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5 023 MAX - Min: Optimization. AP Calculus. 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 2 nd Derivative

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open intervals find the 1 st derivative and the critical numbers
First Derivative Testfor 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 Testfor Max / Min

FIND 2nd Derivative

PLUG IN the critical number

MAX: if the value is negative

MIN: if the value is positive

OPEN INTERVALS:Find the 1stDerivative and the Critical Numbers
example 1 open 1st derivative test
Example 1: Open - 1st Derivative test

VA=1

x=3, x=-1

Undefined at 1

f’(x)

+

+

-

-

-1

1

3

VA

4

-2

0

2

max at -1 min at 3 and a VA at 1 why?

Max of -2 at x=-1 because the first derivative goes from positive to negative at -1.

Min of 6 at x=3 because 1st derivative goes from negative to positive at 3.

A VA at x=1 because 1st derivative and the function are undefined at x=1

closed intervals

Critical #

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]

local

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 1 st derivative and the critical numbers
CLOSED INTERVALS:Find the 1stDerivative 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
Example : Closed Interval Test

Consider endpts.

and x=2

Absolute max

absolute min

optimization problems
OPTIMIZATION PROBLEMS
  • Used to determine Maximum and Minimum Values – i.e.
          • maximum profit,
          • least cost,
          • greatest strength,
          • least distance
method set up
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.

relative maximum and minimum
Relative Maximum and Minimum

DEFN: Relative Extrema are the highest or lowest points in an interval.

Where is x

What is y

1 illustration with method

y-could be 0

y – could be 1000

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:

y

Max area

Generic formula:

x

A=lw

Personalized formula:

A=xy

x+2y=2000

x=2000-2y

x=1000

y=500

Which Test?

To maximize the area the lengths of the sides should be 500 meters.

example 2

Design an open box with the MAXIMUM VOLUME that has a square bottom and surface area of 108 square inches.

Example 2:

y

Statement:

x

x

max volume of rect. prism

Formula:

Personalized Formula:

Plug in 6

Get a -

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 .

Example 3:

ellipse

Major axis y-axis length is 2

Minor axis x-axis length is 1

Secondary equations

Max Area: A=lw

A=2x*2y

A=4xy

CIT:

A(0)=0

A

A(1)=0

Dimensions:

undef: x=1,-1

Zeroes:

example 4

min

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

Example 4:

(0,-1)

=0

example 5

A closed box with a square base is to have a volume of 2000in.3 . the material on the top and bottom will cost 3 cents per square inch and the material on the sides willcost 1cent per square inch. Find the dimensions that will minimize the cost.

Example 5:

Statement:

minimize cost of material

y

x

Formula:

x

example 6
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.

last update

Last Update:

12/03/10

Assignment: DWK 4.4