The last unit showed you some ways the derivative may be used to solve problems in the context of realistic situations. This unit looks at analytical applications of the derivative – that is applications apparently unrelated to any kind of real situation. This is a bit misleading since the things you will learn are meant to be extended to practical problems. It’s just that for now we will study the ideas and techniques in general, not in any context.

The unit begins with two important theorems. The Mean Value Theorem that relates the average rate of change of a function to the instantaneous rate of change (the derivative), The MVT, as it is called, helps prove other important ideas especially the Fundamental Theorem of Calculus at the beginning of the integration.

The other theorem is the Extreme Value Theorem. The EVT tells you about the existence of maximum and/or minimum values of a function on a closed interval.

Both are existence theorems, theorems that tell you that something important or useful exists and what conditions are required for it to exist. More on existence theorems in my next post.

As with all theorems, learn the hypothesis and conclusion. The graphical interpretation makes these easy to understand.

You will learn how to determine where a function is increasing and decreasing. This leads to finding the maximum or minimum points – where the function changes from increasing to decreasing or vice versa: You will learn three “tests” – theorems really – to justify the extreme value.

Along with that you will learn some more about the second derivative and concavity.

These ideas and theorems will help you accurately draw the graph of a function and nail down the precise location of the important points and tell what is happening between them. Yes, your graphing calculator can do that, but you’re taking this course to learn why.

You will be asked to determine information about the function from its derivatives – plural.  The derivative may be given as a function, a graph, or even a table of values.

You will also be asked to justify your reasoning – tell how you can be sure what you say is correct. You do that by citing the theorem that applies and check its hypotheses, not by Paige’s method:

These concepts are tested on the AP Calculus exams and often produce the lowest scores of the six free-response questions. Yet, if you learn these concepts, that question can be the easiest.

P.S. Some books use the Latin words extremum (singular) or extrema (plural). They mean the extreme value(s). Maybe they have hung around so that the uninitiated will think calculus is difficult and confusing. I don’t know. Use them if you like: impress your (uninitiated) friends.

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Course and Exam Description Unit 5 Topics 5.1 through 5.9