CALCULUS WORKSHEET ON APPLICATIONS OF THE DEFINITE INTEGRAL - ACCUMULATION Work the following on notebook paper. Use your calculator on problems 1-8 and give decimal answers correct to three decimal places. 1. A tire factory is located at the center of a town, which etends three miles either side of the factory. Let be the distance from the factory. A straight highway goes through town, and the density of particles of pollutant along the highway is given by p 51 where p is the number of particles of pollutant per mile per day at the point. Find the total number of particles of pollutant per day deposited in town along the highway.. The density function for the population of the small coastal town of Westport, WA, is given p where is the distance along a straight highway from the ocean and p in people per mile. The town etends for two miles from the ocean. Find the population of Westport.. A city in the shape of a circle of radius 1 miles is growing with its population density a function of the distance from the center of the city. At a distance of r miles from the city center, 5 its population density is Pr people/square mile. In 1 r the figure, thin rings have been drawn around the center of the city. The area in square miles of the shaded ring can be approimated by the product r r, where r is the radius of the outer ring. (a) Write an epression to represent the number of people in the shaded ring. (b) Find the population of the city. is measured 4. Greater Seattle can be approimated by a semicircle of radius 5 miles with its center on the shoreline of Puget Sound. Moving away from the center along a radius, the population density can be approimated by.1, e, where p p is measured in people per square mile. Find the population of the city. 5. The density of oil in a circular oil slick on the surface of the ocean at a distance r meters from the center 5 of the slick is given by f r where is measured in kg/m and r is measured in meters. If 1 r the slick etends from r = to r = 1 m, find the eact value of the mass of oil in the slick. f r.1t 6. Water is pumped out of a holding tank at a rate of 5 5e liters/minute, where t is in minutes since the pump is started. If the holding tank contains 1 liters of water when the pump is started, how much water does it hold one hour later? 7. A cup of coffee at 9 C is put into a C room when t =. The coffee s temperature is decreasing at a rate.1t of r t 7e C per minute, with t in minutes. (a) Find the coffee s temperature when t = 1. (b) Find the average rate of change of the temperature of the coffee over the first ten minutes. TURN->>>
Use your calculator on problem 8. 8. Let R be the region bounded by the y-ais and the graphs of y and y as shown in the figure. 1 (a) Find the area of R. (b) Find the volume of the solid generated when R is revolved about the horizontal line y 1. (c) Find the volume of the solid generated when R is revolved about the vertical line 1. (d) The region R is the base of a solid. For this solid, each cross section perpendicular to the -ais is a square. Find the volume of this solid. Do not use your calculator on problem 9. 7. Let R be the region in the first quadrant bounded by the (4, ) -ais and the graphs of y and y 6 as shown in the figure on the right. (a) Find the area of R. R (b) The region R is the base of a solid. For this solid, the cross sections perpendicular to the -ais are rectangles with a height of 5. Wrote, but do not evaluate, an integral epression or the volume of the solid. (c) The region R is the base of a solid. For this solid, the cross sections perpendicular to the y-ais are semicircles. Wrote, but do not evaluate, an integral epression for the volume of the solid. (d) Write, but do not evaluate, an integral epression for the perimeter of R. y R
CALCULUS BC WORKSHEET 1 ON LOGISTIC GROWTH Work the following on notebook paper. Do not use your calculator. 1. Suppose the population of bears in a national park grows according to the logistic differential equation dt (a) Given P (i) Find, where P is the number of bears at time t in years. 5P.P 1. Pt. t (ii) What is the range of the solution curve? (iii) For what values of P is the solution curve increasing? Decreasing? Justify your answer. (iv) Find d P dt and use it to find the values of P for which the solution curve is concave up and concave down. Justify your answer. (v) Does the solution curve have an inflection point? Justify your answer. (vi) Use the information you found to sketch the graph of (b) Given (i) Find t P 15. Pt. Pt (ii) What is the range of the solution curve? (iii) For what values of P is the solution curve increasing? Decreasing? Justify your answer. (iv) For what values of P is the solution curve concave up? Concave down? Justify your answer. (v) Does the solution curve have an inflection point? Justify your answer. (vi) Use the information you found to sketch the graph of (c) Given (i) Find t P. Pt. (ii) What is the range of the solution curve? (iii) For what values of P is the solution curve increasing? Decreasing? Justify your answer. (iv) For what values of P is the solution curve concave up? Concave down? Justify your answer. (v) Does the solution curve have an inflection point? Justify your answer. (vi) Use the information you found to sketch the graph of Pt.. Pt. (d) How many bears are in the park when the population of bears is growing the fastest?. Suppose a rumor is spreading through a dance at a rate modeled by the logistic differential equation P P. What is Pt? What does this number represent in the contet of this problem? dt t TURN->>>
