Name (print):
ID number:
Section (circle):
001
002
004
University of British Columbia
MIDTERM TEST 2 for MATH 110
Date: February 10, 2010
Time: 6:00 p.m. to 7:30 p.m.
Number of pages: 9 (including cover page)
Exam type: Closed book
Aids: No calculators or other electronic aids are allowed
Instructions: Write your name and ID number, and circle your section number, at the top of
this page. Answer the questions in the spaces provided, using the backs of pages for overflow.
The page at the end of this booklet may be used for rough work. Justify your answers, and
show your work. Unsupported answers will receive no credit.
Academic dishonesty: Exposing your paper to another student, copying material from another
student, or representing your work as that of another student constitutes academic dishonesty.
Cases of academic dishonesty may lead to a zero grade in the test, a zero grade in the course,
and other measures, such as suspension from this university.
For examiners’ use only
Question Mark Possible marks
1
8
2
10
3
8
4
8
5
8
6
8
7
3 (bonus)
Total
50
1. Let f (x) = ex+1 .
1. (a) [3 marks] Find f −1 (x), the inverse function of f (x).
1. (b) [1 mark] Find f −1 ◦ f (x).
1. (b) [4 marks] Sketch the curves y = f (x), y = f −1 (x) and y = f −1 ◦ f (x) on the same
axes. Make sure to label the axes and the intercepts.
2
2. Differentiate the given functions.
2. (a) [4 marks] f (x) = e3x ln(sin x)
2. (b) [3 marks] g(x) = (x − 1)2 (x − 2)2 (x − 3)2 (x − 4)2 (x − 5)2 (x − 6)2
2. (c) [3 marks] h(x) = ln (ln (ln x))
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3. (a) [3 marks] Find y 0 , if y = 2x .
3. (b) [2 marks] Find the equation of the line tangent to y = 2x at the point (a, 2a ).
3. (c) [3 marks] Find the equation of the line tangent to y = 2x that passes through the
point (1, 0).
4
4. A certain population of E. Coli bacteria grown in a constrained laboratory environment
satisfies a logistic equation:
1000
P (t) =
1 + M e−kt
where P (t) is the population of bacteria at time t (in hours), and M and k are constants.
(a) [3 marks] A population of 20 bacteria is found to have grown into 100 bacteria after
one hour. Determine the constants M and k.
(b) [3 marks] Find the initial rate of change in the population (that is, at the start of
the experiment).
(c) [2 marks] What is the population after many months (that is, as t gets arbitrarily
large)?
5
5. Actinium-225, or 225 Ac, is a radioactive isotope with a half-life of 10 days; that is, it
decays exponentially, taking 10 days to halve in quantity.
5. (a) [5 marks] Determine how long it takes for a given quantity of 225 Ac to decay to 30%
of its original. (Your answer should be a constant, but you need not simplify it.)
5. (b) [3 marks] Suppose you find a 5 mg sample of 225 Ac. In the absence of other factors,
how much 225 Ac was present 7 days ago? (Again, your answer should be a constant,
but you need not simplify it.)
6
6. (a) [4 marks] A nuclear detonation at ground level emits an explosive blast wave that
expands radially, like a circular ripple, at a constant speed of 30 km/second. Find the
rate of expansion of the area contained within the blast wave 0.5 seconds after the
detonation.
6. (b) [4 marks] A nuclear weapon detonated at an altitude emits a blast wave that expands
radially in three dimensions, like a sphere. The speed of the blast wave is not constant:
it is equal to
30
km/second
r2
where r is the distance (in kilometres) from the centre of detonation. Explain why
the rate of expansion of the volume contained within the blast wave is constant at
any time after the detonation.
7
7. [3 bonus marks] Prove that the curve y = xπ − π x is decreasing at the point (π, 0).
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This page may be used for rough work. It will not be marked.
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