ASSIGNMENT 8
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, January 17. The written assignment is to be handed
in by that time. The online assignment will also close at that time.
1. This question refers to Figure 1 in the 2001 paper “Diagnostic Tests for Renal Artery Stenosis in Patients
Suspected of Having Renovascular Hypertension: A Meta-Analysis” in Annals of Internal Medicine. The
paper is available on the course webpage.
What does the area under each of the curves in Figure 1 represent, and why might you conclude, along with
the authors, that computed tomography angiography is a high-performing diagnostic test for renovascular
hypertension? Explain your answer in one or two paragraphs.
2. Prove that f (x) =
0
1
if x 6= 1
is integrable on [0, 2].
if x = 1
3. (a) Confirm that the following formula holds for n = 1:
n
X
i3 =
i=1
n(n + 1)
2
2
.
(1)
(b) Prove that if the formula (1) holds for n = k, then it also holds for n = k + 1.
(c) Explain why your answers in part (a) and part (b) imply that the formula (1) holds for all positive
integer values of n.
(d) Using Riemann sums, but without using the Fundamental Theorem of Calculus or the fact that
continuous functions are integrable, prove that
Z 2
x3 dx = 4.
0