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Transcript of Differentiation Application
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APPLICATIONS OF DIFFERENTIATIONAPPLICATIONS OF DIFFERENTIATION
4
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APPLICATIONS OF DIFFERENTIATION
The methods we have learned in thischapter for finding extreme values have
practical applications in many areas of life.
! A businessperson wants to minimize costs andmaximize profits.
! A traveler wants to minimize transportation time.
! Fermat’s Principle in optics states that light followsthe path that takes the least time.
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4.7
Optimization Problems
In this section, we will learn:
How to solve problems involving
maximization and minimization of factors.
APPLICATIONS OF DIFFERENTIATION
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In this section (and the next), we solvesuch problems as:
! Maximizing areas, volumes, and profits
! Minimizing distances, times, and costs
OPTIMIZATION PROBLEMS
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OPTIMIZATION PROBLEMS
In solving such practical problems, thegreatest challenge is often to convert the word
problem into a mathematical optimization
problem—by setting up the function that is
to be maximized or minimized.
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OPTIMIZATION PROBLEMS
Let’s recall the problem-solvingprinciples discussed in Chapter 1 and
adapt them to this situation.
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Thus, there are six steps involved insolving optimization problems.
These are as follows.
STEPS IN SOLVING OPTIMIZATION PROBLEMS
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1. UNDERSTAND THE PROBLEM
Read the problem carefully until it isclearly understood.
Ask yourself:! What is the unknown?
! What are the given quantities?
! What are the given conditions?
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2. DRAW A DIAGRAM
In most problems, it is useful to drawa diagram and identify the given and
required quantities on the diagram.
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3. INTRODUCE NOTATION
Assign a symbol to the quantity thatis to be maximized or minimized.
! Let’s call it Q for now.
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Also, select symbols (a, b, c , . . . , x , y)for other unknown quantities and label
the diagram with these symbols.
! It may help to use initials as suggestive symbols.
! Some examples are: A for area, h for height,
and t for time.
3. INTRODUCE NOTATION
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4. EXPRESS Q IN TERMS OF THE VARIABLES
Express Q in terms of some of the other symbols
from Step 3.
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5. EXPRESS Q IN TERMS OF ONE VARIABLE
If Q has been expressed as a function of more than one variable in Step 4, use the
given information to find relationships—in the
form of equations—among these variables.
Then, use the equations to eliminate all but
one variable in the expression for Q.
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Thus, Q will be expressed asa function of one variable x , say,
Q = f ( x ).
! Write the domain of this function.
5. EXPRESS Q IN TERMS OF ONE VARIABLE
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Use the methods of Sections 4.1 and 4.3to find the absolute maximum or minimum
value of f .
! In particular, if the domain of f is a closed interval,then the Closed Interval Method in Section 4.1can be used.
6. FIND THE ABSOLUTE MAX./MIN. VALUE OF f
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OPTIMIZATION PROBLEMS
A farmer has 2400 ft of fencing and wantsto fence off a rectangular field that borders
a straight river. He needs no fence along
the river.
! What are the dimensions of the field thathas the largest area?
Example 1
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OPTIMIZATION PROBLEMS
In order to get a feeling for whatis happening in the problem, let’s
experiment with some special cases.
Example 1
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OPTIMIZATION PROBLEMS
Here are threepossible ways of
laying out the 2400 ft
of fencing.
Example 1
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OPTIMIZATION PROBLEMS
We see that when we try shallow, widefields or deep, narrow fields, we get
relatively small areas.
! It seems plausible that there is some intermediateconfiguration that produces the largest area.
Example 1
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OPTIMIZATION PROBLEMS
This figureillustrates
the general case.
We wish to maximize the area A of
the rectangle.! Let x and y be the depth and width of the rectangle
(in feet).
! Then, we express A in terms of x and y : A = xy
Example 1
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OPTIMIZATION PROBLEMS
We want to express A as a function of just one variable.
! So, we eliminate y by expressing it in terms of x .
! To do this, we use the given information thatthe total length of the fencing is 2400 ft.
! Thus, 2 x + y = 2400
Example 1
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OPTIMIZATION PROBLEMS
From that equation, we have:y = 2400 – 2 x
This gives:
A = x (2400 – 2 x ) = 2400 x - 2 x 2
! Note that x ! 0 and x " 1200 (otherwise A < 0).
Example 1
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OPTIMIZATION PROBLEMS
So, the function that we wish to maximizeis: A( x ) = 2400 x – 2 x 2 0 " x " 1200
! The derivative is: A’ ( x ) = 2400 – 4 x
! So, to find the critical numbers, we solve: 2400 – 4 x = 0
! This gives: x = 600
Example 1
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OPTIMIZATION PROBLEMS
The maximum value of A mustoccur either at that critical number or
at an endpoint of the interval.
