Velocity and Acceleration A car travels in a straight line for 1 hour. Its velocity v in miles per hour at six-minute intervals is shown in the table. t (hours ) 0 0.1 0.2 0.3 0.4 0.5 v (mi/h ) 0 10 20 40 60 50 t (hours ) 0.6 0.7 0.8 0.9 1.0 v (mi/h ) 40 35 40 50 65 (a) Produce a reasonable graph of the velocity function v by graphing these points and connecting them with a smooth curve. (b) Find the open intervals over which the acceleration a is positive. (c) Find the average acceleration of the car (in miles per hour per hour) over the interval [0, 0.4]. (d) What does the integral ∫ 0 1 v ( t ) d t signify? Approximate this integral using the Midpoint Rule with five subintervals. (e) Approximate the acceleration at t = 0.8
Velocity and Acceleration A car travels in a straight line for 1 hour. Its velocity v in miles per hour at six-minute intervals is shown in the table. t (hours ) 0 0.1 0.2 0.3 0.4 0.5 v (mi/h ) 0 10 20 40 60 50 t (hours ) 0.6 0.7 0.8 0.9 1.0 v (mi/h ) 40 35 40 50 65 (a) Produce a reasonable graph of the velocity function v by graphing these points and connecting them with a smooth curve. (b) Find the open intervals over which the acceleration a is positive. (c) Find the average acceleration of the car (in miles per hour per hour) over the interval [0, 0.4]. (d) What does the integral ∫ 0 1 v ( t ) d t signify? Approximate this integral using the Midpoint Rule with five subintervals. (e) Approximate the acceleration at t = 0.8
Solution Summary: The author illustrates the velocity function v by graphing the provided points (hours) and connecting them with a smooth curve.
Velocity and Acceleration A car travels in a straight line for 1 hour. Its velocity v in miles per hour at six-minute intervals is shown in the table.
t (hours)
0
0.1
0.2
0.3
0.4
0.5
v (mi/h)
0
10
20
40
60
50
t (hours)
0.6
0.7
0.8
0.9
1.0
v (mi/h)
40
35
40
50
65
(a) Produce a reasonable graph of the velocity function v by graphing these points and connecting them with a smooth curve.
(b) Find the open intervals over which the acceleration a is positive.
(c) Find the average acceleration of the car (in miles per hour per hour) over the interval [0, 0.4].
(d) What does the integral
∫
0
1
v
(
t
)
d
t
signify? Approximate this integral using the Midpoint Rule with five subintervals.
(e) Approximate the acceleration at
t
=
0.8
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
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