The Mean Value Theorem

Another application of the derivative is the Mean Value Theorem (MVT). This theorem is very important. One of its most important uses is in proving the Fundamental Theorem of Calculus (FTC), which comes a little later in the year. See last Fridays post Foreshadowing the MVT  for an  a series of problems that will get…

The Mean Value Theorem (MVT) is proved by writing the equation of a function giving the (directed) length of a segment from the given function to the line between the endpoints as you can see here. Since the function and the line intersect at the endpoints of the interval this function satisfies the hypotheses of Rolle’s…

L’Hospital’s Rule

Another application of the derivative L’Hospital’s Rule  Locally Linear L’Hospital’s Demonstration of the proof L’Hospital Rules the Graph Good Question An AP Exam question that can be used to delve deeper into L’Hospital’s Rule (2008 AB 6) Revised from a post of November 7, 2017 There will be two extra posts this week! Check tomorrow for some…

Graphing – an Application of the Derivative.

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.…

Summer Fun

Every Spring I have a lot of fun proofreading Audrey Weeks’ new Calculus in Motion illustrations for the most recent AP Calculus Exam questions. These illustrations run on Geometers’ Sketchpad. In addition to the exam questions Calculus in Motion (and its companion Algebra in Motion) include separate animations illustrating most of the concepts in calculus…

A Note on Speed

A quick note on speed. The idea of differentiating speed to determine where it is increasing or decreasing is perfectly reasonable. . Then, Since the denominator is positive, and speed is increasing when  and have the same sign, and and speed is decreasing when they have different signs. As a practical matter, this is the…

Other Derivative Applications

Some final applications of derivatives L’Hospital’s Rule  Locally Linear L’Hospital’s Demonstration of the proof L’Hospital Rules the Graph Good Question An AP Exam question that can be used to delve deeper into L’Hospital’s Rule (2008 AB 6) Related Rate problems Related Rates Problems 1   Related Rate Problems II Good Question 9  Baseball and Related Rates Painting a…