More Chapter 5 Review
Math 504
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)
Speed (ft/sec)
a.
b.
0
2
4
6
8
10
100
80
50
25
10
0
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?
Which statement below can you justify from the information given in the story
and data table? (Choose one and justify it.)
i. The car stopped before getting to the skunk.
ii. The “black box” data is inconclusive. The skunk may or may not
have been hit.
iii. 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 v =
1
,
1+ t
where t is in hours and v is in meters/hour. Use Δt = 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.
a.
If f(t) is measured in meters/second2 and t is measured in seconds, what are the units of
b
" f (t)dt ?
b.
a
If f(t) is measured in dollars per year and t is measured in years, what are the units of
b
" f (t)dt ?
!
a
b
c.
!4.
If f(x) is measured in pounds and x is measured in feet, what are the units of
" f (x)dx ?
a
a.
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.
b. Find the average value of the function over the given interval: G(t) = 1 + t over [0,2]
5.
4
10
" ln xdx
a)
1
x
b)
" e dx
0
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
1
used to approximate
" g(t)dt , with the same number of subdivisions for each rule, are as follows:
0
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?
!
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/second
b. dollars
c. foot-pounds (units of work)
60
4. a. Amount of oil leaked in first hour =
" f (t)dt
0
2
b.
" (1 + t)dt
0
2#0
=2
!
5. a) LHS < TRAP < Actual < RHS
!
b) 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 = left
0.633 = trap
0.632 = midpoint
0.601 = right
b. The true value lies between the midpoint (0.632) and trapezoid (0.633) sums.