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AP Calculus Problem Set 48

Upon completion, circle one of the following to assess your current understanding:

Completely understand / Mostly understand / Sort of understand / Don’t understand

1. The graph of the function f is shown below. Sketch the graph of each approximation method.

a) Approximate using a right Riemann b) Approximate using a left Riemann

sum with 3 subintervals of equal length. sum with 3 subintervals of equal length.

c) Approximate using a midpoint Riemann d) Approximate using a trapezoidal

sum with 3 subintervals of equal length. sum with 3 subintervals of equal length.

2. The graph of the function f is shown below. Sketch the graph of each approximation method.

a) Approximate using a right Riemann b) Approximate using a left Riemann

sum with 4 subintervals of equal length. sum with 4 subintervals of equal length.

c) Approximate using a midpoint Riemann d) Approximate using a trapezoidal

sum with 4 subintervals of equal length. sum with 4 subintervals of equal length.

3. A post office delivered letters between 7 a.m. (t = 0) and 5 p.m. (t = 10). The number of letters delivered t hours after noon is modeled by a differentiable function L for . for all t. Values of , in thousands of letters, at various time t are shown in the table below.

t
(hours) / 0 / 3 / 7 / 8 / 10

(thousands of letters) / 0 / 1 / 4 / 5 / 9

a) Use a trapezoidal sum with the four subintervals given by the table to approximate

Does this approximation overestimate or underestimate the actual value of ?

Give a reason for your answer.

b) Use a left Riemann sum with the four subintervals given by the table to approximate

Using correct units, explain the meaning of , in terms of the number of letters delivered.

4. During the time interval hours, snow accumulates on a driveway at the rate cubic feet per hour. The table below gives values of , a strictly monotonic differentiable function, for selected values of t.

t /
0 / 0
4 / 12
8 / 20
12 / 25
16 / 30
20 / 50
24 / 55

5. The velocity of a particle moving along the x-axis is modeled by a differentiable function, v, in meters per second. Selected values of are given in the table below.

t
(seconds) / 0 / 8 / 10 / 15 / 27 / 47

(meters per second) / 2 / 4 / -8 / -6 / -1 / 3

a) Using correct units explain the meaning of in the context of this problem.

b) Use a trapezoidal sum with the three subintervals indicated by the table to approximate

c) Using correct units explain the meaning of in the context of this problem. Use a right Riemann sum with the three subintervals indicated by the table to approximate

6. Concert tickets went on sale at noon (t = 0) and were sold out within 9 hours. The number of people waiting in line to purchase tickets at time t is modeled by a twice-differentiable strictly monotonic function C for . Values of are various times t are shown in the table below.

t (hours) / 0 / 1 / 3 / 4 / 7 / 8 / 9
(people) / 190 / 180 / 140 / 120 / 100 / 80 / 0

a) Using correct units, explain the meaning of in the context of this problem.

b) Use a left Riemann sum with five subintervals to estimate . Does this approximation overestimate or underestimate the actual value of . Give a reason for your answer.

7. A spherical hot air balloon expands as air inside the balloon is heated. The table below gives selected values of the rate of change of the radius of the balloon over the time interval .

t (minutes) / 0 / 3 / 6 / 9 / 12 / 15 / 18
(feet per minute) / 7 / 5 / 4 / 3 / 2 / 2 / 1

a) Using correct units, explain the meaning of in the context of this problem.

b) Use a right Riemann sum with six subintervals of equal width to approximate .

c) Using correct units, explain the meaning of in the context of this problem.

d) Use a midpoint Riemann sum with three subintervals of equal width to approximate .

8. The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by a twice-differentiable and strictly increasing function A of time t. for all t. A table of selected values of , for the time interval , is shown below.

t
(minutes) /
(gallons per minute)
0 / 10
20 / 50
30 / 60
70 / 80
90 / 85
100 / 87

a) Using correct units, explain the meaning of in the context of the problem.

b) Approximate the value of using a trapezoidal sum with the five subintervals indicated by the data in the table. Is this approximation less than the value of ? Justify your answer

c) Use a right Riemann sum with the five subintervals indicated by the data to approximate Is this approximation greater than the value of ? Justify your answer.

9. If a trapezoidal sum overapproximates , and a right Riemann sum underapproximates which of the following could be the graph of ?

(A) (B)(C)

(D)(E)

10. The graph of the function f is shown below. Of the following, which has the least value?