. (From the 1998 BC Multiple Choice) P The population of a species satisfies the logistic differential equation P, dt 5 Pt where the initial population is P and t is the time in years. What is Pt? (A) 5 (B) (C) 4 (D) 5 (E) 1, 4. Suppose a population of wolves grows according to the logistic differential equation P.1P dt, where P is the number of wolves at time t in years. Which of the following statements are true? I. Pt t II. The growth rate of the wolf population is greatest at P = 15. III. If P >, the population of wolves is increasing. (A) I only (B) II only (C) I and II only (D) II and III only (E) I, II, and III 5. Suppose that a population develops according to the logistic equation.5p.5p dt where t is measured in weeks. (a) What is the carrying capacity? (b) A slope field for this equation is shown at the right. Where are the slopes close to? Where are they largest? Which solutions are increasing? Which solutions are decreasing? (c) Use the slope field to sketch solutions for initial populations of, 6, and 1. What do these solutions have in common? How do they differ? Which solutions have inflection points? At what population level do they occur? 6. (a) On the slope field shown on the right for P P, sketch three dt solution curves showing different types of behavior for the population P. (b) Describe the meaning of the shape of the solution curves for the population. Where is P increasing? Decreasing? What happens in the long run? Are there any inflection points? Where? What do they mean for the population? t
Answers to Worksheet 1 on Logistic Growth 1. (a) (i) 5 (ii) [1, 5) (iii) increasing for [1, 5) (iv) concave up for (1, 15) and concave down for (15, 5) (v) yes, IP when P = 15 (vi) sketch (b) (i) 5 (ii) [15, 5) (iii) increasing for [15, 5) (iv) concave down for (15, 5) (v) no (vi) sketch (c) (i) 5 (ii) (5, ] (iii) decreasing for (5, ] (iv) concave up for (5, ) (v) no (vi) sketch. 6; there are 6 people at the dance.. E 4. C 5. (a) 1 (b) Close to? P = and P = 1 Largest? P = 5 Increasing? Decreasing? P 1 P 1 (c) In common? All have a it of 1. Differ? Two are increasing; one is decreasing. Inflection points? The one with initial condition of. At what pop. level does the inflection point occur? When P = 5. 6. (a) sketch (b) Increasing? Decreasing? In the long run? P 1 P 1 Pt 1 t Any inflection points? Yes Where? When P.5 What do they mean for the population? The population is growing the fastest when P.5.
CALCULUS BC WORKSHEET ON LOGISTIC GROWTH Work the following on notebook paper. Use your calculator on (b) and (c), (c) and (d), 4(b) and (c), and 5(c) and (d) only. 1. Suppose you are in charge of stocking a fish pond with fish for which the rate of population growth is modeled by the differential equation (a) Given (i) Find P 5. Pt. t 8P.P dt. (ii) What is the range of the solution curve? (iii) For what values of P is the solution curve increasing? Decreasing? Justify your answer. (iv) Find d P dt and use it to find the values of P for which the solution curve is concave up and concave down. Justify your answer. (v) Does the solution curve have an inflection point? Justify your answer. (vi) Use the information you found to sketch the graph of (b) Given (i) Find t P. Pt. (ii) What is the range of the solution curve? (iii) For what values of P is the solution curve increasing? Decreasing? Justify your answer. (iv) For what values of P is the solution curve concave up? Concave down? Justify your answer. (v) Does the solution curve have an inflection point? Justify your answer. (vi) Use the information you found to sketch the graph of (c) Given (i) Find t P 5. Pt. (ii) What is the range of the solution curve? (iii) For what values of P is the solution curve increasing? Decreasing? Justify your answer. (iv) For what values of P is the solution curve concave up? Concave down? Justify your answer. (v) Does the solution curve have an inflection point? Justify your answer. (vi) Use the information you found to sketch the graph of Pt Pt.. Pt.. A population of animals is modeled by a function P that satisfies the logistic differential.1p 1 P, where t is measured in years. equation (a) If dt P, solve for P as a function of t. (b) Use your answer to (a) and your graphing calculator to find P when t = years. (c) Use your answer to (a) and your graphing calculator to find t when P = 8 animals.
