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2 Calculus AB Assignment Practice Using Definite Integrals 1 Copyright © 2021 Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the permission of Texas Instruments Incorporated. Copyright © 2011 Texas Instruments Incorporated. 1. Power companies typically bill customers based on the number of kilowatt-hours used during a single billing period. A kilowatt is a measure of how much power (energy) a customer is using, while a kilowatt-hour is one kilowatt of power being used for one hour. For constant power use, the number of kilowatt-hours used is calculated by kilowatt-hours = kilowatts • time (in hours). Thus, if customers use 5 kilowatts for 30 minutes, they’ll have used 5 kilowatts • 1 hours = 2. 5 kilowatt-hours. Suppose the power use of a customer over a 30 -day period is given by the continuous function P = f ( t ) , where P is kilowatts, t is time in hours, and t = 0 corresponds to the beginning of the 30 day period. A. Approximate, with a Riemann sum, the total number of kilowatt-hours used by the customer in the 30 -day period, and explain why your Riemann sum is an approximation of the desired property. Total number of Kilowatt-hours used by customers = 30 day's = 30 X 24 = 720 hours B. Derive an expression representing the total number of kilowatt-hours used by the customer in the 30 -day period, and explain your reasoning. (This expression should not be an approximation.) total number of kilowatt-hours used by customer in = 30 days = 30x 24 = 720 hours According to the left Riemann sum, we get P(0) + P(1) + P(2) + P(3) +...+ P(719) = Total kilo watt hours used by the customer for 30 days C. Consider the following table of data for the function f ( t ) .

72 Calculus AB Assignment Practice Using Definite Integrals 2 Copyright © 2021 Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the permission of Texas Instruments Incorporated. Copyright © 2011 Texas Instruments Incorporated. t f ( t ) 0 2. 3 1 2. 5 2 2. 1 3 3. 9 4 3. 6 5 5. 5 6 4. 5 7 5. 6 8 1. 2 9 1. 0 10 1. 8 Recall that f ( t ) represents the number of kilowatts being used by a customer at time t hours from the beginning of the billing period. Estimate the number of kilowatt-hours the customer uses in this 10 -hour period, and explain your method. Number of kilowatt - hours the customer uses in 10 hours period. p(t) = 2.3+2.5+ 2.1+3.9+3.6+5.5+4.5+5.6+1.2+1.4 = 32-2 kilo-watt/hour. 2. Two separate rabbit populations are observed for 80 weeks, starting at the same time and with the same initial populations. The growth rates of two rabbit populations are modeled as follows, where t = 0 corresponds to the beginning of the observation period: r 1 ( t ) = 4 sin ( 2 π t ) + 0. 1 t + 1 , where r 1 is rabbits per week, and t is time in weeks, r 2 ( t ) = t 1/2 , where r 2 is rabbits per week, and t is time in weeks. Below is a graph of the curves representing the rates of growth of the two populations over the observation period:

Calculus AB Assignment Practice Using Definite Integrals 4 Copyright © 2021 Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the permission of Texas Instruments Incorporated. Copyright © 2011 Texas Instruments Incorporated. B. What’s the physical significance of the area between the two curves from time t = 0 until the first time where the two rates are the same? What does the area represent? The physical significance of that area between the 2 curves is the difference between both rabbits population. That area represents that difference between t=0 and t=31.7 C. Suppose you want to find the first time (call it T ) after the beginning of the observation period at which the two rabbit populations have identical populations. Write an equation to solve for the unknown variable T . For knowing that you first have to know that the area below each curve represents the population of each curve (r1 and r2). So, to know that area you have to use integrals: For r1 For r2 The as you wish to know the time t for which both populations are the same: D. Simplify your equation from part C until you can use your calculator on it. Then use your calculator to solve this equation for T .

4 Calculus AB Assignment Practice Using Definite Integrals 5 Copyright © 2021 Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the permission of Texas Instruments Incorporated. Copyright © 2011 Texas Instruments Incorporated. I think that the equation above is simple enough for any graping and scientific calculator to solve it: To solve this you can: 1) Graph both sides of the equation and see where the curves intercept each other, or, 2) Ask the equation solver in your calculator to solve it. I did the second one: 3. A bucket that weighs 3 lb and a rope of negligible weight are used to draw water from a well that’s 60 feet deep. Suppose the bucket starts with 37 lb of water and is pulled up by a rope at 2 ft/sec, while water leaks out of the bucket at a rate of 1 lb/sec. A. How long does it take for the bucket to get to the top of the well? Write an equation that expresses the total weight of the bucket as a function of time, as the time varies from 0 until the time the buckets gets to the top. 60 feet at 2ft/sec = 30 sec W(f) = 40 – t/4 0≤t≤30

Calculus AB Assignment Practice Using Definite Integrals 7 Copyright © 2021 Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the permission of Texas Instruments Incorporated. Copyright © 2011 Texas Instruments Incorporated. B. Recall that work equals force times distance. Calculate the work done in lifting the bucket to the top of the well, keeping in mind that here force is equal to weight. Dw = w(t) (dx/dt) dt =(40- t/4) x 2dt Dw= (80 – t/2)dt integrate both the side W= 80(30) – (30)^2 / 4 W = 2400 – 225 W= 2175 4. A certain computer algorithm used to solve very complicated differential equations uses an iterative method. That is, the algorithm solves the problem the first time very approximately, and then uses that first solution to help it solve the problem a second time just a little bit better, and then uses that second solution to help it solve the problem a third time just a little bit better, and so on. Unfortunately, each iteration (each new problem solved by using the previous solution) takes a progressively longer amount of time. In fact, the amount of time it takes to process the k -th iteration is given by T ( k ) = 1. 2 k + 1 seconds. A. Use a definite integral to approximate the time (in hours) it will take the computer algorithm to run through 60 iterations. (Note that T ( k ) is the amount of time it takes to process just the k -th iteration.) Explain your reasoning.

Calculus AB Assignment Practice Using Definite Integrals 8 Copyright © 2021 Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the permission of Texas Instruments Incorporated. Copyright © 2011 Texas Instruments Incorporated. B. The maximum error in the computer’s solution after k iterations is given by Error = 2 k – 2 . Approximately how long (in hours) will it take the computer to process enough iterations to reduce the maximum error to below 0. 0001 ?