Functions
Intro
I assume you know what a function is: y = f(x) is what is traditionally shown. A concept as "simple" as this contains many, many intricacies as you study them more. Let the journey begin.
Definition
Math: A function maps a number to another number. y = f(x) is a function that maps a number x to another number y. The function is usually a rule, which can be represented as an equation. "y = 3 +x" maps(associates) any number x to another number y. If x is 4, then it is mapped to 7.
Intuition: These analogies will be helpful when we discuss properties of functions.
A useful way to think of this is to imagine a flat sidewalk and a display. Each step on the pavement is labeled. The number on the display depends on what step on the pavement you are on.
Display = f(step). This means the value of the display is a function (depends on) on what step you are on.
Another analogy is to image the sidewalk on the ground. There is no display, and the value of the function is your altitude.
Now that we have these intuitive understandings of functions, we can use them to understand properties of functions.
Graphs:
Now, when we graph a function, we put the independent variable (x) on the horizontal axis. The equation y = f(x) means "Y is a function of x", or "Y depends on x". Therefore, y is the dependent variable (and goes on the vertical axis). The hill analogy is useful when drawing the graph. Draw each horizontal position (step), and then draw the corresponding height. For the sidewalk/display example, walk down the sidewalk and plot out the value of the display.
Derivatives:
A derivative is the rate of change of a function. You probably know slope as "rise over run", or the change in y divided by the change in x. Let's think more about these definitions.
Think of the derivative as the steepness of the hill. Each point on the hill can have its own steepness. Thus, the derivative is itself a function - for every point on the hill, it gives you tells you the steepness.
Billboard example: current value:
increasing/decreasing:
Obeservations:
The derivative is an accurate short-term prediction. At any point, it tells you whether your function is going up or down (for the short term).
Math: Best linear approximation: f(x + a) ~ f(x)+ a* f'(x)
The derivative is the instantaneous rate of change of the function at a point. Because of this,
The sign of the derivative tells you how to improve the value of your function. If the derivative is positive, it means your function increase if you move forward. Walketh forward!
If your derivative is negative, it means if you go wordward your function will decreasing - move backwards on your sidewalk.
The size (magnitude) of the derivative tells you how much you can change your function. If you derivative is 1, if you increase x by one you increase y by one. "Sure, go ahead, you'll do a little better.". If your derivative is 1 million, the functions is saying "Go go go go go!! Run forward!!" If your derivative is negative one million, the function is saying "Whoa, whoa, what the heck are you doing? Go backwards, now!".
I don't think functions actually talk, but you get the idea.
