Calc HW
flcoman89
Kyle Taitt CSU Webwork
MATH 160 WeBWorK assignment M160-801-FA-2.7 due 10/14/2016 at 11:59pm MDT
1. (1 point) Let f (x) = 6x�2 x+3
. Find the open intervals on which f is concave up (down). Then determine the x- coordinates of all inflection points of f .
1. f is concave up on the intervals 2. f is concave down on the intervals 3. The inflection points occur at x =
Notes: In the first two, your answer should either be a single interval, such as (0,1), a comma separated list of intervals, such as (-inf, 2), (3,4), or the word “none”.
In the last one, your answer should be a comma separated list of x values or the word “none”.
Answer(s) submitted:
• • •
(incorrect)
2. (1 point) Let f (x) = 1
5x2 +3 . Find the open intervals
on which f is concave up (down). Then determine the x- coordinates of all inflection points of f .
1. f is concave up on the intervals 2. f is concave down on the intervals 3. The inflection points occur at x =
Notes: In the first two, your answer should either be a single interval, such as (0,1), a comma separated list of intervals, such as (-inf, 2), (3,4), or the word “none”.
In the last one, your answer should be a comma separated list of x values or the word “none”.
Answer(s) submitted:
• • •
(incorrect)
3. (1 point) Consider the function f (x) = x|x|.
a.) On the interval (�•,0), f 00(x) =
b.) On the interval (0,•), f 00(x) =
Thus,
c.) On the interval (�•,0), f (x) is ?
d.) On the interval (0,•), f (x) is ?
e.) Does f 00(0) exist? ? Answer(s) submitted:
• • • • •
(incorrect)
4. (1 point) Below is the graph of the derivative f 0(x) of a function defined on the interval (0,8). You can click on the graph to see a larger version in a separate window.
Refer to the graph to answer each of the following questions. For part (A), use interval notation to report your answer. (If needed, you use U for the union symbol.)
(A) For what values of x in (0,8) is f (x) concave down? (If the function is not concave down anywhere, enter ”” without the quotation marks.)
Answer: (B) Find all values of x in (0,8) is where f (x) has an inflection point, and list them (separated by commas) in the box below. (If there are no inflection points, enter -1000.)
Inflection Points:
Answer(s) submitted:
• •
(incorrect)
5. (1 point) Find the inflection points of f (x) = 6x4 +58x3 � 30x2 + 14. (Give your answers as a comma separated list, e.g., 3,-2.)
inflection points = Answer(s) submitted:
• (incorrect)
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6. (1 point) Use the given graph of the function f to answer the following questions.
1. Find the open interval(s) on which f is concave upward.
Answer (in interval notation): 2. Find the open interval(s) on which f is concave down-
ward. Answer (in interval notation):
3. Find the coordinates of the points of inflection. List your answers as points in the form (a,b). Answer (separate by commas):
Note: You can click on the graph to enlarge the image.
Answer(s) submitted:
• • •
(incorrect)
7. (1 point) Determine the intervals on which the given function is con-
cave up or down and find the point of inflection. Let
f (x) = x � x�7
p x
�
The x-coordinate of the point of inflection is The interval on the left of the inflection point is ,
and on this interval f is ? . The interval on the right is , and on this interval f is ? .
Answer(s) submitted:
• • • • •
(incorrect)
8. (1 point) Consider the function f (x) = 5x+5x�1. Note that this function has no inflection points, but f 00(x) is un- defined at x = B where B =
For each of the following intervals, tell whether f (x) is concave up (type in CU) or concave down (type in CD). (�•,B): (B,•):
Answer(s) submitted:
• • •
(incorrect)
9. (1 point) Please answer the following questions about the function
f (x) = (x2 +10)(1� x2).
Instructions: If you are asked to find x- or y-values, enter either a number, a list of numbers separated by commas, or None if there aren’t any solutions. Use interval notation if you are asked to find an interval or union of intervals, and enter { } if the interval is empty.
(a) Find the critical numbers of f , where it is increasing and decreasing, and its local extrema. Critical numbers x = Increasing on the interval Decreasing on the interval Local maxima x = Local minima x =
(b) Find where f is concave up, concave down, and has in- flection points. Concave up on the interval Concave down on the interval Inflection points x =
(c) Find any horizontal and vertical asymptotes of f . Horizontal asymptotes y = Vertical asymptotes x =
(d) The function f is ? because ? for all x in the domain of f , and therefore its graph is symmetric about the ?
(e) Sketch a graph of the function f without having a graph- ing calculator do it for you. Plot the y-intercept and the x- intercepts, if they are known. Draw dashed lines for horizon- tal and vertical asymptotes. Plot the points where f has local maxima, local minima, and inflection points. Use what you know from parts (a) and (b) to sketch the remaining parts of
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the graph of f . Use any symmetry from part (d) to your advan- tage. Sketching graphs is an important skill that takes practice, and you may be asked to do it on quizzes or exams.
Answer(s) submitted:
• • • • • • • • • • • • •
(incorrect)
10. (1 point) Choose the best reason that the function f (x) = x79 + x41 + x9 + 7 has neither a local maximum nor a local minimum.
(a) The function f (x) is always positive. (b) The derivative f 0(x) is always negative. (c) The derivative f 0(x) is always positive. (d) The highest power of x in f (x) is odd.
