0 to 10
[0, 10) defines an interval containing the numbers 0 to 9

defines the set of real numbers: (-∞, ∞)
means union.
\ means excluding, for example: ℝ \ 0 is the set of real numbers excluding 0.

Domain

The set the contains the valid inputs is called the function's domain.

For example take the function: ƒ(x) = x2
The domain for this function is: (-∞, ∞) or

Another example: ƒ(x) = 1/x
The domain for this function is: (-∞, 0) ∪ (0, ∞) or ℝ \ {0}

Range

The second set that contains the outputs is called the function's range.

Take the first example used in the domain section: ƒ(x) = x2
The range for this function is: [0, ∞)

And the second example: ƒ(x) = 1/x
The range for this function is ℝ \ {0}

Vertical Line Test

You can visually test if a graphed curve is a function if you can draw a vertical line at any given point and have that line intersect no more than once.
This makes sense if you look back in the function's definition associates to every element of a first set exactly one element of the second set and hence if it intersects more than once the given x has more than one solution.

Piecewise Functions

A piecewise function is a function described in pieces by applying a different formulae on different parts of it's domain.
This can be thought of as applying a condition on the inputs.
An example of this is the absolute function: |x|
This function outputs x if x ≥ 0 otherwise outputs -x if x < 0

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)

Turning Points

A turning point is the point where the gradient is 0, an equation may have multiple, quadratics usually have one, cubic functions usually have 2.

Locating

The turning point can be found by obtaining the first derivative (ƒ(x)).
Then solve for when ƒ(x) = 0.
The results of this are the x values of the turning points, the y values can be found by substituting the x values into the original equation.

Minimum and Maximum

A Minimum is the the turning point that is Concave Up.
A Maximum is the the turning point that is Concave Down.

The Minimum can be found where the first derivative (ƒ(x)) is 0 and the second derivative (ƒ′′(x)) is positive.
ƒ(x) = 0 and ƒ′′(x) > 0

The Maximum can be found where the first derivative (ƒ(x)) is 0 and the second derivative (ƒ′′(x)) is negative.
ƒ(x) = 0 and ƒ′′(x) < 0

Concave Up and Down

Concave Up is the part of the graph that is U shape.
Concave Down is the part of the graph that is n shape.
Cubic functions have one of each, quadratics have one or the other.

To find out if a point is Concave Up or Down, find the second derivative (ƒ′′(x)).
If ƒ′′(x) > 0, then the point is Concave Up.
If ƒ′′(x) < 0, then the point is Concave Down.

Point of Inflection

The Point of Inflection (P.O.I.) is the point on the graph where Concave Up changes to Concave Down.
The P.O.I. is when the second derivative (ƒ′′(x)) is equal to 0 and it changes from a negative value to a positive one immediately beforehand (or vice versa).

If:

ƒ′′(x) = 0
ƒ′′(x - 0.1) < 0
ƒ′′(x + 0.1) > 0

Or:

ƒ′′(x) = 0
ƒ′′(x - 0.1) > 0
ƒ′′(x + 0.1) < 0

Then:

x = P.O.I.

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)