t over the interval 0 t

Size: px

Start display at page:

Download "t over the interval 0 t"

Morris Potter
5 years ago
Views:

1 Student Name: Period: Applications of Riemann Sums (AP Style) 1. A plane has just crashed six minutes after takeoff. There may be survivors, but you must locate the plane quickly without an extensive search. It was traveling due west and the pilot radioed its speed every minute from takeoff to the time it crashed. These data are given in the table below. Determine a reasonable range of distances from the airport to search for the plane. minutes speed (mph) Rocket A has positive velocity vt () after being launched from an initial height of 0 feet at time t 0 seconds. The velocity of the rocket is recorded for selected values of t over the interval 0 t 80 seconds, as shown in the table below. Use a midpoint Riemann sum with 4 subintervals of equal length to approximate the area under the curve. Write a sentence explaining what the area represents. t (seconds) v( (feet per second)

2 3. The temperature of water in a tub at time t is modeled by a strictly increasing, twicedifferentiable function W, where W ( is measured in degrees Fahrenheit and t is measured in minutes. At time t 0, the temperature of the water is 55 F. The water is heated for 30 minutes, beginning at time t 0. Values of W ( at selected times t for the first 20 minutes are given in the table below. Use the data in the table to estimate W '(12). Show the calculations that lead to your answer. Using correct units, interpret the meaning of your answer in the context of this problem. t (minutes) W ( (degrees Fahrenhei A stadium is filling with people from noon until game time at 3:00 p.m. The table below shows the rate of people entering the stadium (measured in people per minute) at particular times. Time of day Noon 1:00 1:30 2:00 2:20 2:40 2:50 2:55 3:00 Rate (people/min) Estimate the total number of people attending the game using a trapezoidal approximation with four partitions, starting at noon, 1:30, 2:20, and 2: Let vt () be the velocity of a particle moving on the x-axis, where v is measured in ft/sec. The starting position of the particle is x = 0 when t = 0. a. When is the particle moving right? Justify your answer. v( t

3 b. When is the particle moving left? Justify your answer. c. When is the particle at its maximum position (farthest righ? Justify your answer d. Write the equation for vt (). e. Determine the equation for the position of the particle, xt (). f. Determine the position of the particle at t = 40 and at t = 80. Include units in your answer. g. Compute the right hand sum for the interval [0, 40] with four equal subintervals. Write a sentence interpreting the meaning of the sum including units. h. Compute the left hand sum for the interval [40, 80] with four equal subintervals. Write a sentence interpreting the meaning of the sum including units

4 6. Concert tickets went on sale at noon ( t 0) and were sold within 9 hours. The number of people waiting in line to purchase tickets at time t is modeled by the twice differentiable function L for 0 t 9. Values of Lt () at various times t are shown in the table below. t (hours) L( (people) a. Use the data in the table to estimate the rate at which the number of people waiting in line was changing at 5:30 P.M. ( t 5.5). Show the computations that lead to your answer. Indicate units of measure. b. Use a trapezoidal sum with three subintervals to estimate the number of people waiting in line during the first 4 hours the tickets were on sale. c. For 0 9 t, what is the fewest number of times at which '( ) L t must equal 0? Give a reason for your answer.

5 8. The twice-differentiable function W models the volume of water in a reservoir at time t, where W ( is measured in gigaliters (GL) and t is measured in days. The table below gives values of W '( sampled at various times during the time interval 0 t 30 days. At time t 30, the reservoir contains 125 gigaliters of water. t (days) W '( (GL per day) a. Use the tangent line approximation to W at time t 30 to predict the volume of water W (, in gigaliters, in the reservoir at t 32. Show the computations that lead to your answer. b. Find the left Riemann sum, with the subintervals indicated by the table. c. The above sum will approximate the change in the volume of water in gigaliters from day zero to thirty. How much is in the reservoir at t 0? d. Explain why there must be at least one time t, other than t 10, such that W '( 0.7GL / day.

AP CALCULUS BC 2008 SCORING GUIDELINES. Question 2

AP CALCULUS BC 2008 SCORING GUIDELINES Question 2 t (hours) 0 1 3 4 7 8 9 Lt ()(people) 120 156 176 126 150 80 0 Concert tickets went on sale at noon ( t = 0) and were sold out within 9 hours. The number