Lesson Series
In this lesson we want to show how we attach a real world meaning to variables and functions.
In this lesson we discuss how a derivative is a rate of change and show how speed is a derivative.
Second derivatives are calculated from first derivatives in the same way that derivatives are calculated from functions. Second derivatives explain how fast something is increasing or decreasing and geometrically describes the concavity of a function.
The Mean-Value Theorem states that given a secant line between two points there is a point where the slope of the function is the same as the slope of the secant line.
As we move from left to right an increasing function will move up and a decreasing function will move down.
Derivatives can be used to find when a function increases and decreases. If between two points $a$ and $b$ $f\,'(x)\gt 0$ then the function is increasing and if $f\,'(x)\lt 0$ then it is decreasing.
A critical point is where $f\,'(x)=0$.
A maximum is the top of a hill and a minimum is the bottom of a valley.
We discuss the relationship between critical points, maxima, and minima. In particular, at every maximum and minimum there is a critical point, however, not every critical point has a maximum or minimum.
At a critical point a maximum is when the slope (first derivative) goes from positive to negative and at a minimum the slope goes from negative to positive.
The second derivative can be used to determine when a function is concave up and concave down. When $f\,''(x)\gt 0$ then the function is concave up and when $f\,''(x)\lt 0$ then the function is concave down.
An inflection point is where a function's concavity changes. We show how second derivatives can be used to find inflection points.
