Name (print):
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Science One number:
University of British Columbia
DECEMBER EXAM: MATHEMATICS
Date: December 13, 2010
Time: 9:00 a.m. to 11:30 a.m.
Number of pages: 12 (including cover page)
Exam type: Closed book
Aids: No calculators or other electronic aids
For examiners’ use only
Question
Rules governing formal examinations:
Each candidate must be prepared to produce, upon request, a
UBC card for identification.
Candidates suspected of any of the following, or similar, dishonest practices shall be immediately dismissed from the examination and shall be liable to disciplinary action:
• Having at the place of writing any books, papers or memoranda, calculators, computers, sound or image players/recorders/transmitters (including telephones), or other memory aid
devices, other than those authorized by the examiners;
Mark
Possible marks
1
6
2
5
3
6
4
5
5
11
6
7
7
6
8
2 (bonus)
Total
50
• Speaking or communicating with other candidates;
• Purposely exposing written papers to the view of other candidates or imaging devices. The plea of accident or forgetfulness
shall not be received.
Candidates must not destroy or mutilate any examination material; must hand in all examination papers; and must not take
any examination material from the examination room without
permission of the invigilator.
Candidates must follow any additional examination rules or directions communicated by the instructor or invigilator.
1. Determine whether each statement is true. If it is, explain why; if not, give a counterexample.
(a) [3 marks] lim xx = 1.
x→0+
(b) [1 mark]
1
d 1
√ = 3/2 .
dx e
2e
(c) [2 marks] A function f which is differentiable on the interval (0, 1) and satisfies f (0) = f (1) = 0
must have a horizontal tangent line on the interval (0, 1).
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2. [5 marks] Three axioms are listed below. From them, provide a formal argument using the Rules of
Inference given below to show that the proposition q is a theorem. (A table of rules of inference may be
found at the end of the exam.)
Proposition
Justification
Axiom
1.
p⇒q
Ax
1
2.
r∨q
Ax
2
3.
p ∨ ¬r
Ax
3
4.
5.
6.
7.
8.
9.
10.
11.
12.
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3. (a) [2 marks] State the definition, using limits, of the derivative of a function f at a point a.
(b) [4 marks] Recall that a function f is even if f (−x) = f (x) for all x, and odd if −f (−x) = f (x) for
all x. Let g be an even function. Use the definition in part (a) to prove that g 0 is odd; that is, that
−g 0 (−x) = g 0 (x).
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4. [5 marks] Let f and g be functions satisfying
f (1) = 2, f 0 (1) = −1, g(1) = 1 and g 0 (1) = −3.
Define h(x) = f (x)g(x). Use a linear approximation to the function h to estimate h(1.1).
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5. Let f (x) = xe−x .
(a) [1 mark] Find the x- and y-intercepts of the function.
(b) [2 marks] Determine if f has any vertical or horizontal asymptotes.
(c) [3 marks] Identify where f is increasing and where it is decreasing.
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(d) [2 marks] Identify where f is concave up and where it is concave down.
(e) [3 marks] Sketch the graph of the function, making sure to label all extrema and inflection points.
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6. (a) [3 marks] Recall that a circle of radius r centred on the origin is described by the equation
x2 + y 2 = r 2 .
Prove that any line through the origin intersects the circle at right angles.
(b) [4 marks] Prove that the ellipse and hyperbola described the equations
x2 + 2y 2 = 2 and 2x2 − 2y 2 = 1,
respectively, intersect at right angles.
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7. In the 1986 Chernobyl reactor explosion, substantial amounts of the isotope strontium-90 (90 Sr) contaminated the area around the nuclear plant. 90 Sr decays at a rate proportional to its quantity: if y is the
quantity of 90 Sr as a function of time,
y 0 = ky
(1)
for some constant k.
(a) [2 marks] Write down the general solution to the differential equation (1).
(b) [2 marks] 90 Sr has a half-life of 29 years; that is, it takes 29 years for a quantity of
by half. Use this information to find the constant k.
(c) [2 marks] What is the proportion of
90
Sr originally released which remains today?
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90
Sr to decrease
8. [2 bonus marks] Consider the harmonic oscillator differential equations
−y1
y2
− y2
= y10
,
= y20
where y1 and y2 are functions of time t. Determine what happens to y1 and y2 as t → ∞.
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This page may be used for rough work. It will not be marked.
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Table 1: Rules of Inference. The notation A ` B means: from statement A we can infer statement B as
a valid argument. The notation A a` B means: from statement A we can infer statement B as a valid
argument and vice versa.
Rule name
Formalism
Modus Ponens (MP)
((p ⇒ q) ∧ p) ` q
Modus Tollens (MT)
((p ⇒ q) ∧ ¬q) ` ¬p
Disjunctive Syllogism (DS)
((p ∨ q) ∧ ¬p) ` q
Projection (P)
(p ∧ q) ` p
Conjunction (C)
p, q ` (p ∧ q)
Addition (A)
p ` (p ∨ q)
De Morgan’s Theorem (1) (DMT)
¬(p ∧ q) ` (¬p ∨ ¬q)
De Morgan’s Theorem (2) (DMT)
¬(p ∨ q) ` (¬p ∧ ¬q)
Distribution (1) (D)
(p ∧ (q ∨ r)) ` ((p ∧ q) ∨ (p ∧ r))
Distribution (2) (D)
(p ∨ (q ∧ r)) ` ((p ∨ q) ∧ (p ∨ r))
Double Negation (DN)
p ` ¬¬p
Double Negation (DN)
¬¬p ` p
Transposition (T)
(p ⇒ q) ` (¬q ⇒ ¬p)
Material Implication (1) (MI)
(p ⇒ q) ` (¬p ∨ q)
Material Implication (2) (MI)
(¬p ∨ q) ` (p ⇒ q)
Material Equivalence (ME)
(p ⇔ q) ` ((p ⇒ q) ∧ (q ⇒ p))
Rule of Deduction (RD)
(A1 , A2 , A3 , · · · AN , p ` q) ` (A1 , A2 , A3 , · · · AN ` p ⇒ q)
Rule of Contradiction (RC)
(A1 , A2 , A3 , · · · AN , p ` q ∧ ¬q) ` (A1 , A2 , A3 , · · · AN ` ¬p)
(For this last rule, A1 , A2 , A3 , · · · AN must be consistent.)
The following rules can be “skipped steps”:
Commutation (1)
(p ∨ q) ` (q ∨ p)
Commutation (2)
(p ∧ q) ` (q ∧ p)
Commutation (3)
(p ⇔ q) ` (q ⇔ p)
Association (1)
(p ∨ (q ∨ r)) ` ((p ∨ q) ∨ r)
Association (2)
(p ∧ (q ∧ r)) ` ((p ∧ q) ∧ r)
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