4.4 Notes.notebook
October 28, 2014
4.4 Modeling and Optimization
Name: ____________________
Objectives: Students will be able to solve application problems
involving finding maximums and minimums of functions.
Optimization problems are one of the oldest application problems of
what we now call "differential calculus".
Examples Solve analytically and support graphically.
1.) Find two numbers whose sum is 20 and whose product is as large
as possible.
Oct 20­6:23 PM
2.) A rectangle is to be inscribed under one arch of the sine curve.
What is the largest possible area the rectangle can have, and what
dimensions give that area?
Oct 20­6:32 PM
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4.4 Notes.notebook
October 28, 2014
3.) An open-top box is to be made by cutting congruent squares of
side length x from the corners of a 20-by-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?
Oct 20­6:34 PM
Let p(x) = profit, r(x) = revenue and c(x) = cost.
p(x) = _____________
4.) Suppose that r(x) = 9x and c(x) = x3 - 6x2 + 15x, where x
represents thousands of units. Is there a production level that
maximizes profit? If so, what is it?
Oct 20­6:36 PM
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4.4 Notes.notebook
October 28, 2014
5.) Let c(x) = the cost function. How can we minimize the average
cost?
Oct 20­6:53 PM
Minimizing Average Cost
The production level (if any) at which average cost is smallest is a
level at which the ___________ ______ _______ ____
____________ _______.
Oct 20­6:54 PM
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4.4 Notes.notebook
October 28, 2014
6.) Suppose c(x) = x3 - 6x2 + 15x, where x represents thousands of
units. Is there a production level that minimizes average cost? If
so, what is it?
Oct 20­6:55 PM
GROUP WORK
Finding Area Show that among all rectangles with an 8 meter
perimeter, the one with the largest area is a square.
Oct 20­7:16 PM
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4.4 Notes.notebook
October 28, 2014
Closing off the First Quadrant You are planning to close off the
corner of the first quadrant with a line segment 20 units long
running from (a,0) to (0,b). Show that the area of the triangle
enclosed by the segment is largest when a = b.
Oct 20­6:58 PM
Minimizing Average Cost Suppose c(x) = xex - 2x2, where x is
measured in thousands of units. Is there a production level that
minimizes average cost? Is so, what is it?
Oct 20­7:21 PM
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4.4 Notes.notebook
October 28, 2014
Fabricating a Box An open-top box is made by cutting squares of
side length x from corners of a 14-by-18 inch sheet of tin and
bending up the sides. How large should the square be to make the
box hold as much as possible? What is the resulting maximum
volume?
Oct 20­7:23 PM
Homework: Pages 226-227: #1-9 odd, 13, 21-25 odd
Sep 17­9:09 AM
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