Mathematics Model Descriptions

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Linear

Linear equation are the simplest form of equation you may deal with; it restricts the problem to a unique unknown variable (usually called x) and no exponent (e.g. f(x) = 2x + 15).

The general form of such equation is: $$\textbf{f(x) = ax + b}$$

Any linear expression can be drawn as a line in a 2D graphic. The graph drawing is very simple: we have y, or f(x), on the vertical axis (the result) and for a range of x values (horizontal abscisse) we compute and plot the points (x, y).


In brief:
Changing a is changing the slope of the line.
Changing b is changing the offset of the line.

Quadratic

Quadratic equation are just a step further from linear equation, we still restrict the problem to a unique unknown variable (usually called ‘x’) and add one term: x² (or x*x). The general form of such equation is: $$\textbf{f(x) = ax² + bx + c } ,\: \: where\: \: a \neq 0$$

They will frequently turn up in many areas and very often make an appearance as part of the overall solution within most of the real world problems in the fields of physics, astronomy, engineering, computing, architecture...

Let's play with it online and we will quickly see that:
Changing a is changing the opening of the parabola.
Changing b is changing the slope of the parabola at x = 0.
Changing c is changing the offset of the parabola.

Translation

Just like transformations in geometry, we can translate / shift a mathematical object by changing its function.

Already seen with the previous models (linear, quadratic), we can move it up or down by adding a constant to the function:
$$f(x) → f(x) + b$$


To move it left or right, we add a constant to the function variable (x-value):
$$f(x) → f(x + a)$$ We can think of this as moving the "abscissa origin" to be more in advance or a bit late.


Putting it together:
$$\textbf{f(x) → f(x + a) + b}$$



Why adding a positive number a to the variable does move the function to the left (the negative direction)?

Well imagine you want to start recording a movie at 8:00 o’clock with the function:
Start(t) = 8:00 o’clock.

If you change your mind to say that we want to record 5 minutes before, we will have the function:
Start(t + 5 mins) = 8:00 o’clock

Adding 5 minutes to the current time will make the record starting 5 minutes earlier (i.e. to the left direction). It's just like putting our watch in advance for an appointment.

Scaling (Stretch / Compress)

If we understand the Linear and Translate functions above, we may imagine what could be the scaling function : instead of playing with additions, we will play with multiplication factors to stretch or compress our mathematical object. Please note that a scale is a non-rigid transformation : it alters the shape and size of the graph function.

We can stretch or compress it in the y-direction by multiplying the whole function by a constant:
$$f(x) → a * f(x)$$



We can stretch or compress it in the x-direction by multiplying the function variable x by a constant:
$$f(x) → f(b * x)$$



Putting it together :
$$\textbf{f(x) → a * f(b * x)}$$

I’m sure you can now imagine why bigger b value cause more compression on the x-scale.
We could say : we put more information within the same base unit.

Inverse - Anti-function - Reverse - Reflect

Commonly written f-1(x), they are extremely useful to solve equations and are therefore a very interesting tool to express several concepts. They allow mathematical operations to be reversed (e.g. minus inverses sum, multiplication reverses division, logarithms inverses exponential etc.). Whenever a mathematical procedure is introduced, one of the most important questions is how to invert it.

Two functions are inverses of one another if they "undo each other" in the following sense: if the output of one is used as input to the other, it results to the original input.

We all naturally know a tons of basic functions and their inverse, here is some you may have been across:


The graph of f-1(x) can be obtained from the graph of f by switching the positions of the x and y axes : this is equivalent to reflecting the graph across the line y = x.

The cool thing about the inverse is that we are able to get back to the original value only with the result. When the function f(door) turns for instance an open door into a close one, then the inverse function f-1(door) turns it back to open. Thoses equations express the same idea:

$$f(x) = y \iff f^{-1}(y) = x$$ $$f^{-1}( f(x) ) = x \quad and \quad f( f^{-1}(x) ) = x$$

Differentiation - Derivative

Unlike a straight line, a curve's slope constantly changes as you move along the graph. Imagine getting prepared for a long hiking on a mountain : the curve is the mountain itself and we want to known what would be the slopes to handle and when we could stop to make our camp (hopefully when there is no slope to be flat).

To get this slope function, mathematics have one of its most wonderful tool : differentiation. Differentiation is used to find the derivative of a function and this derivative, written f’(x), is the exact function that describes the evolution of the slope. This is used to find tons of things such as extrema of a function or describing trajectories (when and how is it accelerating / decelerating) and we are excited to write a full course about it soon.



Shows a curve f(x) in green (the mountain) and its slope evolution f’(x) in blue (the derivative) :
$$f(x) = 0.2x^{3} + x^{2} - 2$$ $$f’(x) = 0.6x² + 2x$$

The evolution of the slope can be read directly along the evolution of our function which is very handy ! We may see in seconds the following interesting information :
- A is an extrema, when the slope of f is equal to 0 (it’s flat). A’ is then point on its derivative (as the slope, f’ = 0)
- B’ is the point when f’ is at its minimum : this is where the slope goes down the most.
- On C, C’ we see the second extrema : same observations as A, A’.

Using the graphing calculator will automatically compute and plot our derivative, it’s an amazing tool to visualize and check the solution of our problems.

Tangent Line

The tangent line on a point of the curve is the straight line (linear function) that runs through that point and has the same slope as the curve at this point. While the tangent line can be computed by calculating the slope on each point, this is “luckily” a direct product of our derivative f’(x) :
$$tangent_{line}(x) = f'(a) (x-a) + f(a)$$



The linear equation of the tangent line can be read at the following :
$$tangent_{line}(x) = ax + b$$ The constant a is f’(a) : the slope of the curve at the point A.
The constant b is f(a) : the value of the curve at the point A (its offset).
The variable x is (x - a) : we simply moved the abscissa origin of the tangent line to the point A (cf. translation above).

Integration - Integral

Integration is one of the two main operations of calculus with its inverse operation : differentiation. The Integral is, in mathematics, either a numerical value equal to the area under the curve (area between abscissa and graph) of a function for some interval (definite integral) or a function F(x) inverse to the derivative (indefinite integral) : finding an integral is the reverse of finding a derivative, this is why it is also called the anti derivative.

The integrals illustrated here are those termed definite integrals : a numerical value equal to the area under the curve between two point A and B (closed interval) and is given by : $$\int _{a}^{b}\,f(x)dx=\left[F(x)\right]_{a}^{b}=F(b)-F(a)$$



The operation of integration leading to F(x) is up to an additive constant, indeed :
$$f(x) = k → f’(x) = 0, \quad thus$$ $$f(x) = 0 → F(x) = any_{constant}$$ In this case, This is the function that refers to an indefinite integral and is simply written : $$F(x)=\int f(x)\,dx$$
In other words the constant may be any area offset that may exists before the point A and therefore the point B, the definite integral cancel this quantity during substraction: $$F(b) - F(a)$$

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