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Lesson Series

A series of lessons detailing the geometry of functions. We discuss extreme values, mean values, maximums, minimums, critical points, concavity and inflection points.
The Intermediate-Value Theorem is the theorem that really proves our original intuition that a continuous function is one that can be drawn on a piece of paper without having to lift the pencil from the paper to draw it.
We provide the intuition and the rigorous mathematical definition of increasing and decreasing functions.
We describe how a function continuous on $[a,b]$ will take on both a maximum value and a minimum value on $[a,b]$.
We discuss the Mean-Value Theorem which states that for a continuous function the slope of a secant line is equal to the derivative at some point.
We show how the first derivative can be used to identify when a function is either increasing or decreasing on an interval.
We provide the definition and graphical intuition behind the concepts of maximum and minimum. A maximum is the top of a hill and a minimum is the bottom of a valley.
We discuss how critical points are defined as those points where $f\,'(x)=0$. We discuss the Critical Point Theorem that discusses the relationship between critical points and maxima and minima of functions.
The First Derivative Test is a method for determining whether there is a maximum or minimum in some interval.
The Second Derivative Test allows us to find out whether a critical point is a maximum or minimum by knowing the sign of the second derivative.
In this lesson we define the concept of concavity and show in pictures when a function is concave up or concave down.
Concavity is defined by whether a function's first derivative is increasing or decreasing. We show how to use a function's second derivative to determine if it's first derivative is increasing or decreasing.
An inflection point is where a function's concavity changes. For example, the point where a function goes from being concave up to concave down is an inflection point. In this lesson we define an inflection point, show how they can be identified, and provide examples.