Math 1311 Lab, Fall 2015
TA: Matteo Altavilla
Name:
uNID:
Worksheet #5
(45 points)
Worksheet #5 is due Thursday, October 1st . 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. (15 points) Consider the inverse trigonometric function f (x) = arcsin(x).
(a) Draw the graph of f (x) with the right domain restrictions (label your axes with all the values!)
√
√
(b) We know that sin(3/4 π) = 2/2. Now use a calculator to find out the value of arcsin( 2/2).
Is there something wrong? Explain what is happening and why.
(c) Prove that
π
2
for −1 ≤ x ≤ 1 [Hint: try and prove that the function on the left hand side is constant first].
arcsin(x) + arccos(x) =
Problem 2. (20 points)
(a) Compute the following derivatives using the appropriate rules:
(i)
d
d
cos(x2 )
(ii)
cos2 (x)
dx
dx
1
d
ln
, for x > 0
(iv)
dx
x
(iii)
(v)
d
1
√
, for |t| < 1
dt 1 − t2
d
(sin(ln(x2 + e)))1/2 .
dt
Now consider our old friend the parametric curve γ(t) defined by
x = cos t
γ(t) =
.
y = sin 2t
(b) Find the exact value of the slope of the tangent line at the point P corresponding to the value
t = π/4, and confirm our previous guess [Hint: use Chain Rule twice, or implicit differentiation].
(c) Find a point
√ Q on the curve such that the value of the slope of the tangent line at that point is
equal to (2 3)/3 [Warning: there’s more than one point with this property. Just choose one].
Write down also the equation for this tangent line.
Problem 3. (10 points) I call a fair n-rate the following compound interest bank plan: divide the
year in n segments, and get an interest of 1/n over your current deposit at the end of each segment.
For example, if n = 2 and you start with 100 $, you get 150 $ after 6 months and 225 $ at the end of
the year.
(a) Let Dn (0) = D be the initial deposit. Find a formula for the function Dn (t), which represents
the amount of money you get after t years under the fair n-rate. Then compute the money you
get after one year if D = 100, for n = 3, 4 and 5.
This looks too good to be true (and in fact it’s not), but even this very profitable plan has its limits.
From item (a), Dn (t) looks promisingly increasing with respect to n. What if I offer you one year of
this plan at n = 100? What if I let you choose n as big as you want? Will you become rich? Let’s
see.
(b) Prove that
lim+ (1 + x)1/x = e
x→0
[Hint: let f (x) = ln(x). Use the fact that f 0 (1) = 1 to prove an equivalent statement.] Conclude
that any possible n-rate will give you less than e · D at the end of the year.