ASSIGNMENT 4
There are two parts to this assignment. The first part is on WeBWorK — the link is available on the course
webpage. The second part consists of the questions on this page. You are expected to provide full solutions
with complete arguments and justifications. You will be graded on the correctness, clarity and elegance of
your solutions. Your answers must be typed or very neatly written. They must be stapled, with your name
and student number at the top of each page.
This assignment is due at 10:00 a.m. on Friday, November 1. The written assignment is to be handed
in by that time. The online assignment will also close at that time.
1. Find the domain, range, derivative and graph of f (x) = arctan(tan x).
2. Recall that we proved the property
ln (xr ) = r ln x for all rational r and all positive x.
In this question, you will extend the property to all real r, and determine its implication for the function
ex .
(a) Let us define
xr = er ln x for all real r and all positive x.
(Note that this definition does not contradict anything, as it is a priori true for rational r.) Prove,
using this definition, the general Power Rule; namely, that
d r
x = rxr−1 for all real r and all positive x.
dx
(b) Prove, using part (a), that
ln (xr ) = r ln x for all real r and all positive x.
You must indicate explicitly why your proof holds for all real r.
(c) Prove, using part (b), that
r
(ex ) = erx for all real r.
You must indicate explicitly why your proof holds for all real r.
3. Note that
aa = bb .
1 1/2
2
=
1 1/4
.
4
Prove that there are infinitely many such distinct pairs (a, b) such that