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…

Type 7 Questions: Miscellaneous

Any topic in the Course and Exam Description may be the subject of a free-response or multiple-choice question. There are topics that are not asked often enough to be classified as a type of their own. The two topics listed here have been the subject of full free-response questions or major parts of them. Other…

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…

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. Here are some previous post on the MVT: Fermat’s Penultimate Theorem   A lemma for Rolle’s Theorem:…

Extreme Values and Linear Motion

Two more applications of differentiation are finding extreme values and the analysis of linear motion. Extreme Values The Marble and the Vase Extremes without Calculus A Standard Problem Far Out! Linear Motion – Motion on a Line  Type 2 Problems Motion Problems: Same Thing, Different Context The Ubiquitous Particle Motion Problem  – a PowerPoint Presentation and…

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.…