Transcript Lesson 4.4 - Coweta County Schools

4.4
Modeling and Optimization
What you’ll learn about

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

Examples from Mathematics
Examples from Business and Industry
Examples from Economics
Modeling Discrete Phenomena with
Differentiable Functions
Essential Questions
How can we use differential calculus to solve
optimization problems?
Quick Review
1. Use the first derivative test to identify the local extrema of
y  x  6 x  12 x  9. none
3
2
2. Use the second derivative test to identify the local extrema of
y  2 x  3 x  12 x  1 Local Max  2, 19, Local Min 1,  8,
3. Find the volume of a cone with radius 4 cm and height 7 cm. 112 cm 3
3
Rewrite the expression as a trigonometric function of the angle  .
3
2
4. sin(  )  sin 
5. cos(  ) cos
6. Use substitution to find the exact solution of the following system
of equations.
x  y  4

 y  3x
2
2
1, 3 ,
 1,  3 
Strategy for Solving Max-Min Problems
1. Understand the Problem Read the problem carefully. Identify the information
you need to solve the problem.
2. Develop a Mathematical Model of the Problem Draw pictures and label the
parts that are important to the problem. Introduce a variable to represent the
quantity to be maximized or minimized. Using that variable, write a function
whose extreme value gives the information sought.
3. Graph the function Find the domain of the function. Determine what values
of the variable make sense in the problem.
4. Identify the Critical Points and Endpoints Find where the derivative is zero or
fails to exist.
5. Solve the Mathematical Model If unsure of the result, support or confirm your
solution with another method.
6. Interpret the Solution Translate your mathematical result into the problem setting
and decide whether the result makes sense.
Example Inscribing Rectangles
1. A rectangle is to be inscribed under one arch of the sine curve. What is the
largest area the rectangle can have, and what dimensions give that area?
Px, sin x 
0  x  /2
    2x
Q  x, sin x 
h  sin x
Ax    2 x sin x
A0  A / 2  0
Ax     2 x cos x  sin x 2  0
Graph it to find solutions. Ax   0 at x  0.71
The area of the rectangle is A(0.71) = 1.12.
The length is 1.72 and the height is 0.65.
Example Inscribing Rectangles
2. Two sides of a triangle have length a and b, and the angle between them is
. What value of  will maximize the triangle’s area?
1


 Hint : A  ab sin  
2


dA 1
 ab cos   0
d 2 cos   0
 

2
A Right Triangle
b

a
Example Inscribing Rectangles
3. You are a designing a rectangular poster to contain 50 in2 of printing with
a 4-in. margin at the top and bottom and a 2-in. margin at each side. What
overall dimensions will minimize the amount of paper used?
A    w  50
50
w

A    8w  4
 50

A    8  4 
 

400
A  4 
 82
 400
A  4  2  0

w4
4
8

2
2
w
4
  10 w  5
  8  18
w4  9
Maximum Profit
Maximim profit (if any) occurs at a production level at which marginal
revenue equals marginal cost.
Example Maximizing Profit
4. Suppose that r ( x)  9 x and c( x)  x  6 x  15 x, where x represents thousands
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2
of units. Is there a production level that maximizes profit? If so, what is it?
r x   9
2

c x   3x  12 x  15
3x  12 x  15  9
2
3x  12 x  6  0
2
x  4x  2  0
x  2 2
2
2  2  0.586
2  2  3.414
The maximum profit occurs at about 3.414.
Minimizing Average Cost
The production level (if any) at which average cost is smallest is a
level at which the average cost equals the marginal cost.
5. Using the following equation where x represents thousands of units,
determine if there is a production level that minimizes cost. If so, what is
3
2
it?
c x  x  6 x  15x

Marginal cost: cx   3x 2  12 x  15


c
x
2
Average cost:
 x  6 x  15
x
2
2
3x  12 x  15  x  6 x  15
2
2x  6x  0
2 x x  3  0
x3
Pg. 226, 4.4
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