DEPARTMENT OF MATHEMATICS AND STATISTICS
Math 1030, W16
Midterm 2
1. This is a 50 minute test. Do NOT start until instructed.
2. Please fill out your personal details on the cover of the Examination Booklet(s). If
you use more than one booklet, please indicate how many you use, e.g. “1 of 3”,
“2 of 3”, “3 of 3” on each booklet.
3. You may quote results from lecture notes without proof, unless asked to do otherwise. But notes or books are not permitted during the exam.
4. Give sufficient working as solutions with no justification will receive little or no
credit.
5. You may use a scientific calculator and/or a ‘graphing calculator’. Note however
that graphical answers require justification. No other electronic equipment is permitted. Turn off or ‘mute’ your cell phone!
6. If you copy the work of your neighbour this is considered Academic Misconduct
and will be reported to the appropriate university authority.
7. There are a total of 38 points to be awarded on this test.
THIS PAGE INTENTIONALLY LEFT BLANK
1. (12 marks)
(a) Producing x units of tacos costs C(x) = 5x + 20 dollars. The revenue earned
is given by R(x) = 15x dollars.
(i) What is the break-even quantity?
(ii) What is the profit from 100 units?
(iii) How many units will produce a profit of $500?
(b) Solve the equation
x
1
9
− 3 = 24
for x without the use of logarithms.
(c) Solve
2 log5 (3x) = 4
for x.
2. (10 points)
(a) Consider the recursively defined sequence
a1 = 1,
a2 = 2,
a3 = 3,
and an+1 =
an + an−1
an−2
n = 3, 4, 5, . . .
List the first 6 terms of this sequence.
(b) Identify the following sequence
3 3
3
{3, − , , − , . . .}
2 4
8
and determine:
(i) the recursive formula for this sequence, (ii) a formula for the nth term, and
(iii) the 10th term.
(c) Evaluate
lim
n→∞
n+2
3n − 1
.
Be careful to justify your answer by showing the rules of limits (or, alternate
arguments) that you use!
Please Turn Over
3. (5 points)
(a) Evaluate
6
X
(1 − k2 ).
k=2
(b) Evaluate
70
X
n.
n=20
(c) Determine whether the infinite geometric series is convergent or divergent. If it
is convergent, find its sum:
2+3+
9
2
+
27
4
+ ...
(Make sure you identify the value of r in this series.)
4. (11 marks)
(a) Find the maturity value (i.e., the future value) of a deposit of $2345 invested at
a simple interest rate of 1.60% invested for 3 months.
(b) Find the present value of $1000 due in 2 years and six months at 6% p.a. compounded monthly.
(c) An amount of $5000 is deposited in a bank paying an annual interest rate of
1.98%, compounded continuously. Find the balance after 2 years.
(d) Suppose we invest $P in an account earning 3.75% compounded quarterly1 .
What is the effective rate of interest? (Give your answer as a percent to 2 d.p.)
END OF TEST
1
Four times per year.