! A(0) = 0; A(600) = 720,000; and A(1200) = 0
! So, the Closed Interval Method gives the maximum
value as: A(600) = 720,000
Example 1
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OPTIMIZATION PROBLEMS
Alternatively, we could have observed that A’’ ( x ) = –4 < 0 for all x
So, A is always concave downward
and the local maximum at x = 600 must be
an absolute maximum.
Example 1
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Thus, the rectangular field shouldbe:! 600 ft deep
! 1200 ft wide
OPTIMIZATION PROBLEMS Example 1
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OPTIMIZATION PROBLEMS
A cylindrical can is to be made tohold 1 L of oil.
! Find the dimensions that will minimizethe cost of the metal to manufacture the can.
Example 2
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OPTIMIZATION PROBLEMS
Draw the diagram as inthis figure, where
r is the radius and h the
height (both incentimeters).
Example 2
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OPTIMIZATION PROBLEMS
To minimize the cost of the metal, we minimize
the total surface area of
the cylinder (top, bottom,
and
sides.)
Example 2
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OPTIMIZATION PROBLEMS
We see that the sides are made froma rectangular sheet with dimensions
2# r and h.
Example 2
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OPTIMIZATION PROBLEMS
So, the surfacearea is:
A = 2# r 2 +
2# rh
Example 2
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OPTIMIZATION PROBLEMS
To eliminate h, we use the fact thatthe volume is given as 1 L, which we take
to be 1000 cm3.
! Thus, # r 2h = 1000
! This gives h = 1000/(# r 2)
Example 2
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OPTIMIZATION PROBLEMS
Substituting this in the expression for A gives:
So, the function that we want to minimize is:
2 2
2
1000 20002 2 2 A r r r
r r ! ! !
!
" #= + = +$ %
& '
2 2000( ) 2 0 A r r r
r
! = + >
Example 2
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OPTIMIZATION PROBLEMS
To find the critical numbers, we differentiate:
! Then, A’ (r ) = 0 when # r 3 = 500
! So, the only critical number is: 3500 /r ! =
Example 2
3
2 2
2000 4( 500)'( ) 4
r A r r
r r
!
!
"
= " =
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OPTIMIZATION PROBLEMS
As the domain of A is (0, $), we can’t use theargument of Example 1 concerning endpoints.
! However, we can observe that A’ (r ) < 0 for
and A’ (r ) > 0 for
! So, A is decreasing for all r to the left of the criticalnumber and increasing for all r to the right.
! Thus, must give rise to an absoluteminimum.
Example 2
3500 /r ! <
3 500 /r ! >
3500 /r ! =
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OPTIMIZATION PROBLEMS
Alternatively, we couldargue that A(r ) ! $
as r % 0+ and A(r ) ! $
as r ! $.
! So, there must bea minimum value of A(r ),which must occur at
the critical number.
Example 2
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OPTIMIZATION PROBLEMS
The value of h corresponding to is:
32 2 3
1000 1000 5002 2
(500 / )h r
r ! ! ! !
= = = =
Example 2
3500 /r ! =
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OPTIMIZATION PROBLEMS
Thus, to minimize the cost of the can,
! The radius should be cm
! The height should be equal to twice theradius—namely, the diameter
Example 2
3 500 /r ! =
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OPTIMIZATION PROBLEMS
The argument used in the exampleto justify the absolute minimum is a variant
of the First Derivative Test—which applies
only to local maximum or minimum values.
! It is stated next for future reference.
Note 1
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Suppose that c is a critical number of acontinuous function f defined on an interval.
"#If f’ ( x ) > 0 for all x < c and f’ ( x ) < 0 for all x > c ,then f (c ) is the absolute maximum value of f .
$#If f’ ( x ) < 0 for all x < c and if f’ ( x ) > 0 for all x > c ,then f (c ) is the absolute minimum value of f .
FIRST DERIV. TEST FOR ABSOLUTE EXTREME VALUES
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An alternative method for solvingoptimization problems is to use implicit
differentiation.
! Let’s look at the example again to illustratethe method.
OPTIMIZATION PROBLEMS Note 2
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IMPLICIT DIFFERENTIATION
We work with the same equations A = 2# r 2 + 2# rh # r 2h = 100
! However, instead of eliminating h,we differentiate both equations implicitlywith respect to r :
A’ = 4# r + 2# h + 2# rh’ 2# rh + # r 2h’ = 0
Note 2
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The minimum occurs at a criticalnumber.
! So, we set A’ = 0, simplify, and arrive atthe equations
2r + h + rh’ = 0 2h + rh’ = 0
! Subtraction gives: 2r - h = 0 or h = 2r
Note 2IMPLICIT DIFFERENTIATION
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Find the point on the parabolay 2 = 2 x
that is closest to the point (1, 4).
