Problem
15: 1998-1999
Below is the graph of a car's velocity with respect
to time once the brakes are applied.
(a) If we call the above curve g(t), create accumulation
function f(x) which represents the distance traveled by the car after the
brakes are applied.
(b) Approximate the car's braking distance
using upper sums and the trapezoidal rule.
Difficulty rating: 
Solution:
(a) Recall that the area beneath a velocity function is exactly
equal to the distance traveled. Therefore, I create function f(x) like
this:
(b) In order to do upper sums, I must first decide how many
intervals to use. For the sake of ease, I will choose n = 6 intervals,
since the domain of the velocity function is [0,6]. Thus, the width of each
interval will be one (second). Note that the higher the n, the closer my approximation
will be to the actual distance traveled. Also, I am going to have to approximate
some of the velocities as my graph does not give each of them exactly. Your
approximations might be different, but that's okay -- it's only an approximation.
To do upper sums, I look at each time interval as the
sides of a rectangle which stretch up to the function. In each rectangle, I
must choose the height (either where the left side or the right side meets the
curve) which is higher. I do the exact opposite for lower sums. Recalling the
method for upper sums (and that my width or "delta-x" is going to be
(b-a)/n or (6-0)/6 = 1, I get the following:
My upper sum tells me that I traveled approximately 193 feet;
don't forget units! Note that in this problem, the upper height is always
the left-hand height, as the car's velocity slows as time passes. This
is not always the case when doing upper sums! My formula may not have the
nice, consecutive progression of 0, 1, 2, 3, 4, 5, etc. Instead, always
take the higher of the two heights.
Trapezoidal Rule is usually simpler by comparison (and
typically more accurate) so it is usually the method o' choice for approximating
area. I will use the same n (number of subintervals) in order that I
may compare my results:
There's quite a difference in dem dar approximations! 158 feet traveled
is likely the closer guess, however. In order to get even a better guess, increase
the n above and (although the number of calculations increase since the number
of things in the brackets do) the answer gets purdier and purdier all the time.
A final note: since n = 6, the number of terms in the trapezoidal rule
quantity is 7, always one more, and a great check to make sure you set it up
correctly.
