Math 1311 Lab, Fall 2015
TA: Matteo Altavilla
Name:
uNID:
Worksheet #2
(50 + 5 points)
Worksheet #2 is due Thursday, September 10th . 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, compute the following limits (if they exist):
x 2 − x4 + x
x→0 2x + 3x4
(a) lim
sin 10x
x→0
x
(c) lim
x3 − 4x2 + x + 6
x→2
x2 − 4x + 4
(b) lim
(d) lim |x − 5|
x→5
Problem 2. (20 points) Doctor Who is flying his TARDIS in space. He moves on a plane, along a
curve γ(t) given by the following parametric equations
x = cos t
γ(t) =
y = sin 2t
(a) Find a cartesian expression for γ(t) [Hint: make cos2 t and sin2 t appear, then try and use some
trigonometry trick].
(b) Is γ(t) a closed circuit? Motivate your answer without the use of a graphic calculator. If γ(t) is
indeed closed, how many laps does the TARDIS complete as t runs from 0 to 4π?
(c) Consider the point P of the graph of γ(t) corresponding to t = π/4. Try and guess the slope of
the tangent line to the graph at P , using approximation by secant lines. [Hint: choose three or
four points near to P , for example the ones corresponding to t = 0, π/8, π/6, . . . , and evaluate
the slope of the lines passing through P and each one of these points. Now you may plot the graph
of γ(t) with some device to help you visualize the problem, but you should try and understand
the shape of the graph by yourself first].
Problem 3. (20 + 5 points)
(a) Given x0 ∈ [0, 1] and f : [0, 1] → R, write down what it means for f to be continuous at the
point x0 .
Now let g : [0, 1] → R be defined as
g(x) =
1
0
if x = 1/2
otherwise
(b) Draw the graph of g, and say if g is continuous at the point x0 = 1/2 using the definition you
gave in item (a). Does the limit limx→1/2 g(x) exist in this case?
Now consider this other function D(x) : [0, 1] → R, called the Dirichlet function:
1
if x ∈ Q ∩ [0, 1]
D(x) =
0
otherwise
(c) Try to draw the graph of D(x), and convince yourself that you can’t. Then, explain why the
limit limx→1/2 D(x) does not exist. [Hint: is the limit equal to 1? Is it equal to 0? Are there
any other possibilities?]. Conclude that D(x) is not continuous at x0 = 1/2.
(d) Bonus question: Let x0 be any point in [0, 1]. Show that limx→x0 D(x) does not exist.
[Hint: split this in two cases, one where x0 is rational and the other one where x0 is irrational,
and adapt your previous argument from item (c)]. Conclude that D(x) is nowhere continuous
(that’s why you couldn’t draw it!).