Derivative Applications – Graphing

Graphing and the analysis of graphs given (1) the equation, (2) a graph, or (3) a table of values of a function and its derivative(s) makes up the largest group of questions on the AP exams. Most of the other applications of the derivative depend on understanding the relationship between a function and its derivatives.…

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Implicit Differentiation and Inverses

Implicit differentiation of relations is done using the Chain Rule.  Implicit Differentiation (from last Friday’s post. I discovered I never did a post on this topic before!) Implicit differentiation of parametric equations A Vector’s Derivative The inverse series  This series of posts reviews and expands what students know from pre-calculus about inverses. This leads to…

Implicit Differentiation

I discovered in doing next week’s post that I apparently never wrote about implicit differentiation. So here goes – an extra post this week! Implicit Differentiation The technique of implicit differentiation allows you to find slopes of relations given by equations that are not written as functions, or may even be impossible to write as…

Teaching and Learning Theorems

Theorems are carefully worded statements about mathematical facts that have been proved to be true. Important (and some not so important) ideas in calculus and all of mathematics are summarized as theorems. When you come across a theorem you need to understand it; the author of your textbook would not have included it and the…

The Chain Rule

Most of the function students are faced with in beginning calculus are compositions of the Elementary Functions. The Chain Rule allows you to differentiate composite functions easily. The posted listed below are ways to introduce and then use the Chain Rule. Experimenting with a CAS – Chain Rule  Using a CAS to discover the Chain…

Differentiation Techniques

So, no one wants to do complicated limits to find derivatives. There are easier ways of course. There are a number of quick ways (rules, formulas) for finding derivatives of the Elementary Functions and their compositions. Here are some ways to introduce these rules; these are the subject of this week’s review of past posts. Why…

Difference Quotients

Difference quotients are the path to the definition of the derivative. Here are three posts exploring difference quotients. Difference Quotients I  The forward and backward difference quotients Difference Quotients II      The symmetric difference quotient and seeing the three difference quotients in action.  Showing that the three difference quotients converge to the same value. Seeing Difference…