Exam 3 Practice Problems

Transcription

1 MATH1214 Exam 3 Practice Problems 1. Find the absolute maximum and absolute minimum values of f(x) = x 3 + 3x 2 9x 7 on each of the following intervals (a) [ 6, 4] (b) [ 4, 2] (c) [ 2, 2] 2. Find the absolute maximum and absolute minimum values of f(x) = x 3 12x on each of the following intervals (a) [ 5, 5] (b) [ 3, 3] (c) [ 3, 1] 3. Find the absolute maximum and absolute minimum values of f(x) = x 3 + x 2 on the interval [ 1, 1]. 4. Find all numbers c guaranteed by Rolle's Theorem for f(x) = x 2 4x + 1 on the interval [0, 4]. (Check the conditions of Rolle's Theorem.) 5. Find all numbers c guaranteed by Rolle's Theorem for f(x) = x 3 3x 2 + 2x + 5 on the interval [0, 2]. (Check the conditions of Rolle's Theorem.) 6. Find all numbers c guaranteed by the Mean Value Theorem (MVT) for f(x) = on the interval x [1, 4]. (Check the conditions of the MVT.) 7. Find all numbers c guaranteed by the Mean Value Theorem (MVT) for f(x) = x 3 x on the interval [0, 2]. (Check the conditions of the MVT.) 8. Let f(x) = 3x 2 x 3. Perform the following (Do not graph) (a) Find the intervals where the function is increasing and decreasing. (b) Find the intervals where the function is concave up and concave down. (c) Find all the local maxima and local minima of f. (d) Find all the inection points of f. 9. Let f(x) = 3x + 4. Perform the following (Do not graph) x 4 (a) Find the intervals where the function is increasing and decreasing. (b) Find the intervals where the function is concave up and concave down. (c) Find all the vertical and horizontal asymptotes of f. 10. Find the local maxima and local minima for each function. Use the second derivative-test when it applies. (a) f(x) = x 3 6x 2 + 9x + 1 (b) f(x) = 1 6 x6 4x x Find the local maxima and local minima for each function. Use the second derivative-test when it applies. (a) f(x) = x 3 9x x 10 (b) f(x) = 10x 6 24x x 4

2 12. For the function f(x) = 1 x x 4 x For the function x 3 f(t) = x 2 4x For the function f(x) = x 1 x For the function f(x) = 2x 1 x 16. The altitude (in feet) of a rocket t seconds into ight is given by s = f(t) = t t t + 6 Find the point of inection of the function f and interpret your result. What is the maximum velocity attained by the rocket? 17. Use the graphing strategy to analyze the function f(x) = x 4 2x 3. State all the pertinent information, 18. Use the graphing strategy to analyze the function f(x) = x 4 + 4x 3. State all the pertinent information, 19. Use the graphing strategy to analyze the function f(x) = x 1. State all the pertinent information, x Use the graphing strategy to analyze the function f(x) = 2x. State all the pertinent information, 1 x 21. A rectangular ower bed is to be constructed with an area of 18 square yards. The bed will be fenced on three sides and the fourth side will be along a long, straight wall. What should the dimensions of the bed be to minimize the amount of fencing used? what is the minimal amount of fencing required to construct the bed? 22. A farmer wants to construct a rectangular pen next to a barn 60 feet long, using all of the barn as part of one side of the pen. Find the dimensions of the pen with the largest area that the farmer can build if: (a) 160 feet of fencing material is available. (b) 250 feet of fencing material is available. 23. A piece of wire 16 inches long is cut into two pieces; one piece is bent to form a square and the other is bent to form a circle. Where should the cut be made so that the sum of the areas of the square and the circle is a minimum? A maximum? (Allow the possibility of no cut.) 24. Patricia wishes to have a rectangular-shaped garden in her backyard. She has 80ft of fencing material with which to enclose her garden. Letting x denote the width of the garden, nd a functionf in the variable x giving the area of the garden. What is its domain?

3 25. Patricia's Neighbor, Juanita, also wishes to have a rectangular-shaped garden in her backyard. But Juanita wants her garden to have an area of 250ft 2. Letting x denote the width of the garden, nd a functionf in the variable x giving length of the fencing material required to construct the garden. What is its domain of the function? 26. By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting aps, an open box may be made. If the cardboard is 15in. long and 8in. wide and the square cutaways have dimensions of xin. by xin., nd a function giving the volume of the resulting box. 27. By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting aps, an open box may be made. If the cardboard is 45 inches long and 24 inches wide and the square cutaways have dimensions of x inches by x inches, for what dimensions does the box have the largest volume? 28. A piece of wire 100 inches long is cut into two pieces; one piece is bent to form a square and the other is bent to form an equilateral triangle. Where should the cut be made so that the sum of the areas of the square and the triangle is a minimum? A maximum? (Allow the possibility of no cut.) 29. A drug is injected into the bloodstream of a patient through the right arm. The concentration of the drug in the bloodstream of the left arm t hours after the injection is approximated by C(t) = 0.14t t 2 0 < t < How many hours after the drug is given will the concentration be maximum? What is the maximum concentration? 30. The concentration C(t), in milligrams per cubic centimeter, of a particular drug in a patient's bloodstream is given by 0.16t C(t) = t 2 0 < t < t + 4 where t is the number of hours after the drug is taken orally. How many hours after the drug is given will the concentration be maximum? What is the maximum concentration? 31. Suppose that the total cost C(x) (in thousands of dollars) for manufacturing x sailboats per year is given by the function C(x) = x 0.2x 2 0 x 50 (a) Find the marginal cost at a production level of x boats per year. (b) Find the marginal cost at a production level of 35 boats, and interpret the results. 32. Suppose that the total cost C(x) (in thousands of dollars) for manufacturing x sailboats per year is given by the function C(x) = x 0.2x 2 0 x 50 (a) Use the marginal cost function to approximate the cost of producing the 41st boat. (b) Use the total cost function to nd the exact cost of producing the 41st boat. 33. The sales (in millions of dollars) of a laser disk recording of a hit movie t years from the date of release is given by S(t) = 5t t How fast are the sales changing at the time the laser discs are released (t = 0)? 34. The total sales S (in thousands of games) for a home video game t months after is introduced are given by S(t) = 150t t + 3

