Description

We next focus our attention on applications of the derivative. We invistigate the idea of rectilinear motion and the derivative; related rates; extrema of functions; using Rolles Theorem and the Mean Value Thorem; Graphing using the first derivative and the second derivative and further applications of extrema including applications of the derivative in economics.

Sections of this topic include:

1. Rectilinear Motion and the Derivative

   One of the first applications of the derivative is for rectilinear motion (motion in a straight line). Groups in this section include: finding position velocity and acceleration at some time t given a position function; exploring rectilinear motion; and determining when a partical is slowing down.

2. Related Rates

   A very useful application of derivatives is to find related rates (when one rate is dependent on another). The group within this section is caleld: solving related rates problems.

3. Extrema of Functions

   In this section, we define extrema to be maximum or minimum values of a function at a point. To find these critical points, we use the derivative. Groups within this section include: finding critical points; and finding absolute extrema on an interval.

4. Rolle's Theorem and the Mean Value Theorem

   In 1691, Michel Rolle provided the first known formal proof of his theorem in Calculus. From there, a more general theorem called the Mean Value Theorem was formulated. Groups within this section include: using Rolle's Theorem; determining why Rolle's Theorem cannot be used from a graph; using the Mean Value Theorem; Determining why the mean value theorem cannot be used from a graph; determining where functions are increasing or decreasing.

5. Graphing and the First Derivative

   We now explore various graphing techniques including using the First Derivative Test to find relative extrema. Groups within this section include: using the first derivative test to find relative extrema.

6. Graphing and the Second Derivative

   Continuing our exploration of graphing techniques, we use the second derivative to explore concavity, inflection points, and use the Second Derivative Test to find extrema. Groups in this section include: concavity, estimating from a graph where the derivative is increasing or decreasing.

7. Further Applications of Extrema

   Using our knowledge of how to find extrema using the first derivative, we explore applications of finding the maximum or minimum values of a scenario. Groups within this section include: Finding extrema of number problems; finding maximum area under a curve.

8. Applications of the Derivative in Economics

   We conclude this topic by appying the derivative to Economics to find maximum revenue, profit and minimum average costs. Groups within this section include: finding the maximum revenue, maximum profit, and minimum average cost; computing marginal revenue; solving applied problems.