AP Calculus AB Riemann Sums Name Intro Activity: The Gorilla Problem A gorilla (wearing a parachute) jumped off of the top of a building. We were able to record the velocity of the gorilla with respect to time twice each second. The data is shown below. Note that he touched the ground just after 5 seconds. Time (sec) Velocity (ft/sec).5 1 1.5 2 2.5 3 3.5 4 4.5 5 5 7 8 11 11.5 12 13 15.5 18 19 1) Approximate how far the gorilla fell during each half second interval and fill in the table below. Time (sec) Distance (ft) -.5.5-1 1-1.5 1.5-2 2-2.5 2.5-3 3-3.5 3.5-4 4-4.5 4.5-5 2) Approximate the total distance the gorilla fell from the time he jumped off of the building until the time he landed on the ground. 3) Is your approximation an overestimate, an underestimate, or is it the exact value? How can you tell?
Riemann Sums Notes 1. A car is traveling so that its speed is never decreasing during a 12-second interval. The speed at various moments in time is listed in the table below. A graph of the data is also provided. Time in Seconds 2 4 6 8 1 12 Speed in ft/sec 5 12 21 32 45 54 65 (a) Using 3 subintervals of equal length, estimate the distance traveled by the car during the 12 seconds by finding the areas of the three rectangles drawn at the heights of the left endpoints. This is called a left Riemann sum. (b) Using 3 subintervals of equal length, estimate the distance traveled by the car during the 12 seconds by finding the areas of the three rectangles drawn at the heights of the right endpoints. This is called a right Riemann sum. (c) Using 3 subintervals of equal length, estimate the distance traveled by the car during the 12 seconds by finding the areas of three rectangles drawn at the heights of the midpoint of each interval. This is called a midpoint Riemann sum. Time in Seconds 2 4 6 8 1 12 Speed in ft/sec 5 12 21 32 45 54 65
(d) Using 3 subintervals of equal length, estimate the distance traveled by the car during the 12 seconds by finding the areas of three trapezoids drawn over in interval. This is called a trapezoidal Riemann sum. Time in Seconds 2 4 6 8 1 12 Speed in ft/sec 5 12 21 32 45 54 65 (e) Which of your answers is an under approximation? (f) Which of your answers is an over approximation? (g) Calculate your average to parts a and b. (h) Which answer is the best?
2. Given the function ( ) 2 the x-axis on [, 2] by using: f x = x + 1, estimate the area bounded by the graph of the curve and (a) a left Riemann sum with n = 4 equal subintervals. Is this an under-estimate or over-estimate? Explain. (b) a right Riemann sum with n = 4 equal subintervals. Is this an under-estimate or over-estimate? Explain. (c) a trapezoidal Riemann sum with n = 4 equal subintervals. Is this an under-estimate or overestimate? Explain. (d) a midpoint Riemann sum with n = 2 equal subintervals.
2 (e) The exact answer for #2 is represented by the integral ( x + 1) 2 dx. Calculate this on the calculator using the F3 menu on the HOME screen. Which Riemann Sums gave the most accurate approximation? 3. In order to determine the average temperature for a day, a meteorologist decides to record the temperature at eight times during the day as indicated by the data in the table. Use a Riemann sum with the intervals indicated in the table to estimate the average temperature during this day. Time 12AM-5AM 5AM-7AM 7AM-9AM 9AM-1PM 1PM-4PM 4PM-7PM 7PM-9PM 9PM-12AM Temp 42 o 57 o 72 o 84 o 89 o 75 o 66 o 52 o
Riemann Sums Homework: 1. A car is travelling with velocity v(t) in feet per second. Selected values for v(t) for t 8 seconds are shown in the table. Time (sec) 2 4 6 8 Velocity (ft/sec) 1 25 4 5 7 a. Produce a graph of the velocity against time. b. Use a left Reimman sum with the four intervals indicated in the table to estimate the distance the car traveled for t 8 seconds. Is this estimate too big or too small? Explain why. c. Use a trapezoidal Reimman sum with the four intervals indicated in the table to estimate the distance the car traveled for t 8 seconds. d. Use a midpoint Reimman sum with two subintervals of equal length to estimate the distance the car traveled for t 8 seconds.
2. Oil is leaking out of a tank. The rate of flow is measured every two hours for a 12-hour period, and the data is listed in the table below. Time (hr) 2 4 6 8 1 12 Rate (gal/hr) 4 38 36 3 26 18 8 (a) Estimate the number of gallons of oil that have leaked out of the tank during the 12-hour period by finding a left Riemann sum with three equal subintervals. (b) Estimate the number of gallons of oil that have leaked out of the tank during the 12-hour period by finding a right Riemann sum with three equal subintervals. (c) Estimate the number of gallons of oil that have leaked out of the tank during the 12-hour period by finding a midpoint Riemann sum with three equal subintervals. (d) Estimate the number of gallons of oil that have leaked out of the tank during the 12-hour period by finding a trapezoidal Riemann sum with three equal subintervals. f( x) 4 2 = x. Consider the region bounded by the graph of f, the x-axis and the line x=2. Divide 3. Let the interval [,2] into 4 equal subintervals. (a) Would a left Riemann sum be an under-estimate or an over-estimate of the area between f(x) and the x-axis on the interval [,2]? Explain your answer. (b) Would a right Riemann sum be an under-estimate or an over-estimate of the area between f(x) and the x-axis on the interval [,2]? Explain your answer. (c) Would a trapezoid Riemann sum be an under-estimate or an over-estimate of the area between f(x) and the x-axis on the interval [,2]? Explain your answer.
