1.
You are given the graph of a function f defined on theinterval (?1, ?). Find the absolute maximum and absolute minimumvalues of f (if they exist) and where they are attained.(If an answer does not exist, enter DNE.)
The x y-coordinate plane is given. A curve, ahorizontal dashed line, and a vertical dashed line are graphed.
- A vertical dashed line crosses the x-axis atx = ?1.
- A horizontal dashed line crosses the y-axis aty = 1.
- The curve enters the window in the third quadrant just to theright of x = ?1, goes up and right becoming less steep,becomes nearly horizontal at the origin, goes up and right becomingmore steep, passes through the approximate point (1, 0.5), goes upand right becoming less steep, and exits the window just belowy = 1.
absolute maximum
(x, y)=
absolute minimum(x, y)=
2.
Find the absolute maximum value and the absolute minimum value,if any, of the function. (If an answer does not exist, enterDNE.)
g(x) =?x2 + 4x + 9
maximum =
minimum=
3.
We first note that the function
f(x) = ?x2 + 2x + 8
is continuous and defined on the closed interval
[3, 6].
Recall that to find the absolute extrema of the given functionwe must first find any critical numbers of the function that lie ininterval
[3, 6].
In other words, we need to find any values of x forwhich
f?'(x) = 0,
or
f?'(x)
does not exist.
Therefore, we must first find
f?'(x).
| f(x) | = | ?x2 + 2x + 8 |
|
| f?'(x) | = | ?2x + |
4.
Find the absolute maximum value and the absolute minimum value,if any, of the function. (If an answer does not exist, enterDNE.)
f(x) = x2 ? x ? 3 on [0, 3]
maximum=
minimum=
5.
Find the absolute maximum value and the absolute minimum value,if any, of the function. (If an answer does not exist, enterDNE.)
f(x) = 4x ?9/X on [9, 11]
maximum=
minimum=
6.
Find the absolute maximum value and the absolute minimum value,if any, of the function. (If an answer does not exist, enterDNE.)
f(x) =
on [?2, 1]
maximum=
minimum=
7.
Average Speed of a Vehicle
The average speed of a vehicle on a stretch of Route 134 between6 A.M. and 10 A.M. on a typical weekday is approximated by thefunction
f(t) = 27t ? 54
+ 64 (0 ? t ? 4)
where f (t) is measured in miles per hour, andt is measured in hours, with t = 0 correspondingto 6 A.M. At what time of the morning commute is the traffic movingat the slowest rate?
=A.M.
What is the average speed of a vehicle at that time?
=mph
8.
Maximizing Profits
The quantity demanded each month of the Walter Serkin recordingof Beethoven's Moonlight Sonata, produced by PhonolaMedia, is related to the price per compact disc. The equation
p = ?0.00054x + 6 (0 ? x ? 12,000)
where p denotes the unit price in dollars andx is the number of discs demanded, relates the demand tothe price. The total monthly cost (in dollars) for pressing andpackaging x copies of this classical recording is givenby
C(x) = 600 + 2x ? 0.00003x2 (0? x ? 20,000).
To maximize its profits, how many copies should Phonola produceeach month? Hint: The revenue is
R(x) = px,
and the profit is
P(x) = R(x) ? C(x).
(Round your answer to the nearest whole number.)
= discs/month
9.
Find the absolute maximum value and the absolute minimum value,if any, of the function. (If an answer does not exist, enterDNE.)
g(x) =
x2 ? 16
on [0, 36]
maximum=
minimum=
10.
Find the absolute maximum value and the absolute minimum value,if any, of the function. (If an answer does not exist, enterDNE.)
g(t) =
on [4, 6]
maximum=
minimum=
11.
Enclosing the Largest Area
The owner of the Rancho Grande has 3,044 yd of fencing withwhich to enclose a rectangular piece of grazing land situated alongthe straight portion of a river. If fencing is not required alongthe river, what are the dimensions (in yd) of the largest area hecan enclose?
A rectangular piece of land has been enclosed along a straightportion of a river. The enclosure is bordered by the river (a longside), another long side of fence, and two short sides offence.
shorter side = yd
longer side= yd
What is this area (in yd2)?
= yd2
12.
Packaging
By cutting away identical squares from each corner of arectangular piece of cardboard and folding up the resulting flaps,an open box may be made. If the cardboard is 16 in. long and 6 in.wide, find the dimensions (in inches) of the box that will yieldthe maximum volume. (Round your answers to two decimal places ifnecessary.)
smallest value = in
= in
largest value = in
13.
Minimizing Packaging Costs
A rectangular box is to have a square base and a volume of 72ft3. If the material for the base costs$0.49/ft2, the material for the sides costs$0.12/ft2, and the material for the top costs$0.15/ft2, determine the dimensions (in ft) of the boxthat can be constructed at minimum cost. (Refer to the figurebelow.)
A closed rectangular box has a length of x, a width ofx, and a height of y.
x= ft
y= ft
14.
Charter Revenue
The owner of a luxury motor yacht that sails among the 4000Greek islands charges $568/person/day if exactly 20 people sign upfor the cruise. However, if more than 20 people sign up (up to themaximum capacity of 100) for the cruise, then each fare is reducedby $4 for each additional passenger.
Assuming at least 20 people sign up for the cruise, determinehow many passengers will result in the maximum revenue for theowner of the yacht.
= passengers
What is the maximum revenue?
$ =
What would be the fare/passenger in this case? (Round youranswer to the nearest dollar.)
= dollars per passenger
15.
Minimizing Packaging Costs
If an open box has a square base and a volume of 103in.3 and is constructed from a tin sheet, find thedimensions of the box, assuming a minimum amount of material isused in its construction. (Round your answers to two decimalplaces.)
height = in
length = in
width= in