OPTIMIZATION PROBLEMS Example 3
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OPTIMIZATION PROBLEMS
The distance betweenthe point (1, 4) and
the point ( x , y ) is:
! However, if ( x , y ) lies onthe parabola, then x = " y 2.
! So, the expression for d
becomes:
2 2( 1) ( 4)d x y= ! + !
Example 3
2 2 212
( 1) ( 4)d y y= ! + !
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Alternatively, we could havesubstituted to get d in terms
of x alone.
OPTIMIZATION PROBLEMS Example 3
2 y x=
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OPTIMIZATION PROBLEMS
Instead of minimizing d , we minimizeits square:
! You should convince yourself that the minimum of d
occurs at the same point as the minimum of d 2.
! However, d 2 is easier to work with.
( ) ( )
2 22 21
2( ) 1 4d f y y y= = !
+ !
Example 3
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OPTIMIZATION PROBLEMS
Differentiating, we obtain:
So, f’ (y ) = 0 when y = 2.
( )2 312
'( ) 2 1 2( 4) 8 f y y y y y= ! + ! = !
Example 3
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OPTIMIZATION PROBLEMS
Observe that f’ (y ) < 0 when y < 2 and f’ (y ) > 0when y > 2.
So, by the First Derivative Test for Absolute
Extreme Values, the absolute minimum
occurs when y = 2.
! Alternatively, we could simply say that, due tothe geometric nature of the problem, it’s obvious thatthere is a closest point but not a farthest point.
Example 3
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OPTIMIZATION PROBLEMS
The corresponding valueof x is:
x = " y 2 = 2
Thus, the point on y 2 =
2 x
closest to (1, 4) is (2, 2).
Example 3
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OPTIMIZATION PROBLEMS
A man launches his boatfrom point A on a bank of
a straight river, 3 km
wide,
and wants to reach point
B
(8 km downstream on
the opposite bank) as
quickly as possible.
Example 4
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OPTIMIZATION PROBLEMS
He could proceed inany
of three ways:
! Row his boat directly acrossthe river to point C and thenrun to B
! Row directly to B
! Row to some point D
betweenC and B and then run to B
Example 4
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OPTIMIZATION PROBLEMS
If he can row 6 km/h andrun 8 km/h, where should
he land to reach B as
soon as possible?
! We assume that the speedof the water is negligiblecompared with the speed at
which he rows.
Example 4
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OPTIMIZATION PROBLEMS
If we let x be the distance from C to D,then:
! The running distance is: |DB| = 8 – x
! The Pythagorean Theorem gives the rowingdistance as: | AD| =
Example 4
29 x +
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OPTIMIZATION PROBLEMS
We use the equation
! Then, the rowing time is:
! The running time is: (8 – x )/8
! So, the total time T as a function of x is:
2 9 8( )6 8
x xT x
+ != +
Example 4
distance
time= rate
29 / 6 x +
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OPTIMIZATION PROBLEMS
The domain of this function T is [0, 8].
! Notice that if x = 0, he rows to C , and if x = 8,
he rows directly to B.
! The derivative of T is:2
1'( )
86 9
xT x
x
= !
+
Example 4
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OPTIMIZATION PROBLEMS
Thus, using the fact that x!
0,we have:
! The only critical number is:
2
2
2 2
2
1'( ) 0
86 9
4 3 916 9( 9)
97 81
7
xT x
x
x x
x x
x x
= ! =
+
! = +
! = +
! = ! =
Example 4
9 / 7
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OPTIMIZATION PROBLEMS
To see whether the minimum occurs atthis critical number or at an endpoint of
the domain [0, 8], we evaluate T at all three
points: (0) 1.5
9 71 1.33
87
73(8) 1.42
6
T
T
T
=
! "= + #$ %
& '
= #
Example 4
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OPTIMIZATION PROBLEMS
Since the smallest of
these values of T
occurs when x = ,
the absolute minimum
value of T must occur
there.
! The figure illustratesthis calculation byshowing the graphof T .
Example 4
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OPTIMIZATION PROBLEMS
Thus, the man should
land
the boat at a point
(# 3.4 km) downstream
from his starting point.
Example 4
9 / 7
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Find the area of the largest rectanglethat can be inscribed in a semicircle
of radius r .
OPTIMIZATION PROBLEMS Example 5
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OPTIMIZATION PROBLEMS
Let’s take the semicircle
to be the upper half of
the circle x 2 + y 2 = r 2 with
center the origin.
! Then, the wordinscribed meansthat the rectanglehas two vertices
on the semicircleand two verticeson the x -axis.