4. Bernard Riemann s odometer broke. Being the descendant of the mathematician George Friedrich Bernhard Riemann (1826-1866), he knows how to do a Reimann sum! The table contains the data he collects during a road trip. Each speedometer reading (in miles per hour) is made at some point during the time interval recorded. What is his average speed over the three hour trip in miles per hour? (Hint: Convert the time to hours.) Time Interval (minutes) 15 25 3 15 2 35 4 Speed (mph) 3 45 5 25 45 55 5 5. The rate at which water is being pumped into a tank is given by the function R( t ). A table of selected values of R( t ), for the time interval t 2 minutes, is shown below. t (min.) 4 9 17 2 R( t ) (gal/min) 25 28 33 42 46 (a) Use data from the table and four subintervals to find a left Riemann sum to approximate the value 2 of R ( t ) dt. (b) Use data from the table and four subintervals to find a right Riemann sum to approximate the value 2 of R ( t ) dt. 2 (c) Using correct units, explain the meaning of R ( t ) dt in the context of the situation.
6. A student is speeding down Route 11 in his fancy red Porsche when his radar system warns him of an obstacle 4 feet ahead. He immediately applies the brakes, starts to slow down and spots a skunk in the road directly ahead of him. Suppose that the black box in the Porsche records the car s speed every two seconds producing the following table. Assume that the speed decreases throughout the 1 seconds it takes to stop, although not necessarily at a uniform rate. Time since brakes applied (sec) 2 4 6 8 1 Speed (ft/sec) 1 8 5 25 1 a. Using the information in this table, what is your best estimate of the total distance that the student s car traveled before coming to rest? (You may either average the left and right estimates or use the trapezoid method.) b. Which statement below can you justify from the information given in the story and data table? i) The car stopped before getting to the skunk. ii) The black box data is inconclusive. The skunk may or may not have been hit. iii) The unfortunate skunk was hit by the car.
AP style problems 1. The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by a twice-differentiable and strictly increasing function R of time t. The graph of R and a table of selected values of R(t), for the time interval t 9 minutes, are shown. (a) Use data from the table to find an approximation for R '( 45). Show the computations that lead to your answer. Indicate units of measure. t (minutes) 2 3 3 4 4 5 55 7 65 9 7 R(t) (gallons per minute (b) The rate of fuel consumption is increasing fastest at time t=45 minutes. What is the value of ( ) R '' 45? Explain your reasoning. (c) Approximate the value of 9 R() t dt using a left Riemann sum with the five subintervals indicated by the data in the table. Is this numerical approximation less than the value of 9 R() t dt? Explain. (d) For < b 9 minutes, explain the meaning of plane. Explain the meaning of of measure in both answers. b b R() t dt in terms of fuel consumption for the 1 R() t dt b in terms of fuel consumption for the plane. Indicate units
2. The graph of the velocity v(t), in ft/sec, of a car travelling on a straight road, for t 5 A table of values for v(t), at 5 second intervals of time t, is shown as well., is shown. (a) During what intervals of time is the acceleration of the car positive? Give a reason for your answer. (b) Find the average acceleration of the car, in ft/sec 2, over the interval t 5. (c) Find one approximation for the acceleration of the car, in ft/sec 2, at t=4. Show the computations that you used to arrive at your answer. (d) Approximate 5 v() t dt with a Riemann sum, using the midpoints of five subintervals of equal length. Using correct units, explain the meaning of this integral. (e) Based on the values in the table, what is the fewest number of times that the acceleration of the car could equal zero on the interval < t < 5? Justify your answer.
3. A test plane flies in a straight line with positive velocity v(t), in miles per minute at time t minutes, where v(t) is a differentiable function. Selected values of v(t) for t 4 are shown in the table. (a) Use a midpoint Riemann sum with four subintervals of equal length and values from the 4 table to approximate v() t dt. Show the computations that lead to your answer. Using correct units, explain the meaning of 4 v() t dt in terms of the plane s flight. t (minutes) v(t) (miles per min) 5 1 15 2 25 3 35 4 7. 9.2 9.5 7. 4.5 2.4 2.4 4.3 7.3 (b) Find the average acceleration of the car, in ft/sec 2, over the interval 5 t 25. (c) Find an approximation for the acceleration of the car, in ft/sec 2, at t=18. Show the computations that you used to arrive at your answer. (d) Based on the values in the table, what is the smallest number of instances at which the acceleration of the plane could equal zero on the open interval < t < 4? Justify your answer. t 7t = 6 + cos + 3sin, is used to model the velocity of the 1 4 plane, in miles per minute, for t 4. According to this model, what is the acceleration of the plane at t=23? Indicate units of measure. (e) The function f, defined by f ( t) (f) According to the model f, given in part (e), what is the average velocity of the plane, in miles per minute, over the time interval t 4?