4.3: Optimization
This activity is intended to supplement Calculus, Chapter 3, Lesson 7.
ID: 9609
Time required: 45 minutes
Activity Overview
Students will learn how to use the second derivative test to find maxima and minima in word problems and solve optimization problems in parametric functions.
Topic: Application of Derivatives

Find the maximum or minimum value of a function in an optimization problem by finding its critical points and applying the second derivative test. Use Solve (in the Algebra menu) to check the solution to
f′(x)=0 .  Use the command fMin or fMax to verify a manually computed extremum.
 Solve optimization problems involving parametric functions.
Teacher Preparation and Notes
 This investigation uses FMax and fMin to answer questions. Students will have to take derivatives and solve on their own.

Before starting this activity, students should go to the home screen and select
F6 :Clean Up > 2:NewProb, then press ENTER. This will clear any stored variables, turn off any functions and plots, and clear the drawing and home screens.  This activity is designed to be studentcentered with the teacher acting as a facilitator while students work cooperatively.
Associated Materials
 Student Worksheet: Optimization http://www.ck12.org/flexr/chapter/9728, scroll down to third activity.
Problem 1 – Optimization of distance and area
Students will graph the equation
They are to rewrite the function using one variable:
To find the exact coordinates of the point, students will take the first derivative (Menu > Calculus > Derivative), and solve for the critical value (Menu > Algebra > Solve), and take the second derivative. Since the second derivative is always positive, there a minimum at the critical value of
To find the
Students are to maximize the function
Students will take the first derivative and solve to find the critical value is
The maximum area is
Problem 2 – Optimization of time derivative problems
Remind students to use
The boat heading north is going from the right angle to the point northward. Its position equation is
The boat heading west is going to the right angle. At 1 pm, it is one hour from the arrival time 2 pm so it is 15 km away. Its position equation is
Students are to minimize the distance function
There is a restriction of
The time at which the distance between the boats is minimized is
Extension – Parametric Function
To rewrite the parametric equations, students will need to know that
To find when the projectile hits the ground, students are to set
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Date Created:
Feb 23, 2012Last Modified:
Nov 04, 2014If you would like to associate files with this section, please make a copy first.