E. g. 5—Solution 1
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OPTIMIZATION PROBLEMS
Let ( x, y ) be the vertexthat lies in the first
quadrant.
! Then, the rectanglehas sides of lengths 2 x and y.
! So, its area is:
A = 2 xy
E. g. 5—Solution 1
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OPTIMIZATION PROBLEMS
To eliminate y , we use the fact that ( x, y )lies on the circle x 2 + y 2 = r 2.
! So,
! Thus, 2 22 A x r x= !
2 2
y r x=
!
E. g. 5—Solution 1
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OPTIMIZATION PROBLEMS
The domain of this function is 0$ x$ r .
Its derivative is:
! This is 0 when 2 x 2 = r 2, that is x = ,
(since x ! 0).
2 2 22 2
2 2 2 2
2 2( 2 )' 2
x r x A r x
r x r x
!
= ! ! =
! !
E. g. 5—Solution 1
/ 2r
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OPTIMIZATION PROBLEMS
This value of x gives a maximum value of A,since A(0) = 0 and A(r ) = 0 .
Thus, the area of the largest inscribed
rectangle is:2
2 22
22 2
r r r A r r
! "= # =$ %
& '
E. g. 5—Solution 1
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A simpler solution is possibleif we think of using an angle as
a variable.
OPTIMIZATION PROBLEMS Example 5
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OPTIMIZATION PROBLEMS
Let& be the angle
shown here.
! Then, the area of the
rectangle is: A(& ) = (2r cos & )(r sin & )
= r 2(2 sin & cos & ) = r 2 sin 2&
E. g. 5—Solution 2
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OPTIMIZATION PROBLEMS
We know that sin 2& has a maximumvalue of 1 and it occurs when 2& = # /2.
! So, A(& ) has a maximum value of r 2
and it occurs when & = # /4.
E. g. 5—Solution 2
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OPTIMIZATION PROBLEMS
Notice that this trigonometric solutiondoesn’t involve differentiation.
! In fact, we didn’t need to use calculus at all.
E. g. 5—Solution 2
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Let us now look atoptimization problems in business
and economics.
APPLICATIONS TO BUSINESS AND ECONOMICS
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MARGINAL COST FUNCTION
In Section 3.7, we introduced the idea of marginal cost.
! Recall that if C ( x ), the cost function, is the cost of producing x units of a certain product, then the marginalcost is the rate of change of C with respect to x .
! In other words, the marginal cost function is
the derivative, C’ ( x ), of the cost function.
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DEMAND FUNCTION
Now, let’s consider marketing.
! Let p( x ) be the price per unit that the company
can charge if it sells x units.
! Then, p is called the demand function(or price function), and we would expect itto be a decreasing function of x .
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If x units are sold and the price per unitis p( x ), then the total revenue is:
R ( x ) = xp( x )
! This is called the revenue function.
REVENUE FUNCTION
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The derivative R’ of the revenuefunction is called the marginal revenue
function.
! It is the rate of change of revenue with respectto the number of units sold.
MARGINAL REVENUE FUNCTION
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If x units are sold, then the total profitis P ( x ) = R ( x ) – C ( x )
and is called the profit function.
The marginal profit function is P’ ,
the derivative of the profit function.
MARGINAL PROFIT FUNCTION
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MINIMIZING COSTS AND MAXIMIZING REVENUES
In Exercises 53–58, you are asked to usethe marginal cost, revenue, and profit
functions to minimize costs and maximize
revenues and profits.
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A store has been selling 200 DVD burnersa week at $350 each. A market survey
indicates that, for each $10 rebate offered to
buyers, the number of units sold will increaseby 20 a week.
! Find the demand function and the revenue function.
! How large a rebate should the store offer to maximizeits revenue?
Example 6MAXIMIZING REVENUE
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If x is the number of DVD burners soldper week, then the weekly increase in sales
is x – 200.
! For each increase of 20 units sold, the priceis decreased by $10.
Example 6DEMAND & REVENUE FUNCTIONS
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So, for each additional unit sold, the decreasein price will be 1/20 x 10 and the demand
function is:
p( x ) = 350 – (10/20)( x – 200) = 450 – " x
DEMAND FUNCTION Example 6
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The revenue function is:R ( x ) = xp( x )
= 450 x – " x 2
Example 6REVENUE FUNCTION
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Since R’ ( x ) = 450 – x , we see thatR’ ( x ) = 0 when x = 450.
! This value of x gives an absolute maximumby the First Derivative Test (or simply by observingthat the graph of R is a parabola that opensdownward).
Example 6MAXIMIZING REVENUE
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The corresponding price is: p(450) = 450 – "(450) = 225
The rebate is: 350 – 225 = 125
! Therefore, to maximize revenue, the store
should offer a rebate of $125.
Example 6MAXIMIZING REVENUE