Math 1311 Lab, Fall 2015
TA: Matteo Altavilla
Name:
uNID:
Worksheet #7
(50 points)
Worksheet #7 is due Thursday, October 29th . You are encouraged to work with other people to
solve these problems, but you have to write the solutions down individually. Please turn in these
sheets only. You can write at the bottom of each page and/or on the back. Show all the steps
and explain your reasoning when necessary: correct solutions with no explanation do not count as
complete. Partial credit will be assigned to incomplete answers.
Problem 1. (10 points) An important tool in the study of a function is the following Extreme Value
Theorem, due to Weierstrass.
Theorem (Weierstrass). Let f be a continuous function whose domain is a closed interval [a, b]. Then
f attains an absolute maximum and an absolute minimum at some points c, d ∈ [a, b].
Given the following four functions f1 , . . . , f4 and their respective domains, decide for each one
whether the EVT is applicable or not, and why. Then, find the values for the absolute maximum and
minimum of f1 , . . . , f4 if they exist [Warning: they might exist even if the theorem is not applicable].
f1 (x) = x − ln(2x + 1),
f3 (x) = −x2 ,
D = [0, 2]
D = (−1, 1)
f2 (x) = 2 sin(x) + cos(2x), D = [1, 3]
x
for 0 ≤ x < 1
f4 (x) =
D = [0, 2]
x − 2 for 1 ≤ x ≤ 2
Problem 2. (20 points) A crocodile is stalking a zebra along a river. The river is 6 meters wide and
the distance between the prey and the point opposite to the crocodile on the other bank is 20 meters
(see picture below).
The crocodile moves at different (constant) speeds on land and water: when it walks, its speed is
1/4 m/s, while it slows to 1/5 m/s when swimming.
(a) Assuming the poor zebra is unaware of the danger and won’t flee, compute the time in which
the crocodile reaches its prey in the two extreme cases: if it swims the less possible distance or
if it only swims diagonally towards the zebra.
(b) Assuming now that the crocodile can only move along straight lines, find a point P on the other
side of the river to which the crocodile has to swim in order to minimize the time that it takes
to reach the zebra.
Problem 3. (20 points) This problem is devoted to exploring the possibility of failure of two methods
you learned in class: L’Hôpital’s Rule for limits and Newton’s Method for finding zeroes of functions.
(a) Compute
ex
x→∞ x2 + 5
applying L’Hôpital’s Rule twice. Explain why you can apply it twice.
lim
Now consider
x3 − 4x2 + x + 6
,
x→2
x2 − 4x + 4
and see what happens if you blindly apply L’Hôpital’s Rule twice. After that, reconsider what
you did and find the actual limit, explaining why L’Hopital’s got it wrong.
lim
(b) Use Newton’s Method with initial approximation x1 = 1 to find the solution to x3 − x − 1 = 0
correct to the sixth decimal place. Now try it again with x1 = 0.6 and x1 = 0.57 and explain
why the method is so sensitive around these values. You can use your calculator to graph the
function and get an idea of what’s happening.