Math 1311 Lab, Fall 2015
TA: Matteo Altavilla
Name:
uNID:
Worksheet #3
(40 + 5 points)
Worksheet #3 is due Thursday, September 17th . 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) Using the appropriate rules (no de L’Hôpital, please!), compute the following
limits (if they exist):
sin 2x
(a) lim
x→+∞ sin x
sin x
(b) lim
x→+∞ x
2x2 − x4 + 5
(c) lim
x→−∞ x2 + x3 − 3
f (5 + 3h) − f (5)
, where f is a differentiable function
h→0
h
(d) lim
Problem 2. (20 points) Consider the function
3
x − 2x + 1
fa,b (x) =
ax + b
x ∈ (−∞, 0]
,
x ∈ (0, +∞)
depending on the parameters a and b.
(a) Find a and b such that fa,b (x) is differentiable at any point of R. Whenever you have to compute
a (left or right) derivative, use the definition and do not use differentiation rules.
(b) Using the values of a and b found in the previous item, prove that there exists a point x0 ∈ [−2, 0]
such that fa,b (x0 ) = 0.
(c) Compute the value of x0 with a precision of at least three digits.
To do so, you can use the so-called bisection method. The goal of this method is to split
the interval in half and recognize in which of the two halves the point x0 has to fall. It goes
like this: if you solved the previous item, you know why the function assumes the value of zero
in [−2, 0]. Now consider the two intervals [−2, −1] and [−1, 0], and repeat the same argument
for both of them: now you know where x0 is, and you can replace the old interval with this
new one, split it in two and go on. In the meanwhile, the values of the two extrema of each
interval will be approaching to each other, and you can stop when they are less than 0.001 apart.
Can you name a situation in real life in which you may use the bisection method?
Problem 3. (10 points) A racing car is testing on a circuit. At t = 0 the car is not moving, then it
starts accelerating. The following graph (t in seconds versus a(t) in meters per square second) shows
how the acceleration changes in the interval [0, 20]
Find out as much information as possible about the velocity of the car from the graph of its acceleration
[Hint: remember that a(t) = v 0 (t)]. At what time does the car reach the maximum velocity?
Problem 4. (Bonus Problem, 5 points) A Buddhist monk must reach a temple located on the top
of a mountain. There’s only one path that leads to the temple; it starts from the very bottom of the
mountain and goes up to the top, but nobody knows how long it is. The monk starts his journey
at noon, and follows the path straight to the top; he stops for lunch and meditation, and he does
not keep a uniform pace due to the asperities of the rocks. Finally, he reaches the temple exactly
at midnight. The following day, the monk has to go back; he leaves the temple at noon going down
along the same path, and he reaches the foot of the mountain exactly at midnight.
Show that there exists at least one time t between noon and midnight at which the monk has occupied
the exact same spot both during the ascent and the descent.