Math 1311 Lab, Fall 2015
TA: Matteo Altavilla
Name:
uNID:
Worksheet #8
(40 points)
Worksheet #8 is due Thursday, November 5th . You are encouraged to work with
to solve these problems, but you have to write the solutions down individually. Please
sheets only. You can write at the bottom of each page and/or on the back. Show
and explain your reasoning when necessary: correct solutions with no explanation do
complete. Partial credit will be assigned to incomplete answers.
other people
turn in these
all the steps
not count as
Problem 1. (20 points) Consider the function f (x) = −x3 + x2 + x + 2, whose plot in the interval
[0, 2] is reported in the figure below.
(a) Express the area of the region delimited by the function and the x axis as a limit of left Riemann
sums. Do not evaluate the limit.
(b) Find an approximation of the integral using a partition of the interval in n = 5, 10, 20
subintervals. You don’t have to write down all the calculations in your solution, just set up the
solution method and write down the final results.
(c) Using the data from the endpoints of the subintervals in item (b), draw a sketch of the function
A(x), which assigns to each x the value of the area under f (x) between 0 and x.
Problem 2. (20 points) Marty McFly and Doc Emmett Brown are stuck in 1955 and need to go
Back to the Future. The two are sitting inside their DeLorean, and Doc calculated that starting with
cold engine and pressing the accelerator at maximum power, the car will move at a velocity which
depends on time and is governed by the following formula
√
√
v(t) = t + t3 m/s
(a) Compute the correspondent acceleration a(t) and approximate the area under its graph on the
interval [1, 5] using rectangles, dividing the interval in a number of subintervals of your choice
(make it reasonable). Compare your result with the exact value of v(5) − v(1), verifying the
Evaluation Theorem.
(b) As in every action movie, there’s a big problem: because of a previous crash, the car cannot be
moved from its starting position and can only go ahead. This is a problem because there’s a
big wall standing x = 185 meters ahead of the car. Marty and Doc can only hope that, while
speeding up towards the wall, the car hits 88 mph and is automatically transfered in 1985 (when
the wall is not there anymore: think quadrimensionally!).
Given the above formula for v(t), will the DeLorean reach 88 mph before hitting the wall? [Hint:
find the time at which the velocity is the one required, and compute the distance that the car
covers in that time using the Evaluation Theorem]