MAX - Min: Optimization

AP Calculus

OPEN INTERVALS:

Find the 1st Derivative and the Critical Numbers

• First 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 +

• Second Derivative Test for Max / Min– 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

2 3( )

1

xf x

x

Example 2: Open - 2nd Derivative Test

5 3( ) 3 5f x x x

LHE p. 186

CLOSED INTERVALS:

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 valueMIN: if the Smallest value

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 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

52( ) ( )3 3( ) 5 1,4f x x x on

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?

Statement:

Generic formula:

Personalized formula:

Which Test?

Figure:

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 in such a way that the sides are parallel to the axes .

2 24 4x y

Example 4:

Find the point on closest to the point (0, -1).

3 4x y

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. 2

2( ) 50 and ( ) 4000 40 0.02100

xR x x C x x x

2

2( ) 50 4000 40 0.02100

xP x x x x

2( ) 4000 90 0.03P x x x

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.

15in.120 in.

1) Find the size of the vertex angle that will give the maximum capacity of the trough.

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