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Section 4.7
Optimization Problems
AP Calculus
October 30, 2009
Berkley High School, D2B2
[email protected]
Example

Find two numbers whose
difference is 100 and
whose product is a
minimum.
x =first number
y =second number
x  y  100
P( x, y )  xy
Calculus, Section 4.7, Todd Fadoir, CASA, 2003
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Example
x  100  y
50  y
P ( y )  (100  y ) y
x  100  ( 50)
P ( y )  100 y  y
P( y )  100  2 y
Answer:
2
0  100  2 y
100 2 y

2
2
50  y
x  50
when one number is -50 and
the other is 50, the product
is minimized.
Calculus, Section 4.7, Todd Fadoir, CASA, 2003
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Summary





Find relationship between unknown values
Write a function for the quantity needing to be
maximized or minimized.
Reduce the function to one variable.
Take the first derivative of the function, and find
value that maximizes or minimizes the quantity.
Answer the question posed.
Calculus, Section 4.7, Todd Fadoir, CASA, 2003
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Example
A farmer has 2400 ft of fencing and wants
to fence off a rectangular field that borders
a straight river.
 He needs no fence along the river.
 What are the dimensions of the field that
has the largest area? What is the area?

Calculus, Section 4.7, Todd Fadoir, CASA, 2003
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w
h
2400  2h  w
A(h, w)  h  w
Calculus, Section 4.7, Todd Fadoir, CASA, 2003
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2400  2h  w
w
2400  2h  w
h
A(h, w)  h  w
The sides perpendicular
A(h)  h  (2400  2h)
to the river are 600 ft.
The side parallel
A(h)  2400h  2h
A(h)  2400  4h
to the river is 1200 ft.
0  2400  4h
2
The area is 720,000 ft .
2
4h  2400
h  600
Calculus, Section 4.7, Todd Fadoir, CASA, 2003
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Assignment

Section 4.7, 1-17, odd
Calculus, Section 4.7, Todd Fadoir, CASA, 2003
8