**Lesson Title: Maxima and Minima Problems**

Topic/Focus Area: A.P. Calculus (Application of Derivatives)

**Subject(s):** Mathematics

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__Lesson Overview__

__Student Learning Objectives__

- Many problems that arise in science and mathematics require finding the largest or smallest values that a differentiable function can assume on a given domain. In the previous lesson, students were introduced to the Maxima and Minima Theory, and learned how to use derivatives to determine where functions take on minimum or maximum values. In this lesson, students will utilize the Minima and Maxima Theory and develop a five step strategy to solve applied optimization (maxima and minima) problems.

__Activities__

- See Attachments: Lecture Notes (Sec 3.5) Minima and Maxima Application Problems, for detailed description of this three day activity.

__Resources__

Course Textbook: Thomas-Finney, "Elements of Calculus and Analytic Geometry": Addison-Wesley Publishing, 1989, p.135-203.

Supplemental Text: Leithold, "The Calculus 7 of a Single Variable" : Harper Collins College Publishers, 1996, p.219-228.

Supplemental Text: Larson-Hostetler-Edwards, "Calculus of a Single Variable", Houghton Mifflin Company, 1998, p.205-214.

Supplemental Text: Larson-Hostetler, "Calculus With Alalytic Geometry", D.C.Heath and Company, 1986, p.215-223.

Supplemental Reference: REA's Problem Solvers 'Calculus' p.239-294.

URL: (fourier.math.temple.edu)

URL: (147.4.150.5)

URL: (www.math.montana.edu)

URL: (www.exambot.com)

Computer, large screen(36 inch) TV with converter or Projector, Ti-83 Graphing Calculator w/o'head viewscreen, Chalkboard, Ruler, Vernier Caliper, String, Clear Plastic Sphere(must be able to separate into two halves), Scissors, Sand, Balance Beam, Scanner/Copier, Digital Camera.

Miscellaneous Software: Power Point, Word, Math Type, Photo Delux, Ti-GraphLink, Geometer's SketchPad.

__File Attachments__

Additional Comments (Lecture 3.5) Maxima and Minima Application Problems

—*CommentsSec3.5.doc*
(21.5 KB)

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Students will be assessed on mastery of this topic via a series of evaluations as follows:

(1) Textbook Homework Assignment. (Pg.199, problems 1, 2, 11,

12, 22, and 24)

(2) Lab Construction: Maximum Volume of a Cone inside a

Sphere.

(3) (Optional) Development of Power Point presentation of Lab

Construction project. ((See sample evaluation rubric in

attachments.)) ((See Sample Student ppt Presentations in

attachments.))

(4) Graphing Calculator determination of a maximum value,

derived from graphs of Primary Function and Derivative, using

the calculator's maximum and zero menus.

(5) Solution of Optimization Application Problems located at

Teacher assigned Internet Sites.

(6) Formal topic test covering Chapter 3.1 through 3.5. (See

copy of AP Calculus Test (Ch 3.1 thru 3.5) in attachments.)

(7) (Optional) Individual Student generated Optimization

Application Proplem (maximum volume of an inscribed shape),

using Power Point Presentation and physical construction of

the calculated maximum dimensions/shape. (Note: This makes

a good weekend follow-up assignment.) ((See sample student

ppt presentations in attachments.))

(8) Student competition. Class sets up onto groups to judge

student power point presentatations, based on a specific,

teacher designed, judging rubrics.

See Additional Comments: (Lecture 3.5) Maxima and Minima Application Problems, in the attachments.