Math 504More Chapter 5 Review

1. A student is speeding down Route 1 in his fancy red Porsche when his radar system warns him of an obstacle 400 feet ahead. He immediately applies the brakes, starts to slow down, and spots a skunk in the road directly ahead of him.

Suppose that the “black box” in the Porsche records the car’s speed every two seconds, producing the following table. Assume that the speed decreases throughout the 10 seconds it takes to stop, although not necessarily at a uniform rate.

Time since brakes applied (sec) / 0 / 2 / 4 / 6 / 8 / 10
Speed (ft/sec) / 100 / 80 / 50 / 25 / 10 / 0
  1. Using the information in this table, what is your best estimate of the total distance that the student’s car traveled before coming to rest?
  2. Which statement below can you justify from the information given in the story and data table? (Choose one and justify it.)
  3. The car stopped before getting to the skunk.
  4. The “black box” data is inconclusive. The skunk may or may not have been hit.
  5. The unfortunate skunk was hit by the car.

2. For 0 t 1, a bug is crawling at a velocity, v, determined by the formula ,

where t is in hours and v is in meters/hour. Use t = 0.2 to estimate the distance that the bug crawls during this hour. Find an overestimate and an underestimate. Then average the two to get a new estimate.

3.

  1. If f(t) is measured in meters/second2 and t is measured in seconds, what are the units of ?
  2. If f(t) is measured in dollars per year and t is measured in years, what are the units of ?
  3. If f(x) is measured in pounds and x is measured in feet, what are the units of ?

4.

  1. Oil is leaking out of a ruptured tanker at a rate of r = f(t) gallons per minute, where t is in minutes. Write a definite integral expressing the total quantity of oil which leaks out of the tanker in the first hour.
  1. Find the average value of the function over the given interval: G(t) = 1 + t over [0,2]

5.

a)b)

For each integral, arrange the RHS, LHS, and TRAP approximations and the true value in ascending order. Explain, using diagrams, how you predicted this ordering without actually calculating the values.

6. The width, in feet, at various points along the fairway of a hole on a golf course in given below. If one pound of fertilizer covers 200 square feet, estimate the amount of fertilizer needed to fertilize the fairway.

7. The graph of g is shown below. The results from the left, right, trapezoid, and midpoint rules used to approximate , with the same number of subdivisions for each rule, are as follows: 0.601, 0.632, 0.633, 0.664.

a)Match each rule with its approximation.

b)Between which two approximations does the true value of the integral lie?

8. Find .

(Hint: Break up the area under the curve from x = 0 to x = 1 into two pieces: a sector of a circle and a right triangle.)

Answer Key: More Chapter 5 Review

1. a. LHS = 530 feet and RHS = 330 ft. The average, or trapezoid sum, is 430 feet and probably the best estimate.

b. Since the speed is decreasing throughout the interval, the LHS overestimates the distance and provides an upper bound. The RHS underestimates the distances and provides a lower bound. So, the distance has to be between 330 and 530 feet, but we cannot conclude for certain anything else. Therefore, since the skunk was 400 feet in front of the car, the black box data is inconclusive.

2. Since v(t) is decreasing on the interval, I calculated a LHS for an overestimate = .7456m and a RHS for an underestimate = .6456 m. The average is .6956. Since the function is also concave up on this interval, you could calculate the trapezoid sum (same as the average above) for an overestimate and the midpoint sum as an underestimate.

3. a. meters/secondb. dollarsc. foot-pounds (units of work)

4. a. Amount of oil leaked in first hour =

b.

5. a) LHS < TRAP < Actual < RHSb) LHS < Actual < TRAP < RHS

The trapezoid rule is an overestimate if f is concave up, and an underestimate if f is concave down. Since ln x is concave down, the trapezoidal estimate is too small. Since ex is concave up, the trapezoidal estimate is too large. In each case, however, the trapezoidal estimate should be better than the left- or right-hand sum, since it is the average of the two. Both functions are increasing, so the left sum underestimates and the right sum overestimates.

6. Approximate are of the playing field using LHS = RHS = TRAP = 89,000 sq ft. Divide 89,000 by 200 to get approximately 445 lbs of fertilizer.

7. a. 0.664 = left0.633 = trap0.632 = midpoint0.601 = right

b. The true value lies between the midpoint (0.632) and trapezoid (0.633) sums.

8.