4 (a) Find S (t) = (b) Find S(12) and S (12). Write a brief interpretation of these results. (c) Use the results from part (b) to estimate the total sales after 13 months. 35. According to the South Coast Air Quality Management District, the level of nitrogen dioxide, a brown gas that impairs breathing, present in the atmosphere on a certain May day in downtown Los Angeles, is approximated by A(t) = 0.03t 3 (t 7) (0 t 7) where A(t) is measured in pollutant standard index and t is measured in hours, with t = 0 corresponding to 7A.M. (a) Find A (t) = (b) Find A (1), A (3), and A (4) and interpret your results. 36. The number of subscribers to CNC Cable Television in the town of Randolph is approximated by the function N(x) = 1000(1 + 2x) 1/2 (1 x 30) where N(x) denotes the number of subscribers to the service in the xthe week. Find the rate of increase in the number of subscribers at the end of the 12th week. 37. The marketing department of Telecom Corporation has determined that the demand for their cordless phones obeys the relationship p = 0.02x (0 x 30, 000) where p denotes the phone's unit price (in dollars) and x denotes the quantity demanded. (a) Find the revenue function R. (b) Find the marginal revenue function R. (c) Compute R (10, 000) and interpret your result. 38. The market research department of a company recommends that the company manufacture and market a new transistor radio. After suitable tests marketing, the research department presents the following price demand equation p = x where x is the number of radios retailers are likely to buy at $p per radio. Also the nancial department provides the following cost function C(x) = 7, x 0 x 10, 000 where 7, 000 is the estimate of xed costs (tooling and overhead) and $2 is the estimate of variable costs per radio (materials, labor, marketing, etc.) (a) Find R (3, 000) and R (6, 000) and interpret the results. (b) Find P (2, 000) and P (7, 000) and interpret the results. 39. The Williams Commuter Air Service realizes a monthly revenue of R(x) = 8000x 100x 2 dollars when the price charged per passenger is x dollars. (a) Find the marginal revenue R (x) = (b) Compute R (39), R (40), and R (41). What do your results imply?

5 40. According to the Census Bureau, the number of Americans aged 45 to 54 will be approximately N(t) = t t t t + 25 million people in year t, where t = 0 corresponds to the beginning of Compute N (10) and N (10) and interpret your results. 41. Find the area bounded by the graph of the following equations (a) y = x y = x + 1 (b) y = x 3 3x 2 9x + 12 y = x Evaluate the following integrals (a) x x+5 (b) 4 x 1 x+5 (c) x x + 5 (d) 4 1 x x + 5 (e) 3 0 x x + 1 (f) 1 1 e5x (g) t dt = (h) 3 ( x2) (i) x x 2 9 (j) t dt = (k) 1 2x+2 (l) e ln x 1 x (m) x x+2 (n) ( e 3x + 4x x 3 ) (o) ( x + 1 x ) (p) e 3x 2 (q) x 2 +x x 3 (r) e x3 x 2 (s) (2 cos x) 4 sin x (t) π 2 0 (2 cos x)4 sin x (u) (2 + sin x) 4 cos x (v) π 2 0 (2 + sin x)4 cos x (w) 6xe 3x (x) x 2 ln 2x (y) ln x (z) x 3 e x 43. Show that the indicated function is a solution of the given dierential equation; that is, substitute the indicated function for y to see that it produces an equality.

6 (a) d2 y dx 2 + y = 0; y = c 1 sin x + c 2 cos x ( ) 2 dy (b) + y 2 = 1; y = sin(x + c) and y = ±1. dx 44. Solve the following dierential equations (a) dy x, (1, 4). y (b) dy y2 x(x 2 + 2) 4, (0, 1). 45. An object is moving along a coordinate line subject to the indicated acceleration (in centimeters per second per second) with initial velocity v 0 (in centimeters per second) and directed distance s 0 (in centimeters). Find both the velocity v and directed distance s after 2 seconds. (a) a = t; v 0 = 3 s 0 = 0 (b) a = (1 + t) 4 ; v 0 = 3 s 0 = A ball is thrown upward from the surface of a planet where the acceleration of gravity is k (a negative constant) feet per second per second. If the initial velocity is v 0, show that the maximum height is v 2 0/2k 47. The rate of change of volume V of a melting snowball is proportional to the surface area S of the ball; that is, dv/dt = ks, where k is a positive constant. If the radius of the ball at t = 0 is r = 2 and at t = 10 is r = 0.5, Show that r = 3 20 t A certain rocket shot straight up has an acceleration of 6t meters per second every second during the rst 10 seconds after blast-o, after which the engine cuts out and the rocket is subject only to gravitational acceleration of 10 meters per second every second. How high will the rocket go? 49. An oil tanker aground on a reef is losing oil and producing an oil slick that is radiating outward at a rate given approximately by dr dt = 60 t 0 t + 9 where R is the radius (in feet) of the circular slick after t minutes. Find the radius of the slick after 16 minutes if the radius is 0 when t = Find the average value of the given function on the given interval. (a) f(x) = cos x; [0, π] (b) f(x) = sin 2 x cos x; [0, π/2]