1-Consider the following.
(a) Use six rectangles to find estimates of each type for the area under the given graph of f from x = 0 to x = 24.
(i) Sample points are le
ft
endpoints.
L6
=
(ii) Sample points are right endpoints.
R6
=
(iii) Sample points are midpoints.
M6 =
(b) Is L6 an
underestimate
or
overestimate
of the true area?
overestimate
underestimate
(c) Is R6 an underestimate or overestimate of the true area?
overestimate
underestimate
(d) Which of the numbers gives the best estimate?
R6
M6
L6
2-Do the following.
Estimate the area under the graph of f(x) = 2 cos(x) from x = 0 to x = π/2. (Round the answer to four decimal places.)
(a) Use four approximating rectangles and right endpoints.
R4 =
Is your estimate an underestimate or an overestimate?
underestimate
overestimate
(b) Use four approximating rectangles and left endpoints.
L4 =
Is your estimate an underestimate or an overestimate?
underestimate
overestimate
3-Speedometer readings for a motorcycle at 12-second intervals are given in the table.
t (s)
0
12
24
36
48
60
v (ft/s)
32
27
25
22
24
27
(a) Estimate the distance traveled by the motorcycle during this time period using the velocities at the beginning of the time intervals.
ft
(b) Give another estimate using the velocities at the end of the time periods.
ft
(c) Are your estimates in parts (a) and (b) upper and lower estimates? Explain.
(a) is a lower estimate and (b) is an upper estimate since v is an increasing function of t.
(b) is a lower estimate and (a) is an upper estimate since v is a decreasing function of t.
(a) and (b) are neither lower nor upper estimates since v is neither an increasing nor decreasing function of t.
4-Oil leaked from a tank at a rate of r(t) liters per hour. The rate decreased as time passed, and values of the rate at two hour time intervals are shown in the table. Find lower and upper estimates for the total amount of oil that leaked out.
V
≈
L (lower estimate)
V
≈
L (upper estimate)
t (h)
0
2
4
6
8
10
r(t) (L/h)
8.6
7.9
6.8
6.4
5.7
5.1
5-The velocity graph of a braking car is shown. Use it to estimate the distance, d traveled by the car while the brakes are applied. (Use M6 to get the most precise estimate.)
d = ft
a = 30
6-The area A of the region S that lies under the graph of the continuous function is the limit of the sum of the areas of approximating rectangles.
Use this definition to find an expression for the area under the graph of f as a limit. Do not evaluate the limit.
7-The area A of the region S that lies under the graph of the continuous function is the limit of the sum of the areas of approximating rectangles:
(a) Use this definition to find an expression for the area under the curve y = x3 from 0 to 3 as a limit.
(b) Use the following formula for the sum of the cubes of the first integers to evaluate the limit in part (a).
8-If f is continuous on [a, b], then “the definite integral from a to b of f” is the area bounded by the graph of f, the x-axis, and the lines x = a, x = b.
True
False
9-Let f(t) be the growth rate of a population at time t. Then the definite integral from t1 to t2 of f(t) is
the change in growth rate between t1 and t2.
the total population at t2.
the change in the size of the population between t1 and t2.
none of the above.
1-Consider the following.
(a) Use six rectangles to find estimates of each type for the area under the given graph of f from x = 0 to x = 24.
(i) Sample points are le
ft
endpoints.
L6
=
(ii) Sample points are right endpoints.
R6
=
(iii) Sample points are midpoints.
M6 =
(b) Is L6 an
underestimate
or
overestimate
of the true area?
overestimate
underestimate
(c) Is R6 an underestimate or overestimate of the true area?
overestimate
underestimate
(d) Which of the numbers gives the best estimate?
R6
M6
L6
2-Do the following.
Estimate the area under the graph of f(x) = 2 cos(x) from x = 0 to x = π/2. (Round the answer to four decimal places.)
(a) Use four approximating rectangles and right endpoints.
R4 =
Is your estimate an underestimate or an overestimate?
underestimate
overestimate
(b) Use four approximating rectangles and left endpoints.
L4 =
Is your estimate an underestimate or an overestimate?
underestimate
overestimate
3-Speedometer readings for a motorcycle at 12-second intervals are given in the table.
t (s)
0
12
24
36
48
60
v (ft/s)
32
27
25
22
24
27
(a) Estimate the distance traveled by the motorcycle during this time period using the velocities at the beginning of the time intervals.
ft
(b) Give another estimate using the velocities at the end of the time periods.
ft
(c) Are your estimates in parts (a) and (b) upper and lower estimates? Explain.
(a) is a lower estimate and (b) is an upper estimate since v is an increasing function of t.
(b) is a lower estimate and (a) is an upper estimate since v is a decreasing function of t.
(a) and (b) are neither lower nor upper estimates since v is neither an increasing nor decreasing function of t.
4-Oil leaked from a tank at a rate of r(t) liters per hour. The rate decreased as time passed, and values of the rate at two hour time intervals are shown in the table. Find lower and upper estimates for the total amount of oil that leaked out.
V
≈
L (lower estimate)
V
≈
L (upper estimate)
t (h)
0
2
4
6
8
10
r(t) (L/h)
8.6
7.9
6.8
6.4
5.7
5.1
5-The velocity graph of a braking car is shown. Use it to estimate the distance, d traveled by the car while the brakes are applied. (Use M6 to get the most precise estimate.)
d = ft
a = 30
6-The area A of the region S that lies under the graph of the continuous function is the limit of the sum of the areas of approximating rectangles.
Use this definition to find an expression for the area under the graph of f as a limit. Do not evaluate the limit.
7-The area A of the region S that lies under the graph of the continuous function is the limit of the sum of the areas of approximating rectangles:
(a) Use this definition to find an expression for the area under the curve y = x3 from 0 to 3 as a limit.
(b) Use the following formula for the sum of the cubes of the first integers to evaluate the limit in part (a).
8-If f is continuous on [a, b], then “the definite integral from a to b of f” is the area bounded by the graph of f, the x-axis, and the lines x = a, x = b.
True
False
9-Let f(t) be the growth rate of a population at time t. Then the definite integral from t1 to t2 of f(t) is
the change in growth rate between t1 and t2.
the total population at t2.
the change in the size of the population between t1 and t2.
none of the above.
