Warsame Warsame MATH 160 CSU Webwork WeBWorK assignment M160-801-SM-2.6 due 07/12/2017 at 11:59pm MDT 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\". 1. (1 point) Let f (x) = 2x3 + 3. Find the open intervals on which f is increasing (decreasing). Then determine the xcoordinates 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 = Answer(s) submitted: 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\". (incorrect) 4. (1 point) The function f (x) = 5x +9x1 has one local minimum and one local maximum. with value This function has a local maximum at x = Answer(s) submitted: and a local minimum at x = with value Answer(s) submitted: (incorrect) 2. (1 point) Let f (x) = 6 x 4x 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 = (incorrect) 5. (1 point) The function f (x) = 6x3 45x2 108x 4 is decreasing on the interval . Enter your answer using the interval notation for open intervals. It is increasing on the interval(s) . The function has a local maximum at . 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: Answer(s) submitted: (incorrect) 6. (1 point) Find the critical points, A and B, of the following polynomial (with A < B). (incorrect) f (x) = 4x3 + 24x2 252x + 5 3. (1 point) Let f (x) = x3 12x2 + 45x 9. 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 = A= B= Answer(s) submitted: (incorrect) Notes: In the first two, your answer should either be a single 1 Next is the interval . On this interval f is ? while f 0 is ? . . On this interval f Finally, the right-most interval is 0 is ? while f is ? . 7. (1 point) The function f (x) = 2x3 + 9x2 + 168x 6 is increasing on the interval ( , ). It is decreasing on the interval ( , , ). The function has a local maximum at ) and the interval ( Answer(s) submitted: . Answer(s) submitted: (incorrect) 8. (1 point) Find the critical points and determine if the function is increasing or decreasing on the given intervals. y = 3x4 + 6x3 Left critical point: c1 = Right critical point: c2 = The function is: ? on (, c1 ). ? on (c1 , c2 ). ? on (c2 , ). (incorrect) 10. (1 point) The raccoon population on a small island is observed to be given by the function Answer(s) submitted: P(t) = 120t 0.4t 4 + 800 where t is the time (in months) since observations of the island began. (incorrect) 9. (1 point) Find the critical points and the interval on which the given function is increasing or decreasing, and apply the First Derivative Test to each critical point. Let 2 f (x) = 42 x4 + 83 x3 + 18 2 x 72x 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 ? 0 while f is ? . The next interval (going left to right) is . On this interval f is ? while f 0 is ? . Note: you can get a larger view of the graph by clicking on it (a) The maximum population attained at 2 t= months. 13. (1 point) Consider the function f (x) = 2 8x2 on the interval [3, 4]. (A) Find the average or mean slope of the function on this interval, i.e. The maximum population is P(t) = raccoons. (b) When does the raccoon population disappear from the island? t= f (4) f (3) = 4 (3) (B) By the Mean Value Theorem, we know there exists a c in the open interval (3, 4) such that f 0 (c) is equal to this mean slope. For this problem, there is only one c that works. Find it. c= months Answer(s) submitted: Answer(s) submitted: (incorrect) f (x) = 2x3 6x2 90x + 11. (1 point) Consider the function 9 on the interval [5, 7]. Find the average or mean slope of the function on this interval. (incorrect) 14. (1 point) Consider the function graphed below. By the Mean Value Theorem, we know there exists at least one c in the open interval (5, 7) 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: 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] f (b) f (a) At what point c is f 0 (c) = ? ba ? c=m c=n c=p there is no such point (incorrect) 12. (1 point) In this problem you will use Rolle's theorem to determine whether it is possible for the function f (x) = 2x7 + 8x 15 to have two or more real roots (or, equivalently, whether the graph of y = f (x) crosses the x-axis two or more times). Answer(s) submitted: 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) = . (incorrect) 15. (1 point) Suppose that 4 f 0 (x) 6 for all values of x. Use the Mean Value Theorem to find values for the inequality below. f (8) f (2) Answer: 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: Answer(s) submitted: (incorrect) (incorrect) 3 Answer(s) submitted: 16. (1 point) At 2:00pm a car's speedometer reads 50mph, and at 2:10pm it reads 70mph. Use the Mean Value Theorem to find an acceleration the car must achieve. Answer( in mi/h2 ): (incorrect) Answer(s) submitted: 20. (1 point) (incorrect) 17. (1 point) Find a point c satisfying the conclusion of the Mean Value Theorem for the function f (x) = x7 on the interval [1, 2]. 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 Referring to the graph above, which of the following statements is correct: 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 = 12 6 sec on the interval [0, 3 ] fmin = fmax = A. B. C. D. f 00 (x) < 0 for all x f 00 (x) > 0 for all x f 00 (x) changes sign from + to f 00 (x) changes sign from - to + Answer(s) submitted: (incorrect) c Generated by WeBWorK, http://webwork.maa.org, Mathematical Association of America 4