ASSIGNMENT 8
There are two parts to this assignment. The first part is on WeBWorK — the link is available on the course
webpage. WeBWorK questions can be submitted directly online. They should be completed before you attempt
the second part of the assignment, which consists of the questions on this page. For these questions, 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 typeset or very neatly written.
They must be stapled, with your name and student number at the top of each page.
1. Plasma drug concentration curves, depicting the concentration of a drug in the bloodstream of a patient with respect to time, play a major role in pharmacology. One such
curve, taken from the 1996 paper “Increased Plasma Rifabutin Levels with Concomitant Fluconazole Therapy in
HIV-Infected Patients” in Annals of Internal Medicine, is
shown to the right.
Let f (t) be the concentration of Rifabutin at time t, represented above by the
dotted line. Explain, in a few sentences,
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what the integral 0 f (t)dt represents pharmacologically.
2. Let f (x) = x3 . In this question you will determine
2
Z
f (x) dx directly.
0
(a) In this part of the question, you will prove the formula
n
X
i3 =
i=1
n(n + 1)
2
2
.
(1)
i. Confirm that (1) holds when n = 1.
ii. Assume that (1) holds for n = k. Using this assumption, show that (1) holds for n = k + 1.
iii. Explain why your results in the previous two parts imply that (1) holds for n = 100.
(b) Consider a partition of the interval [0, 2] into n equal subintervals. Find the upper and lower sums
corresponding to this partition.
(c) Explain why x3 is integrable on [0, 2], and find its integral.
3. Prove that the function
f (x) =
1
0
if x is rational
if x is irrational
is not integrable on [0, 1].
Bonus: prove that
(
f (x) =
1
q
if x is a rational number
0
if x is irrational
is integrable on [0, 1], and find its integral.
p
q
in lowest terms with q > 0
.