Lesson Title: Maxima and Minima Problems
Topic/Focus Area: A.P. Calculus (Application of Derivatives)

Subject(s): Mathematics

Name: Ken Smith
Taught: Mathematics(Geometry, Algebra-2, AP Calculus)
Phone:
E-mail: ihs03@icoe.k12.cs.us
School: Imperial High

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Lesson Overview

Standards

Subject
: Mathematics
: Eight through Twelve
Strand
: Calculus

When taught in high school, calculus should be presented with the same level of depth and rigor as are entry-level college and university calculus courses. These standards outline a complete college curriculum in one variable calculus. Many high school programs may have insufficient time to cover all of the following content in a typical academic year. For example, some districts may treat differential equations lightly and spend substantial time on infinite sequences and series. Others may do the opposite. Consideration of the College Board syllabi for the Calculus AB and Calculus BC sections of the Advanced Placement Examination in Mathematics may be helpful in making curricular decisions. Calculus is a widely applied area of mathematics and involves a beautiful intrinsic theory. Students mastering this content will be exposed to both aspects of the subject.

Substrand:
11.0
Students use differentiation to solve optimization (maximum-minimum problems) in a variety of pure and applied contexts.

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

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

Resources

Content Resources (books, articles, etc.)
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.

Web Resources
URL: (fourier.math.temple.edu)

URL: (147.4.150.5)

URL: (www.math.montana.edu)

URL: (www.exambot.com)

Hardware/Software Resources (computers, CD-ROMs, TV, VCR, etc.)
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

Lecture Notes (Sec 3.5) Maxima and Minima Application Problems
— MaxMinLectureCalc3-5.doc   (38.5 KB)

Optimizing an open box (power point demo presentation)
— MaxBoxVol.ppt   (651 KB)

Maximum volume of cone in sphere (power point lab)
— CalcVolLab.ppt   (1004 KB)

AP Calculus Test (Ch 3.1 thru 3.5)
— CalcCh3-1to5.doc   (487 KB)

Photos of Maximum Volume Lab
— MaxVolLabPhoto.ppt   (2.96 MB)

Student Power Point presentation of Volume Lab
— DENISSE'S_OPTIMIZATION_LAB.ppt   (45.5 KB)

Evaluation Rubric for Optimization Power Point Lab
— EvalofOptimizationLab.doc   (41.5 KB)

OptimizationPriscila
— PriscilaOptimization.ppt   (41.0 KB)

Optimization David
— DavidOptimization.ppt   (98.0 KB)

Solana Optimization PPT Presentation
— KimberOptimization.ppt   (228 KB)

Student Optimization Project Presentation Photos
— StudentOptPres.ppt   (895 KB)

Williams Optimization PPT Presentation
— Presentation1.ppt   (225 KB)

Morales Optimization PPT Presentation
— DenieseOptimization.ppt   (64.5 KB)

Burk Optimization PPT Presentation
— AmberOptimization.ppt   (125 KB)

Assessment
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.