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AP Calculus Exam 4 Correctives Spring, 2010 E.C. Hedstrom Objective 8: The Fundamental Theorem of Calculus. First, some notes: The basic idea comes from the Fundamental Theorem of Calculus (See text P. 181): “Let the function f be continuous on [a, b] with derivative f . Then b total change in f = f (b) f (a) f (t )dt. ” a Examples: 1) After t hours, a population of bacteria is growing at a rate given by r (t ) 2 t hundred bacteria per hour. If there were initially 10 bacteria at time t = 0, how many bacteria were there at time t = 6 hours? 2 2 0 10 ln 2 hundreds of bacteria. 6 1 1 6 63 Solution (non-calculator) :10 + 2 t dt 6 2t 0 ln 2 0 ln 2 6 Solution (calculator): 10 + r (t )dt 90.890 10 101 hundreds of bacteria. 0 2. Water is pouring into a tank at the constant rate of 30 gal per hour. Unfortunately, the tank has a leak and is leaking at the rate of r (t ) 4t 2 gal per hour. If there are initially 10 gallons of water in the tank at time t = 0, how much water will be in the tank at time t = 10? 10 Solution (non-calculator): 10 30 10 4t 2dt 310 2t 2 2t 310 220 0 90 gallons. 10 0 0 10 Solution (calculator): 10 30 10 r (t )dt = 90 gallons. 0 3. A company purchases a new machine for which the rate of depreciation (loss of value) is dV 500(t 6), 0 t 5 where V is the value of the machine in dollars after t years. dt If the machine was purchased for $15,500, what is its value after 3 years? 3 Solution (calculator): $15,500 500(t 6)dt $8750 . 0 Questions: 1. For 0 t 30 , the rate of change of the number of mosquitoes on Tropical Island at time t days is modeled t by R (t ) 5 t sin mosquitoes per day. There are 500 mosquitoes on Tropical Island at time t = 0. 5 a) Set up a definite integral to find the total change in the number of mosquitoes on the island over the 30-day period. b) Evaluate the integral from part (a) and round your answer to the nearest whole number. You do not need to show an antiderivative. c) How many mosquitoes are on the island after 30 days? 2. A tank contains 150 gallons of heating oil at time t 0. During the time interval 0 t 12 hours, heating oil is pumped into the tank at the rate 11 H (t ) 3 gallons per hour. 1 ln t 1 During the same time interval, heating oil is removed from the tank at the rate t2 R(t ) 13 sin gallons per hour. 45 a) Set up a definite integral to compute the amount of heating oil removed from the tank from t = 0 to t = 12 hours. b) Evaluate the integral from part (a). You do not need to show an antiderivative. c) Set up a definite integral to compute the amount of heating oil pumped into the tank from t = 0 to t = 12 hours. d) Evaluate the integral from part (c). You do not need to show an antiderivative. e) How much oil is in the tank at t = 12 hours? 4. The number of gallons, P(t ) , of a pollutant in a lake changes at the rate of P(t ) 1 3e 0.2 t gallons per day, where t is measured in days. There are 50 gallons of the pollutant in the lake at time t = 0. The lake is considered to be safe when it contains 40 gallons or less of pollutant. a) Write a definite integral to compute the total change in the amount of pollutant in the lake over 30 days. b) Evaluate the integral you made in part (a). Round your answer to the nearest gallon. c) Is the lake safe after 30 days? Justify your answer. 5. The tide removes sand from Sandy Point Beach at a rate modeled by the function R, given by 4t R (t ) 2 5 sin . 25 A pumping station adds sand to the beach at a rate modeled by the function S given by 15t S (t ) . 1 3t Both R(t ) and S (t ) have units of cubic yards per hour and t is measured in hours for 0 t 6 . At time t 0 , the beach contains 2500 cubic yards of sand. You do not need to show any antiderivatives for this question. a) How much sand will be removed during the 6 hours? b) How much sand will be added during the 6 hours? c) How much sand is on the beach after 6 hours? 6. (Calculator Allowed) Traffic flow is defined as the rate at which cars pass through an intersection, measured in cars per minute. The traffic flow at a particular intersection is modeled by the function F defined by t F (t ) 82 4 sin for 0 t 30 where F (t ) is measured in cars per minute and t is measured in minutes. 2 30 a) Evaluate F (t ) dt . 0 Your work must include an antiderivative and substitution. b) Using correct units, explain the meaning of the integral you found in part (a). 7. A swimming pool is being drained for cleaning with an electric pump. The rate at which water is being removed is summarized in the following table where the rate, r (t ) is in gallons per minute and the time, t , is in minutes. gal 55 65 62 75 70 88 96 122 133 r (t ) min t min 0 2 4 6 8 10 12 14 16 16 a) Estimate r (t ) dt using a midpoint Riemann sum with 4 subintervals of equal length. 0 Show all work. b) Using correct units, explain the meaning of the quantity you found in part (a). 8. t (minutes) 0 5 10 15 20 25 30 35 40 miles 7.0 9.2 9.5 7.0 4.5 2.4 2.4 4.3 7.3 v (t ) minute A test plane flies in a straight line with positive velocity v (t ) , in miles per minute at time t minutes, where v is a differentiable function of t . Selected values of v (t ) for 0 t 40 are shown in the table above. 40 a) Estimate v(t ) dt using a midpoint Riemann sum with 4 subintervals of equal length. 0 Show all work. b) Using correct units, explain the meaning of the number you found in part (a). Objective 9: Definite Integral Properties. CSV: P. 273 (18, 35), P. 278 (1 – 17, 28) Objective 10: Antiderivatives. CSV P. 361 (1 – 13, 21 – 23, 25 , 26, 28, 30, 32, 34, 36, 37)

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