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

Lesson Overview
Content Standards
File Attachments
Additional Comments

Subject(s): Mathematics

Grade Level(s): 11, 12

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


Lesson Overview


: Mathematics
: Eight through Twelve
: 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.

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

Student Learning Objectives


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


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: (

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

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

download the file Additional Comments (Lecture 3.5) Maxima and Minima Application Problems
  — CommentsSec3.5.doc   (21.5 KB)

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

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

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

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

download the file Student Power Point presentation of Volume Lab

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

download the file OptimizationPriscila
  — PriscilaOptimization.ppt   (41.0 KB)

download the file Optimization David
  — DavidOptimization.ppt   (98.0 KB)

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

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

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

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

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

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
(3) (Optional) Development of Power Point presentation of Lab
Construction project. ((See sample evaluation rubric in
attachments.)) ((See Sample Student ppt Presentations in
(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.
Additional Comments
See Additional Comments: (Lecture 3.5) Maxima and Minima Application Problems, in the attachments.