Marianne Kemp
Assignment HW8 Derivative Applications II due 03/08/2012 at 10:00pm MST
math1210spring2012-3
Is f a maximum or minumum at the critical point? ?
The interval on the left of the critical point is
On this interval, f is ? while f 0 is ? .
1. (1 pt) Suppose that
f (x) = 5x3 − 9x − 8.
The interval on the right of the critical point is
On this interval, f is ? while f 0 is ? .
(A) Find the average of the x values of all local maxima of
f.
Note: If there are no local maxima, enter -1000.
Average of x values =
.
.
5. (1 pt) Let f (x) be the function shown in the graph below.
(B) Find the average of the x values of all local minima of f .
Note: If there are no local minima, enter -1000.
Average of x values =
(C) Find the average of the x values of all inflection points of f .
Note: If there are no inflection points, enter -1000.
Average of x values =
2. (1 pt) Suppose that
f (x) = x3 − 4x2 + 14.
(A) List the x values of all local maxima of f . If there are no
local maxima, enter ’NONE’.
x values of local maximums =
(B) List the x values of all local minima of f . If there are no
local minima, enter ’NONE’.
x values of local minimums =
(C) List the x values of all the inflection points of f . If there
are no inflection points, enter ’NONE’.
x values of inflection points =
Click on the graph to enlarge it.
Determine the absolute minimum of the function shown in
the graph.
Answer: x =
, f (x) = .
Determine the absolute maximum of the function shown in
the graph.
Answer: x =
, f (x) =
.
3. (1 pt) Let
2
f (x) = 6x + .
x
Use either the first derivative test or the second derivative test to
find the following:
Select maximum or minimum for the following:
The function attains a local ? at x = 2.
The function attains a local ? at x = 4.
The function attains a local ? at x = 5.
(A) The average of the x values of all local maxima of f .
Note: If there are no local maxima, enter -1000.
Average of x values =
(B) The average of the x values of all local minima of f .
Note: If there are no local minima, enter -1000.
Average of x values =
7. (1 pt) The function
f (x) = 2x3 + 3x2 − 180x − 2
4. (1 pt) From Rogawski ET 2e section 4.3, exercise 37.
Find the critical point and the interval on which the given
function is increasing or decreasing, and apply the First Derivative Test to the critical point. Let
f (x) = − x25+7
Critical Point =
is decreasing on the interval (
,
).
It is increasing on the interval ( −∞,
, ∞ ).
and the interval (
The function has a local maximum at
1
)
.
8. (1 pt) Answer the following questions for the function
p
f (x) = x x2 + 16
11. (1 pt) Find two positive numbers A and B (with A ≤ B)
whose sum is 52 and whose product is maximized.
A=
defined on the interval [−7, 7].
A.
B.
C.
D.
E.
B=
to
f (x) is concave down on the region
f (x) is concave up on the region
to
The inflection point for this function is at
The minimum for this function occurs at
The maximum for this function occurs at
12. (1 pt) A car rental agency rents 190 cars per day at a rate
of 28 dollars per day. For each 1 dollar increase in the daily rate,
5 fewer cars are rented. At what rate should the cars be rented
to produce the maximum income, and what is the maximum
income?
Rate =
9. (1 pt) A box is to be made out of a 8 cm by 20 cm piece of
cardboard. Squares of side length x cm will be cut out of each
corner, and then the ends and sides will be folded up to form a
box with an open top.
(a) Express the volume V of the box as a function of x.
Maximum Income =
cm3
V=
(b) Give the domain of V in interval notation. (Use the fact
that length and volume must be positive.)
13. (1 pt) A civil engineer wants to estimate the maximum
number of cars that can safely travel on a particular road at a
given speed. She assumes that each car is 19 feet long, travels
at speed s, and follows the car in front of it at a safe distance for
that speed. She finds that the number N of cars that can pass a
given spot per minute is modeled by the function
(c) Find the length L, width W , and height H of the resulting
box that maximizes the volume. (Assume that W ≤ L).
L=
W=
cm
82s
s 2
19 + 19
23
At what speed can the greatest number of cars travel safely
on that road?
N(s) =
cm
cm
H=
(d) The maximum volume of the box is
cm3 .
s=
10. (1 pt) A javalina rancher wants to enclose a rectangular
area and then divide it into four pens with fencing parallel to
one side of the rectangle (see the figure below). He has 690
feet of fencing available to complete the job. What is the largest
possible total area of the four pens?
14. (1 pt) From Rogawski ET 2e section 4.7, exercise 26.
Find the maximum area of a triangle formed in the first quadrant by the x-axis, y-axis and a tangent line to the graph of
f = (x + 1)−2 .
Area =
15. (1 pt) From Rogawski ET 2e section 4.7, exercise 33.
A box is contructed out of two different types of metal. The
metal for the top and bottom, which are both square, costs $5
per square foot and the metal for the sides costs $4 per square
foot. Find the dimensions that minimize cost if the box has a
volume of 20 cubic feet.
Length of base x =
Height of side z =
16. (1 pt) A fence 6 feet tall runs parallel to a tall building at
a distance of 4 feet from the building. What is the length of the
shortest ladder that will reach from the ground over the fence to
the wall of the building?
Note: you can click on the image to get a enlarged view.
Largest area =
ft2
Answer:
2
The second approximation x2 is
and the third approximation x3 is
17. (1 pt) If 2400cm2 of material is available to make a box
with a square base and an open top, find the largest possible
volume of the box.
20. (1 pt) Use Newton’s method to approximate a root of the
equation x3 + x + 3 = 0 as follows.
cm3
Volume =
Let x1 = −1 be the initial approximation.
The second approximation x2 is
and the third approximation x3 is
18. (1 pt) A rectangle is inscribed with its base on the x-axis
and its upper corners on the parabola y = 7 − x2 . What are the
dimensions of such a rectangle with the greatest possible area?
Width =
21. (1 pt) Use Newton’s method to approximate the value of
√
3
51
Height =
as follows:
Let x1 = 3 be the initial approximation.
The second approximation x2 is
and the third approximation x3 is
19. (1 pt) Use Newton’s method to approximate a root of the
equation 5x7 + 7x4 + 4 = 0 as follows.
Let x1 = 3 be the initial approximation.
c
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