The table shows speedometer readings at 10-second intervals during a 1-minute period for a car racing at the Daytona International Speedway in Florida.

Question

Time (s)Velocity (mi/h)
0	183.9
10	167.0
20	107.6
30	98.8
40	125.5
50	177.1
60	176.6

(a) Estimate the distance (in mi) the race car traveled during this time period using the velocities at the beginning of the time intervals. (Round your answer to three decimal places.)

(b) Give another estimate (in mi) using the velocities at the end of the time periods. (Round your answer to three decimal places.)

(c)

Are your estimates in parts (a) and (b) upper and lower estimates? Explain.

1-The velocity is decreasing on the given interval, so the estimate in part (a) is a lower estimate and the estimate in part (b) is an upper estimate.

2-The velocity is increasing on the given interval, so the estimate in part (a) is an upper estimate and the estimate in part (b) is a lower estimate. 

3-The velocity is neither increasing nor decreasing on the given interval, so the estimates in parts (a) and (b) are neither upper nor lower estimates.

4-The velocity is decreasing on the given interval, so the estimate in part (a) is an upper estimate and the estimate in part (b) is a lower estimate.

5-The velocity is increasing on the given interval, so the estimate in part (a) is a lower estimate and the estimate in part (b) is an upper estimate.

Answer 1

This is a Riemann sum approximation problem. Use areas of rectangles to approximate the area under the curve. The width of each rectangle is 10 sec = 1/36 hrs and the lengths are the velocities.

a) Find the left hand approximation:

L_H = 1/36 ( 183.9 + 167 + 107.6 + 98.8 + 125.5 + 177.1) ≈ 23.886 mi

b) Find the right hand approximation:

R_H = 1/36 (167 + 107.6 + 98.8 +125.5 +177.1 + 176.6) ≈ 23.683 mi

c) If a function is increasing, then L_H is an underapproximation or lower estimate and R_H is an over

approximation or upper estimate.

If a function is decreasing, then the opposite is true. L_H = upper and R_H = lower.

1. This means that choices 1 and 2 are wrong because they are impossible.
2. Option 3 is incorrect because, over the 1 minute interval, the velocity decreases over [0,30) and then increases over (30, 60].
3. The most correct option ( between choices 4 and 5) is option 4 because L_H ≈ 23.886 mi is greater than R_H ≈ 23.683 mi, so L_H must be the upper estimate.

Hope this helps!