Transcript Lesson presentation 4.4 Optimization

Lesson 4.4
Modeling and Optimization
What you’ll learn about
•Using derivatives for practical
applications in finding the maximum and
minimum values in business, economics,
and mathematical contexts
Strategy for Solving Max-Min Problems
• Understand the problem – write down what you are looking for and
identify needed information
• Develop a mathematical model –
Draw and label pictures as needed. Write 2 equations. The 1st will
find the extreme value needed. The 2nd is a constraint and can be
used to write the 1st equation in terms of 1 variable. Use substitution
to get a function whose extreme value answers your question.
• Graph the function. Find the domain and determine what values
make sense in the problem.
• Find f’ of your model & identify the critical points and
endpoints. Test to determine if you have maximum or minimum
values.
• Interpret the Solution. Translate your mathematical result into a
sentence, confirm that it makes sense.
Example 1: Using the Strategy
Find 2 non-negative numbers whose sum is 20 and whose product is
as large as possible.
What do we want to find?
• Identify the variables and write 2 equations. Use substitution to get
a function to maximize.
• Simplify, find f ‘, critical points (including endpoints!)
• Verify max or min value
• Answer question in sentence
You try:
Find 2 numbers whose sum is 20 and the sum of their squares is as
large as possible.
What if we want the sum of the squares as small as possible?
Example 2 Inscribing Rectangles
•
A rectangle is to be inscribed under one arc of y = sin x. What is the largest area
the rectangle can have, and what dimensions give that area?
Draw and label graph.
We know A(x) = x • y and y = sin x
Use substitution to get a function for the area of the rectangle.
What dimensions maximize area?
Can’t solve algebraically, too messy.
Using A(x), find a maximum value on calculator.
Using A’(x) = 0, find a maximum value.
THEY SHOULD BE THE SAME!!!
Interpret your findings: Give maximum length, height, and area.
Example 3: Fabricating a Box
An open top box is to be made by cutting congruent squares of side length “x” from the
corners of a 20 x 25 inch sheet of tin and bending up the sides.
How large should the squares be to make the box hold as much as possible?
What is the resulting maximum volume?
•
Draw a diagram, label lengths
•
•
V(x) = length • width • height
define length, width, height in terms of x, define domain
•
•
Solve graphically, Max of V = Zeroes of V’, confirm analytically
Take derivative, find critical points, find dimensions (use 2nd derivative test to
confirm max and min values)
•
Find volume, answer question in a sentence, including units.
Example 4: Designing a Can
You have been asked to design a one liter oil can shaped like a right
circular cylinder. What dimensions will use the least material?
What are we looking for?
Given: Volume of can = 1000 cm3
Volume formula:
Surface Area formula:
Use substitution to write an equation.
Solve graphically, confirm analytically. Use 2nd derivative test to confirm min or max.
Answer question in a sentence, include units!
The one liter can that uses the least amount of material has height equal to
____, radius equal to ______ and a surface area of ____________.
Homework
Page 226
Examples 2, 6, 20, 27,
31, 36, 40, 41
Warm Up
What is the smallest perimeter possible for a
rectangle whose area is 16 in2, and what
are its dimensions?
Examples from Economics
Big Ideas
• r(x) = the revenue from selling x items
• c(x) = the cost of producing x items
• p(x) = r(x) – c(x) = the profit from selling x items
Marginal Analysis
Because differentiable functions are locally linear, we use the marginals to approximate
the extra revenue, cost, or profit resulting from selling or producing one more item. We
find the marginal analysis by taking the derivative of each function.
Theorem 6
Maximum profit (if any) occurs at a production level at which marginal revenue equals
marginal cost.
• p’(x) = r’(x) – c’(x) is used to find the production level at which maximum profit
occurs (Theorem 6)
Example 5: Maximizing Profit
3
2
c
(
x
)

x

6
x
 15 x, where x represents
Suppose that r(x) = 9x and
thousands of units. Is there a production level that maximizes profit?
If so, what is it?
• Find r’(x) = c’(x), simplify
• Find critical points, determine max or min using 2nd derivative test
Interpret
Maximum profit occurs when production level is at ________ , where x
represents thousands of units.
Maximum loss occurs when production level is at __________, where x
represents thousands of units.
Theorem 6
Maximizing Profit
Maximum profit (if any) occurs at a
production level at which marginal
revenue equals marginal cost.
r’(x) = c’(x)
Theorem 7
Minimizing Cost
The production level (if any) at which
average cost is smallest is a level at
which the average cost equals the
marginal cost.
c’(x) = c(x) / x
Example 6: Minimizing Average Cost
Suppose c( x)  x  6 x  15x , where x represents thousands of units.
Is there a production level that minimizes average cost? If so, what is
it?
We want c’(x) = c(x) / x
3
2
Solve for x,
Use 2nd derivative test to determine if you’ve found a max or min.
Interpret
The production level to minimize average cost occurs at x = ____, where x
represents thousands of units.
Summary
How can we solve an optimization
problem?
• Identify what we want to find and the information we are
given to find it.
• Draw a picture, write equations, use substitution to get a
function in terms of the variable needed.
• Solve graphically, confirm analytically
• Find max / min points on the graph, don’t forget to consider
endpoints. Use 2nd derivative test to confirm max or min.
• Use the values you’ve found to answer the original question
in a sentence. Make sure your answer makes sense!
Homework
Page 226
Exercises 3, 7, 9, 10, 12,
18, 22, 23, 39
Today’s Agenda
• Present homework problems on the
document camera in teams
• Teamwork: Page 229 #13 work out as a
Free Response Question
• Homework: P230 51-56