# MATHEMATICS

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”.

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2. (1 point) Let f (x) = 1

5×2 +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”.

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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:

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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:

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5. (1 point) Find the inflection points of f (x) = 6×4 +58×3 � 30×2 + 14. (Give your answers as a comma separated list, e.g., 3,-2.)

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1

Kyle Taitt

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-

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.

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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 ? .

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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,•):

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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.

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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.