Review of Reimann Sums/Average Value

AP Calculus ABReview of Riemann Sums/Average Value

1973 #42

Calculate the approximate area of the shaded region in the figure by the trapezoidal rule, using divisions at and .

a) b) c) d) e)

1993 #36 (calculator question)

If the definite integral is first approximated by using 2 inscribed rectangles of equal width and then approximated by using the trapezoidal rule with n = 2, the difference between the two approximations is

a) 53.60b) 30.51c) 27.80d) 26.80e) 12.78

1997 #24

The expression is a Riemann sum approximation for:

a) b) c) d) e)

1997 #89 (calculator question)

A table of values for a continuous function f is shown above. If four equal subintervals of [0, 2] are used, which of the following is the trapezoidal approximation of ?

a) 8b) 12c) 16d) 24e) 32

1998 #85 (calculator question)

The function f is continuous on the closed interval [2, 8] and has values that are given in the table above. Using the subintervals [2, 5], [5, 7], and [7, 8], what is the trapezoidal approximation of ?

a) 110b) 130c) 160d) 190e) 210

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1996 #3 (calculator question)

The rate of consumption of cola in the United States is given by S(t) = Cekt, where S is measured in billions of gallons per year and t is measured in years from the beginning of 1980.

a)The consumption rate doubles every 5 years and the consumption rate at the beginning of 1980 was 6 billion gallons per year. Find C and k.

b)Find the average rate of consumption of cola over the 10-year time period beginning January 1, 1983. Indicate units of measure.

c)Use the trapezoidal rule with four equal subdivisions to estimate

d)Using correct units, explain the meaning of in terms of cola consumption.

The temperature, in degrees Celsius (˚C), of the water in a pond is a differentiable function W of time t. The table to the right shows the water temperature as recorded every 3 days over a 15-day period.

1. Use data from the table to find an approximation for W '(12). Show the computations that lead to your answer. Indicate units of measure.
2. Approximate the average temperature, in degrees Celsius, of the water over the time interval

0 ≤ t ≤ 15 days by using a trapezoidal approximation with subintervals of length Δt = 3 days.

1. A student proposes the function P, given by , as a model for the temperature of the water in the pond at time t, where t is measured in days and P(t) is measured in degrees Celsius. Find P'(12). Using appropriate units, explain the meaning of your answer in terms of water temperature.
2. Use the function P defined in part c to find the average value, in degrees Celsius, of P(t) over the time interval 0 ≤ t ≤ 15 days.

2003 #3 (calculator question)

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 0 ≤ t ≤ 90 minutes, are shown above.

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.

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 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? Explain your reasoning.

d)For 0 < b ≤ 90 minutes, explain the meaning of in terms of fuel consumption for the plane. Explain the meaning of in terms of fuel consumption for the plane. Indicate units of measure in both answers.

2005 #3 (calculator question)

A metal wire of length 8 centimeters (cm) is heated at one end. The table above gives selected values of the temperature T(x), in degrees Celsius (˚C), of the wire x cm from the heated end. The function t is decreasing and twice differentiable.

a)EstimateT'(7). Show the work that leads to your answer. Indicate units of measure.

b)Write an integral expression in terms of T(x) for the average temperature of the wire. Estimate the average temperature of the wire using a trapezoidal sum with the four subintervals indicated by the data in the table. Indicate units of measure.

c)Find , and indicate units of measure. Explain the meaning of in terms of the temperature of the wire.

d)Are the data in the table consistent with the assertion thatT''(x) > 0 for every x in the interval 0 < x < 8? Explain your answer.

2006 #4

Rocket A has positive velocity v(t) after being launched upward 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 above.

1. Find the average acceleration of rocket A over the time interval 0 ≤ t ≤ 80 seconds. Indicate units of measure.
2. Using correct units, explain the meaning of in terms of the rocket's flight. Use a midpoint Riemann sum with 3 subintervals of equal length to approximate .
3. Rocket b is launched upward with an acceleration of feet per second per second. At time t = 0 seconds, the initial height of the rocket is 0 feet, and the initial velocity is 2 feet per second. Which of the two rockets is traveling faster at time t = 80 seconds? Explain your answer.

2007 #5

The volume of a spherical hot air balloon expands as the air inside the balloon is heated. The radius of the balloon, in feet, is modeled by a twice-differentiable function r of time t, where t is measured minutes. For 0 < t < 12, the graph of r is concave down. The table above gives selected values of the rate of change, r '(t), of the radius of the balloon over the time interval 0 ≤ t ≤ 12. The radius of the balloon is 30 feet when t = 5. (Note: the volume of a sphere of radius r is given by )

1. Estimate the radius of the balloon when t = 5.4 using the tangent line approximation at

t = 5. Is your estimate greater than or less than the true value? Give a reason for your answer.

1. Find the rate of change of the volume of the balloon with respect to time when t = 5. Indicate units of measure.
2. Use a right Riemann sum with the five subintervals indicated by the data in the table to approximate . Using correct units, explain the meaning of in terms of the radius of the balloon.
3. Is your approximation in part c greater than or less than ? Give a reason for your answer.