MAX - Min: Optimization
AP Calculus
OPEN INTERVALS:
Find the 1st Derivative and the Critical Numbers
• First Derivative Test
for Max / Min
• Second Derivative Test
for Max / Min
– TEST POINTS on
either side of the
critical numbers
– MAX:if the value
changes from + to –
– MIN: if the value
changes from – to +
– FIND 2nd Derivative
– PLUG IN the critical
number
– MAX: if the value is
negative
– MIN: if the value is
positive
Example 1: Open - 1st Derivative test
x 3
f ( x) 
x 1
2
Example 2: Open - 2nd Derivative Test
f ( x)  3x  5x
5
3
LHE p. 186
CLOSED INTERVALS:
EXTREME VALUE THEOREM:
If f is continuous on a closed interval [a,b],
then f attains an absolute maximum f(c) and
an absolute minimum f(d) at some points
c and d in [a,b]
Closed Interval Test
Find the 1st Derivative and the Critical Numbers
Plug In the Critical Numbers and the
End Points into the original equation
MAX: if the Largest value
MIN: if the Smallest value
CLOSED INTERVALS:
Find the 1st Derivative and the Critical Numbers
• Closed Interval Test
• Plug In the Critical Numbers and the
End Points into the original equation
• MAX: if the Largest value
• MIN: if the Smallest value
Example : Closed Interval Test
f ( x)  5 x
(2 )
3
x
(5 )
3
on
 1, 4
LHE p. 169
OPTIMIZATION PROBLEMS
• Used to determine Maximum and Minimum
Values – i.e.
»maximum profit,
»least cost,
»greatest strength,
»least distance
METHOD: Set-Up
Make a sketch.
Assign variables to all given and to find quantities.
Write a STATEMENT and PRIMARY (generic) equation to be
maximized or minimized.
PERSONALIZE the equation with the given information.
Get the equation as a function of one variable.
< This may involve a SECONDARY equation.>
Find the Derivative and perform one the tests.
1
ILLUSTRATION : (with method)
A landowner wishes to enclose a rectangular field that borders a
river. He had 2000 meters of fencing and does not plan to fence the
side adjacent to the river. What should the lengths of the sides be to
maximize the area?
Figure:
Statement:
Generic formula:
Personalized formula:
Which Test?
Example 2:
Design an open box with the MAXIMUM VOLUME that has a
square bottom and surface area of 108 square inches.
Example 3:
Find the dimensions of the largest rectangle that can be inscribed in
the ellipse 4 x 2  y 2  4 in such a way that the sides are parallel
to the axes .
Example 4:
Find the point on 3 x  y  4
closest to the point (0, -1).
Example 5:
A closed box with a square base is to have a volume of 2000in.3 . the
material on the top and bottom is to cost 3 cents per square inch and
the material on the sides is to cost 1.5 cents per square inch. Find the
dimensions that will minimize the cost.
Example 6:
Suppose that P(x), R(x), and C(x) are the profit, revenue, and cost
functions, that P(x) = R(x) - C(x), and x represents thousand of units.
Find the production level that maximizes the profit.
x2
R( x)  50 x 
and C ( x)  4000  40 x  0.02 x 2
100

x2 
2
P( x)   50 x 
   4000  40 x  0.02 x 
100 

P( x)  4000  90 x  0.03x 2
Example 7:AP Type Problem:
At noon a sailboat is 20 km south of a freighter. The sailboat is
traveling east at 20 km/hr, and the freighter is traveling south at 40
km/hr. If the visibility is 10 km, could the people on the ships see
each other?
Example 8:AP - Max/min - Related Rates
The cross section of a trough has the shape of an inverted
isosceles triangle. The lengths of the sides of the cross section
are 15 in., and the length of the trough is 120 in.
1) Find the size of the vertex angle that
will give the maximum capacity of the
trough.
15in.
120 in.
2) If water is being added to the trough at
36 in3/min, how fast is the water level
rising when the level is 5 in high?
Last Update:
12/03/10
Assignment:
DWK 4.4