Calculus HW

Kyle Taitt CSU Webwork

MATH 160 WeBWorK assignment M160-801-FA-2.6 due 10/11/2016 at 11:59pm MDT

1. (1 point) Let f (x) = 4x3 + 3. Find the open intervals on which f is increasing (decreasing). Then determine the x- coordinates of all relative maxima (minima).

1. f is increasing on the intervals 2. f is decreasing on the intervals 3. The relative maxima of f occur at x = 4. The relative minima of f 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 two, 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) = 4 p

x� 6x for x > 0. Find the open intervals on which f is increasing (decreasing). Then determine the x-coordinates of all relative maxima (minima).

1. f is increasing on the intervals 2. f is decreasing on the intervals 3. The relative maxima of f occur at x = 4. The relative minima of f 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 two, your answer should be a comma separated list of x values or the word “none”.

Answer(s) submitted:

• • • •

(incorrect)

3. (1 point) Let f (x) = x3 �15x2 +63x+13. Find the open intervals on which f is increasing (decreasing). Then determine the x-coordinates of all relative maxima (minima).

1. f is increasing on the intervals 2. f is decreasing on the intervals 3. The relative maxima of f occur at x = 4. The relative minima of f 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 two, your answer should be a comma separated list of x values or the word “none”.

Answer(s) submitted:

• • • •

(incorrect)

4. (1 point) The function f (x) = 6x+8x�1 has one local min- imum and one local maximum. This function has a local maximum at x= with value

and a local minimum at x = with value Answer(s) submitted:

• • • •

(incorrect)

5. (1 point) The function

f (x) = 2x3 +3x2 �336x+7

is decreasing on the interval . Enter your answer using the interval notation for open inter-

vals.

It is increasing on the interval(s) . The function has a local maximum at .

Answer(s) submitted:

• • •

(incorrect)

6. (1 point) Find the critical points, A and B, of the following polynomial (with A < B).

f (x) = 6x3 +27x2 �180x+8

A = B =

Answer(s) submitted:

• •

(incorrect)

1

7. (1 point) The function

f (x) =�2x3 +9x2 +168x+2 is increasing on the interval ( , ).

It is decreasing on the interval ( �•, ) and the interval ( , • ). The function has a local maximum at . Answer(s) submitted:

• • • • •

(incorrect) 8. (1 point)

Find the critical points and determine if the function is increas- ing or decreasing on the given intervals. y = 9x4 +8x3

Left critical point: c1 = Right critical point: c2 = The function is: ? on (�•,c1). ? on (c1,c2). ? on (c2,•).

Answer(s) submitted:

• • • • •

(incorrect) 9. (1 point) Find the critical points and the interval on which the given

function is increasing or decreasing, and apply the First Deriva- tive Test to each critical point. Let f (x) = 34 x

4 + 63 x 3 + �32 x

2 �6x There are three critical points. If we call them c1,c2, and c3,

with c1 < c2 < c3, then c1 = c2 = and c3 = .

Is f a maximum or minumum at the critical points? At c1, f is ? At c2, f is ? At c3, f is ?

These three critical give us four intervals. The left-most interval is , and on this interval f is ? while f 0 is ? . The next interval (going left to right) is . On this interval f is ? while f 0 is ? .

Next is the interval . On this interval f is ? while f

0 is ? . Finally, the right-most interval is . On this interval f is ? while f 0 is ? .

Answer(s) submitted:

• • • • • • • • • • • • • • • • • •

(incorrect)

10. (1 point) The lynx population on a small island is ob- served to be given by the function

P(t) = 120t �0.4t4 +700

where t is the time (in months) since observations of the island began.

Note: you can get a larger view of the graph by clicking on it (a) The maximum population attained at

2

t = months.

The maximum population is

P(t) = lynx.

(b) When does the lynx population disappear from the island?

t = months Answer(s) submitted:

• • •

(incorrect)

11. (1 point) Consider the function f (x) = 2x3 �6x2 �90x+ 10 on the interval [�5,9]. Find the average or mean slope of the function on this interval.

By the Mean Value Theorem, we know there exists at least one c in the open interval (�5,9) such that f 0(c) is equal to this mean slope. Find all values of c that work and list them (separated by commas) in the box below. If there are no values of c that work, enter None .

List of numbers: Answer(s) submitted:

• •

(incorrect)

12. (1 point) In this problem you will use Rolle’s theorem to determine whether it is possible for the function

f (x) =� ⇣

6x5 +3x+6 ⌘

to have two or more real roots (or, equivalently, whether the graph of y = f (x) crosses the x-axis two or more times).

Suppose that f (x) has at least two real roots. Choose two of these roots and call the smaller one a and the larger one b. By applying Rolle’s theorem to f (x) on the interval [a,b], there exists at least one number c in the interval (a,b) so that f 0(c) =

.

The values of the derivative f 0(x) = are always ? , and therefore it is ? for f (x) to have two or more real roots.

Answer(s) submitted:

• • • •

(incorrect)

13. (1 point) Consider the function f (x) = 6� 2x2 on the interval [�4,5]. (A) Find the average or mean slope of the function on this inter- val, i.e.

f (5)� f (�4) 5� (�4) =

(B) By the Mean Value Theorem, we know there exists a c in the open interval (�4,5) such that f 0(c) is equal to this mean slope. For this problem, there is only one c that works. Find it.

c = Answer(s) submitted:

• •

(incorrect)

14. (1 point) Consider the function graphed below.

Does this function satisfy the hypotheses of the Mean Value Theorem on the interval [a,b]? [?/yes/no]

Does it satisfy the conclusion? [?/yes/no]

At what point c is f 0(c) = f (b)� f (a)

b�a ? • ? • c=m • c=n • c=p • there is no such point Answer(s) submitted:

• • •

(incorrect)

15. (1 point) Suppose that 1  f 0(x)  3 for all values of x. Use the Mean Value Theorem to find values for the inequality below.

Answer:  f (4)� f (�2)

Answer(s) submitted:

• •

(incorrect)

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16. (1 point) At 2:00pm a car’s speedometer reads 30mph, and at 2:10pm it reads 40mph. Use the Mean Value Theorem to find an acceleration the car must achieve.

Answer( in mi/h2): Answer(s) submitted:

• (incorrect) 17. (1 point)

Find a point c satisfying the conclusion of the Mean Value Theorem for the function f (x) = x�8 on the interval [1,5].

c = Answer(s) submitted:

• (incorrect) 18. (1 point) If a and b are positive numbers, find the maximum value of

f (x) = xa(1� x)b, 0  x  1

Your answer may depend on a and b. maximum value = Answer(s) submitted:

• (incorrect) 19. (1 point)

Find the minimum and maximum values of y= p

14q� p

7secq on the interval [0, p3 ]

fmin = fmax =

Answer(s) submitted:

• •

(incorrect)

20. (1 point)

Referring to the graph above, which of the following statements is correct:

• A. f 00(x) changes sign from + to - • B. f 00(x)> 0 for all x • C. f 00(x) changes sign from - to + • D. f 00(x)< 0 for all x

Answer(s) submitted:

•

(incorrect)

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