TURN->>>. The rate at which a rumor spreads through a high school of students can be modeled by the differential equation.p P, where P is the number of students who have heard the rumor t hours dt after 9AM. (a) How many students have heard the rumor when it is spreading the fastest? Justify your answer. (b) If P 5, solve for P as a function of t. (c) Use your answer to (b) and your graphing calculator to determine how many hours have passed when half the student body has heard the rumor. (d) Use your answer to (b) and your graphing calculator to determine how many students have heard the rumor after hours. 4. Suppose a rumor is spreading at a dance attended by students. The rumor is spreading at a rate that is directly proportional to both the number of students who have heard the rumor and the number of students who have not heard the rumor. Let P be the number of students who have heard the rumor, and let t be the time in minutes since the rumor began to spread. (a) Write a differential equation to model this rate of change. (b) If P P 1 and 15 5, solve for P as a function of t. (c) Use your solution to (b) to find the number of students who have heard the rumor after 1 hour. (d) Use your solution to (b) to find the time it takes for 175 students to hear the rumor.
Answers to Worksheet on Logistic Growth 1.(a) (i) 4 (ii) [5, 6) (iii) increasing for [5, 4) (iv) concave up for (5, ) and concave down for (, 4) (v) yes, IP when P = (vi) sketch (b) (i) 4 (ii) [, 6) (iii) increasing for [, 4) (iv) concave down for (, 4) (v) no (vi) sketch (c) (i) 4 (ii) (4, 5] (iii) decreasing for (4, 5] (iv) concave up for (4, 5) (v) no (vi) sketch t 1e 1. (a) P or P t e 4 14 (b) 8.9 animals (c).77 years. (a) 1 students 6t t e e (b) P or P 6t 6 e 99 199e (c).998 hours (d) 1995.189 so 1995 people t kp P dt 19 kt ln e (b) P or P where k kt kt 19 e 19e 1 4. (a)
CALCULUS BC WORKSHEET ON L HOPITAL S RULE Work the following on notebook paper. No calculator. On problems 1, find the it by: (a) using techniques from Chapter 1 (b) using L Hopital s Rule. 6 1 5 1 1... 9 5 Evaluate by using L Hopital s Rule, if possible. 4. 1. ln 5. 4 14. ln 6. e 1 15. 1 sin 7. sin sin 16. 1 8. arcsin 17. 1 1 9. 1 8 18. 4 1. 1 1 19. ln 1 1 11. 1. e 5 1. ln 1. e 1 e
Answers to Worksheet on L Hopital s Rule 1... 1 1 4 5 8. 1 15. 1 9. 1. 4. 11. 1 (L Hop. doesn t work) 18. 16. 1 17. e 5. 1. 19. 6. 1. / 7. 14. 1. (Not a L Hop. problem)
CALCULUS BC WORKSHEET ON L HOPITAL S RULE AND IMPROPER INTEGRALS Work the following on notebook paper. No calculator. Evaluate. 1 cos 1. 5. cos 1 e 1 1 1. e 6. 1. 1 1 1 ln 1 7. cos cos 4. 1 sin 1 cos Evaluate. 8. d 1. d 1 9. 1 d 14. 1 1 4 d 1. 1 ln e d 15. d 1 5 6 11. d 1 16. 8 e d 1. 4 d f e, and let R be the unbounded region between f and the 17. Let for -ais. Find the volume of the solid generated when R is revolved about the -ais.
Answers to Worksheet on L Hopital s Rule and Improper Integrals 1 1... 1 8 5. 1 6. e 7. 1 8. converges to 1 9. converges to 11. converges to 9 1. converges to 4 1. diverges to 1. converges to ln 4. 1 4 14. converges to 17. converges to 4 15. converges to ln 16. converges to 1