# UTPA STEM/CBI Courses/Calculus/Applied Optimization Problems

Course Title: Calculus 1
Lecture Topic: Using the first Derivative to solve optimization Problems
Instructor: Hanan Amro
Institution:South Texas College

## Backwards Design

Course Objectives
* Primary Objectives- By the next class period students will be able to:
** Solve applied Maximum and Minimum Problems using Calculus
* Sub Objectives- The objectives will require that students be able to:
** Identify the knowns and unknowns for each problem.
** Write the primary equation from the given information.
** Determine the feasible domain of the primary equation.
* Difficulties- Students may have difficulty:
** Coming up with equations to set up the the problems
** Determining the derivative of trigonometric function, particularly when using methods such as the Chain Rule
* Real-World Contexts- There are many ways that students can use this material in the real-world, such as:
** Minimizing the cost
** Maximize the profit
** Desiging a suit case
** Minimize the dimension of a poster
** Maximize the sensitivity to medicine.
Model of Knowledge
* Concept Map
** Set up the equations from known and unkown informations
** Determine the critical values
** Set the first derivative equal to zero
** Determine the Local Extrema (Maximum and Minimum)
* Content Priorities
** Enduring Understanding
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** Important to Do and Know
*** First Derivative Test for Absolute Extrema
Let I be the interval of all possible optimal values of f(x). Further suppose that f(x) is continuous on I , except possibly at the endpoints. Finally suppose that x = c is a critical point of f(x), and that c is in the interval I. If we restrict x to values from I (i.e. we only consider possible optimal values of the function) then,
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** Worth Being Familiar with
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Assessment of Learning
* Formative Assessment
** In Class (groups)
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** Homework (individual)
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* Summative Assessment
** In class quiz over the subject
** Test involve topic similar to the challenge

## Legacy Cycle

OBJECTIVE
By the next class period, students will be able to:
* Convert the word problem into mathematical equation
* Solve the equation using first derivative test.
The objectives will require that students be able to:
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THE CHALLENGE
A farmer plans to fence a rectangular pasture adjacent the river seen in the diagram. The pasture must contain 245,000 square meters in order to provide enough grass for the herd. What dimensions will require the least amount of fencing if no fencing is needed along the river
GENERATE IDEAS
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MULTIPLE PERSPECTIVES
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RESEARCH & REVISE
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TEST YOUR METTLE
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GO PUBLIC
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## Pre-Lesson Quiz

Various material can be tested here, depending on time constraints.

## Test Your Mettle Quiz

* Suppose that c(x) = x$^3$ -20x$^2$ +20,000x is the cost of the manufacturing x items. Find the level of a production level that will minimize the average cost of making x items.
* A rectanluar plot of farmland will be bounded on one side by the river and on the other three sides by a single-strand electric fence. With 800 m of wire at disposal, what is the largest area you can enclose, and what are its dimension?
* We want to construct a box with a square base and we only have 10 m2 of material to use in construction of the box. Assuming that all the material is used in the construction process determine the maximum volume that the box can have.
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