Download Calculus Fall 2010 Lesson 26 _Optimization problems_

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Lesson Plan #26
Class: Intuitive Calculus
Date: Wednesday November 10th, 2010
Topic: Optimization Problems
Aim: How do we solve optimization problems?
Objectives:
1) Students will be able to solve problems where they have to maximize or minimize a value.
HW# 26:
1) The sum of one number and two times a second number is 24. What numbers should be selected so that their
product is as large as possible?
2) The product of two positive numbers is 192. What numbers should be chosen so that the sum of the first plus three
times the second is a minimum?
Do Now:
You run a small tutoring school. The graph at
right represents the amount of profit you take in
per week depending on the number of students
you have. Based on the graph, approximately how
many students should you take in to maximize
your profit?
With respect to the topics we have discussed, what
do we call the x-coordinate at which this
maximum value occurs?
In general, what do we do to find this relative
maximum?
Procedure:
Write the Aim and Do Now
Get students working!
Take attendance
Give back work
Go over the HW
Collect HW
Go over the Do Now
Example:
Find two positive numbers that minimize the sum of twice the first number plus the second if the product of the two numbers is
288.
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Example:
The difference between two numbers is 50. Find the two numbers so that their product is as small as possible
Example:
An open box is to be constructed from 108 square inches. What dimensions will produce the box with the maximum volume?
Exercise:
Find two positive numbers whose sum is 110 and whose product is a maximum
Exercise:
The sum of one positive number and twice a second positive number is 100. Find the two numbers so that their product is a
maximum
Exercise:
A rectangle has a perimeter of 100 feet. What length and width should it have so that its area is maximum?
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Note: Recall the definition of
Definition of the Derivative of a Function: The derivative of a f at
the derivative of a function:
x is given by
f ( x  x)  f ( x)
x 0
x
provided the limit exists
f ' ( x)  lim
It is often extremely useful to
evaluate a limit by recognizing
that it is merely an expression
for the definition of the derivative for a specific function.
You can think of lim
h0
f (c  h)  f (c )
as f ' (c )
h
Evaluate:
( 2  h)  2 4
1) lim
You should be able to recognize that the function is f ( x)  x 4 and you have to find the
h0
h
value of the derivative at x  2 . f ' ( x)  4 x3
f ' (2)  4(2)3  32
9 h 3
h
2) lim
h0
First, recognize the function. What is the function?______________
What is the derivative of the function? ___________
Now evaluate the derivative of the function at x  9 . What is f ' (9) ? _______________________
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1

3) lim 2  h 2
h0
h
Ans.
(1  h)  1
is
h0
h
A) 0
B) 1
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4) lim
3
5) lim
h 0
A) 0
C) 6
D) 
E) nonexistent
C) 1
D) 192
E) 
8h 2
h
B)
1
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Sample Test Questions:
1) The slope of the curve y 2  xy  3x  1 at the point (0,-1) is
A) -1
B) -2
C) +1
D) 2
E) -3
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2) The total number of local maximum and minimum points of the function whose derivative, for all x , is
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given by f ' ( x)  x( x  3) 2 x  1 is
A) 0
B) 1
C) 2
D) 3
E) None of the other choices
3) What is the minimum value of the slope of the curve y  x5  x3  2 x ?
A) 0
B) 2
C) 6
D) -2
E) None of the other choices
4) The slope of the curve y 3  xy2  4 at the point where y  2 is
1
1
1
A) -2
B)
C) 
D)
E) 2
4
2
2
5) A line is drawn through the point (1,2) forming a right triangle with the positive x- and y-axes. The slope
of the line forming the triangle of least area is
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A) -1
B) -2
C) -4
D) 
E) -3
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6) The area of the largest isosceles triangle that can be drawn with one vertex at the origin and with the
others on a line parallel to and above the x-axis and on the curve y  27  x 2 is
A) 12 3
B) 27
C) 24 3
D) 54
E) 108