MATH1115 ASSIGNMENT 5,
DUE MONDAY MARCH 27 AT 11PM
Homework Policy
The following rules apply to collaboration and the use of outside resources:
• Collaboration with other students is allowed – encouraged, in fact – but if
you collaborate with other students you must acknowledge the collaboration
in writing at the top of your assignment. For example, “On problems 1
and 4 I benefitted from conversations with John Doe, and my solution to
problem 5 was obtained in collaboration with Jane Doe.” Assignments
without a collaboration statement will not be graded.
• The use of outside resources is also permitted, but be careful. For example,
you are allowed to read about Venn diagrams on Wikipedia. You are not
allowed to read the solution to an assignment problem someone posted on
Stackexchange. If in doubt, ask before you look. Any such use must also
be acknowledged in writing in the assignment.
• Unless otherwise noted, the problems should be solved without the assis-
tance of computer software.
• Regardless of which sources you use, the writing of all solutions must be
your own.
Note that this is only half of the assignment. The other half is a MATLAB
Grader assignment.
(0) (a) Write a collaboration statement, detailing who you worked with and
any outside resources you may have used. (You don’t need to mention
help you received from the lecturers, your demonstrator, or the drop-in
centre.) This part is mandatory.
(b) Write a reflection on how the course is going. Did you go to lectures,
and to your workshop? Did you spend time with a study group, or use
any of the other resources (like the drop-in centre or office hours) that
are available to you? How much time did you spend on the course in
the past week? This part is optional.
(1) (4 points) Suppose that A,B are non-empty subsets of R. Show that if
A ⊂ B and B is bounded from above, then supA exists and supA ≤ supB.
(2) (7 points)
(a) (3 points) Let f : R → R be a function which is continuous at x = 0.
Use the definition of differentiability to prove that the function g :
R → R defined by g(x) = x(f(x)) is differentiable at x = 0 and find
g′(0).
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2 MATH1115 ASSIGNMENT 5, DUE MONDAY MARCH 27 AT 11PM
(b) (4 points) Again using the definition of the derivative, calculate the
derivative of f(x) = 12+5x on its natural domain (−∞,− 25 )∪(− 25 ,∞) =
R\{− 25}. You may not use any differentiation rules (like the product
rule, the quotient rule, etc).
(3) (4 points) The final calculus question will be posted after the Modelling
Workshop on Thursday, 23/3.
(4) (3 points) Without computing any inverse matrices, solve the following
equation for X:
X ·

2 5 1 3
42 11 −165 13
6 13 −12 22
7 17 6 5
 =

2 5 1 3
42 11 −165 13
8 18 −11 25
1 2 3 −4

(5) (Total of 8 points) Recall that the transpose AT of a matrix is defined by
interchanging the rows and columns: (AT )ij = Aji.
We say a square matrix A is skew symmetric if AT = −A.
(a) (2 points) Show that whenever AB is defined, BTAT is also defined
and (AB)T = BTAT . (For this part A and B do not have to be skew
symmetric.)
(b) (3 point) If A =
 b cf
 is a skew symmetric 3 × 3 matrix, what
are the remaining entries of A? Is A invertible? (Your answer may
depend on b, c and f .)
(c) (3 points) Show that if A is skew symmetric and invertible then A−1
is also skew symmetric. (To be clear: A is not the 3 × 3 matrix from
part (b) but rather a skew symmetric matrix of arbitrary dimension.)
(6) (4 points) Prove that there is exactly one degree at most 2 polynomial
p(x) = a0+a1x+a2x
2 passing through 3 points (x1, y1), (x2, y2) and (x3, y3)
as long as x1, x2 and x3 are distinct. (You may now use determinants if
you want to.)