Determining the Indeterminate

When determining the limit of an expression the first step is to substitute the value the independent variable approaches into the expression. If a real number results, you are all set; that number is the limit. When you do not get a real number often expression is one of the several indeterminate forms listed below:…

Foreshadowing the FTC

This is an example to help prepare students to tackle the Fundamental Theorem of Calculus (FTC). Use it after the lesson on Riemann sums and the definition of the definite integral, but before the FTC derivation. Consider the area, A, between the graph of  and the x-axis on the interval . Set up a Riemann sum using…

Power Rule Implies Chain Rule

Having developed the Product Rule  and the Power Rule  for derivatives in your class, you can explore similar rules for the product of more than two functions and suddenly the Chain Rule will appear. For three functions use the associative property of multiplication with the rule above: So expanding with a slight change in notation: For…

The Mean Value Theorem II

The Rule of Four suggests that mathematics be studied from the analytical, graphical, numerical and verbal points of view. Proof can only be done analytically – using symbols and equations. Graphs, numbers and words aid in that, but do not by themselves prove anything. On the other hand numbers and especially graphs can make many…

The Chain Rule

Except for the simplest functions, a procedure known as the Chain Rule is very helpful and often necessary to find derivatives. You can start with an example such as finding the derivative of  .  Most students will expand the binomial to get and differentiate the result to get . They will try the same approach…

Difference Quotients II

The Symmetric Difference Quotient In the last post we defined the Forward Difference Quotient (FDQ) and the Backward Difference Quotient (BDQ). The average of the FDQ and the BDQ is called the Symmetric Difference Quotient (SDQ): You may be forgiven if you think this might be a better expression to use to find the derivative.…

Difference Quotients I

Difference Quotients & Definition of the Derivative In the second posting on Local Linearity II, we saw that what we were doing, finding the slope to a nearby point, looked like this symbolically: This expression is called the Forward Difference Quotient (FDQ). It kind of assumes that h > 0. There is also the Backwards…