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4.7 Optimization Problems. Steps to follow for max and min problems. (a) Draw a diagram, if possible. (b) Assign symbols to unknown quantities. (c) Assign a symbol, Q , to the quantity to. be maximized or minimized. (d) Express Q in terms of the other symbols.

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4.7

Optimization Problems

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Steps to follow for max and min problems.

(a) Draw a diagram, if possible.

(b) Assign symbols to unknown quantities.

(c) Assign a symbol,Q, to the quantity to

be maximized or minimized.

(d) Express Q in terms of the other symbols.

(e) Eliminate all but one unknown symbol,

say x, from Q.

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(f) Find the absolute maximum or minimum

of Q= f (x).

(iii) Use the first derivative

(iv) Does the absolute max or min

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1. Example:

Solution:

=˃p(x)=x(x-100)

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(i) Domain of P(x):

all real numbers.

(ii) Critical numbers of P(x)

P′(x)=

(iii) First Derivative Test

P′(x)

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2. Example:

A farmer wants to fence a

rectangular enclosure for his horses

and then divide it into thirds with fences

parallel to one side of the rectangle. If

he has 2000m of fencing, find the area

of the largest rectangle that can be

enclosed.

Solution:

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(ii) Critical numbers of A(l)

By the first derivative test we can show that A(500) is the max value of A,

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When l=500,

max area of the rectangle =

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3. Example:

x=y2 that is closest to the poit (0,3).

Let D denote the distance from (0,3) to

any point (x,y) on the parabola.

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D(y)=

Find the value of y which makes D(y)

(i) Domain of D(y):

all real numbers > 0.

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(iii) First Derivative Test

D′(y)

By the first derivative test D(y)

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(ii) Critical numbers of V(y)

2π r2 -3y2

By the first derivative test

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5. Example: Mountain Beer sells its beer

in an aluminum can in the shape of a right-

circular cylinder. The volume of each can

What should the dimension of

the can be in order to minimize the amount

of aluminum used?

Solution:

and

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r

h

Surface Area,

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S(r)=

(i) Domain of S(r)

(ii) Critical numbers of S(r)

S′(r)=

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S′(r)=0 =˃

(iii) First Derivative Test

S′(r)=

S′(r)

S(r)

S(r) has its absolute minimum value