Answer:
Note: Input a, b, c, or d . Answer(s) submitted:
•
(incorrect)
11. (1 point) Use a graph below of f (x) = ln(8x2 +1) to es- timate the x-values of any critical points and inflection points of f (x).
critical points (enter as a comma-separated list): x =
inflection points (enter as a comma-separated list): x =
Next, use derivatives to find the x-values of any critical points and inflection points exactly. critical points (enter as a comma-separated list): x =
inflection points (enter as a comma-separated list): x =
Answer(s) submitted:
• • • •
(incorrect) 12. (1 point) The figure below gives the behavior of the de-
rivative of g(x) on �2 x 2.
Graph of g
0(x) (not g(x)) (Click on the graph to get a larger version.)
Sketch a graph of g(x) and use your sketch to answer the following questions.
A. Where does the graph of g(x) have inflection points? x = Enter your answer as a comma-separated list of values, or enter
none if there are none.
B. Where are the global maxima and minima of g on [�2,2]? minimum at x = maximum at x =
C. If g(�2) =�8, what are possible values for g(0)? g(0) is in (Enter your answer as an interval, or union of intervals, giving
the possible values. Thus if you know �10 < g(0) �5, enter (-10,-5]. Enter infinity for •, the interval [0,0] to indicate a single point).
How is the value of g(2) related to the value of g(0)? g(2) g(0) (Enter the appropriate mathematical equality or inequality, =, <, >, etc.)
Answer(s) submitted:
• • • • •
(incorrect) 3
13. (1 point) The graph of f 0 (not f ) is given below.
(Note that this is a graph of f
0 , not a graph of f .)
At which of the marked values of x is A. f (x) greatest? x = B. f (x) least? x = C. f
0(x) greatest? x = D. f
0(x) least? x = E. f
00(x) greatest? x = F. f
00(x) least? x = Answer(s) submitted:
• • • • • •
(incorrect)
14. (1 point) Given the graph of y = f (x) below, at which of the marked x-values can the following statements be true?
(For each question, enter your answer as a comma-separated
list, e.g., x1,x3,x5. Enter none if no points satisfy the given con-
dition)
A. f (x)> 0 at x =
B. f
0(x)> 0 at x =
C. f (x) is increasing at x =
D. f
0(x) is increasing at x =
E. The slope of f (x) is negative at x =
F. The slope of f 0(x) is negative at x = Answer(s) submitted:
• • • • • •
(incorrect)
15. (1 point) At exactly two of the labeled points in the fig- ure below, which shows a function f , the derivative f 0 is zero; the second derivative f 00 is not zero at any of the labeled points. Select the correct signs for each of f , f 0 and f 00 at each marked point.
Point A B C D E f [?/positive/zero/negative] [?/positive/zero/negative] [?/positive/zero/negative] [?/positive/zero/negative] [?/positive/zero/negative] f
0 [?/positive/zero/negative] [?/positive/zero/negative] [?/positive/zero/negative] [?/positive/zero/negative] [?/positive/zero/negative] f
00 [?/positive/zero/negative] [?/positive/zero/negative] [?/positive/zero/negative] [?/positive/zero/negative] [?/positive/zero/negative]
Answer(s) submitted:
• • • • • • • • • • • • • • •
(incorrect)
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16. (1 point) Consider the function f (x) graphed below.
For this function, are the following nonzero quantities posi- tive or negative? f (0.5) is [?/positive/negative] f
0(0.5) is [?/positive/negative] f
00(0.5) is [?/positive/negative] (Because this is a multiple choice problem, it will not show
which parts of the problem are correct or incorrect when you
submit it.)
Answer(s) submitted:
• • •
(incorrect)
17. (1 point) Use the given graph of the function on the in- terval [0,8] to answer the following questions.
1. For what values of x does the function f have a local maximum on (0,8)? Answer (separate by commas): x =
2. For what values of x does the function f have a local minimum on (0,8)? Answer (separate by commas): x =
3. Find the absolute maximum for the function f on the interval [0,8]. Answer:
4. Find the absolute minimum for the function f on the in- terval [0,8]. Answer:
Note: You can click on the graph to enlarge the image.
Answer(s) submitted:
• • • •
(incorrect)
18. (1 point)
You are given the following graph of the function f (x):
Find the point where the second derivative changes sign from negative to positive?
x= Answer(s) submitted:
• (incorrect)
19. (1 point) The picture below shows the graph y = f 0(x) of the derivative of a function y = f (x).
For each of the labelled points on the graph, classify the corre- sponding point on the graph of y = f (x) as on of the following: MAX, MIN, INFL, INT (short for maximum, minimum, inflec- tion point, x-intercept) A: B: C:
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D: E:
For each of the following intervals, classify whether the graph of y = f (x) is INC or DEC over that interval (short for increasing or decreasing). (�•,A) (A,B) (B,C) (C,D) (D,E) (E,•)
For each of the following intervals, classify whether the graph of y = f (x) is CU or CD over that interval (short for con- cave up or concave down). (�•,A) (A,B) (B,C) (C,D) (D,E) (E,•)
Answer(s) submitted:
• • • • • • • • • • • • • • • • •
(incorrect) 20. (1 point) Function Analysis The graph of the function f is given below. Assume that f is
as smooth as the graph allows.
Fill in the function analysis table.
x x <�4 x =�4 �4 < x < 3 x = 3 3 < x < 6 x = 6 6 < x
f ? ? ? ? ? ? ? ? ? ? ?
f
0 ? ? ? ? ? ? ? ? ? ? ?
f
00 ? ? ? ? ? ? ?
Answer(s) submitted:
• • • • • • • • • • • • • • • • • • • • • • • • • • • • •
(incorrect)
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