A race car is running practice laps in preparation for an upcoming race. To judge how the car is performing, the crew takes measurements of the car's speed S(t) (in miles per hour, or mph) every minute. The measurements are given in the table below.

t (min) --- S(t) (mph
0 --- 201
1 --- 205
2 --- 208
3 --- 214
4 --- 218
5 --- 212
6 --- 219
7 --- 223
8 --- 220
9 --- 221
10 --- 217
11 --- 218
12 --- 216

A. Use the trapezoid rule with 4 equal subdivisions to approximate the total distance the car traveled (in miles) over the first 12 minutes.

B. Find one approximation for S''(6), including the units. Explain what this quantity means in the context of the problem.

C. What was the car's average speed in mph over the first 12 minutes? If the car needs to have an average speed of 210 mph to qualify for the race, is it currently fast enough to qualify?

A. The total distance the car traveled over the first 12 minutes can be approximated using the trapezoid rule with 4 equal subdivisions as follows:

d = (1/2)(201 + 205 + 208 + 214 + 218 + 212 + 219 + 223 + 220 + 221 + 217 + 218 + 216) * (12/4) = 2,722 miles

B. One approximation for S''(6) is 4.5 mph/min. This quantity represents the rate of change of the car's acceleration at t = 6 minutes.

C. The car's average speed over the first 12 minutes is 214.5 mph. Since this is greater than 210 mph, the car is currently fast enough to qualify for the race.

A. To approximate the total distance using the trapezoid rule with 4 equal subdivisions, we need to calculate the sum of the areas of the trapezoids formed by the data points.

Let's divide the time interval [0,12] into 4 equal subdivisions: [0,3], [3,6], [6,9], [9,12].

For each subdivision, the average speed can be approximated as the average of the two speeds at the endpoints of the subdivision.

Subdivision 1: [0,3]
Average speed = (S(0) + S(3))/2 = (201 + 214)/2 = 207.5 mph
Distance traveled in this subdivision = average speed * (3-0) = 207.5 * 3 = 622.5 miles

Subdivision 2: [3,6]
Average speed = (S(3) + S(6))/2 = (214 + 219)/2 = 216.5 mph
Distance traveled in this subdivision = average speed * (6-3) = 216.5 * 3 = 649.5 miles

Subdivision 3: [6,9]
Average speed = (S(6) + S(9))/2 = (219 + 221)/2 = 220 mph
Distance traveled in this subdivision = average speed * (9-6) = 220 * 3 = 660 miles

Subdivision 4: [9,12]
Average speed = (S(9) + S(12))/2 = (221 + 216)/2 = 218.5 mph
Distance traveled in this subdivision = average speed * (12-9) = 218.5 * 3 = 655.5 miles

Total distance traveled = sum of distances in all subdivisions = 622.5 + 649.5 + 660 + 655.5 = 2587.5 miles

Therefore, the car traveled approximately 2587.5 miles over the first 12 minutes.

B. To find an approximation for S''(6), we can use the central difference approximation.

S''(6) = (S'(7) - S'(5))/(7-5), where S'(t) represents the derivative of S(t).

Using the given data, S'(7) = (S(8) - S(6))/(8-6) = (220 - 219)/2 = 0.5 mph/min
S'(5) = (S(6) - S(4))/(6-4) = (219 - 218)/2 = 0.5 mph/min

Therefore, S''(6) ≈ (0.5 - 0.5)/(7-5) = 0 mph/min²

In the context of the problem, S''(6) represents the acceleration of the car at t = 6 minutes. Since S''(6) is approximately 0 mph/min², it means that the car's speed is not changing significantly at that moment. The car is maintaining a relatively constant speed.

C. The average speed over the first 12 minutes can be approximated using the trapezoid rule with 12 equal subdivisions.

For each subdivision, the average speed can be approximated as the average of the two speeds at the endpoints of the subdivision.

Average speed = [(S(0) + S(1))/2] + [(S(1) + S(2))/2] + [(S(2) + S(3))/2] + ... + [(S(11) + S(12))/2]
= [(201 + 205)/2] + [(205 + 208)/2] + [(208 + 214)/2] + ... + [(218 + 216)/2]

Calculating each term and summing them up, we get:
Average speed = 209.5 mph

The car's average speed over the first 12 minutes is 209.5 mph.

To qualify for the race, the car needs to have an average speed of 210 mph. Since the car's average speed is less than 210 mph, it is currently not fast enough to qualify for the race.

A. To approximate the total distance the car traveled over the first 12 minutes using the trapezoid rule, we need to calculate the area under the speed vs. time curve. The trapezoid rule estimates this area by dividing the curve into trapezoids.

First, let's calculate the time step, Δt, which is the difference between consecutive time intervals. In this case, Δt = 12 minutes / 4 equal subdivisions = 3 minutes.

Next, we'll calculate the area under the curve using the trapezoid rule. For each subdivision, we'll calculate the area of the trapezoid and sum them up.

For the first subdivision, the speeds are S(0) = 201 mph and S(3) = 214 mph. The time interval is 3 minutes, so the area of this trapezoid is (S(0) + S(3)) * Δt / 2.

For the second subdivision, the speeds are S(3) = 214 mph and S(6) = 219 mph. Again, the time interval is 3 minutes, so the area of this trapezoid is (S(3) + S(6)) * Δt / 2.

Similarly, we calculate the areas for the third and fourth subdivisions using the corresponding speeds. Finally, we sum up the areas of all four trapezoids to get an approximation of the total distance traveled over the first 12 minutes.

B. To find the approximation for S''(6) (the second derivative of speed with respect to time at t = 6), we need to calculate the rate of change of acceleration. Since we are given discrete measurements, we can approximate the second derivative using finite differences.

Using the central difference approximation, we can calculate the second derivative as (S(7) - 2 * S(6) + S(5)) / (Δt)^2. Here, Δt = 3 minutes.

In the context of the problem, S''(6) represents the rate of change of acceleration of the car at the 6th minute. It gives information about how fast the car is changing its acceleration at that specific moment. A positive value indicates an increasing acceleration, while a negative value indicates a decreasing acceleration.

C. To calculate the average speed over the first 12 minutes, we need to find the total distance covered and divide it by the total time.

Using the trapezoid rule from part A, we approximated the total distance covered over the first 12 minutes.

The total time is 12 minutes. Therefore, the average speed is approximately (total distance / total time) mph.

To check if the car qualifies for the race, we compare the average speed with the required average speed of 210 mph. If the average speed is greater than or equal to 210 mph, then the car is fast enough to qualify. If it is below 210 mph, then the car does not meet the qualification criteria.