Name (print):
Student number:
University of British Columbia
MATH 100 (Vantage College): Final exam
Date: December 5, 2014
Time: 3:30 p.m. to 6:00 p.m.
Number of pages: 14 (including cover page)
Exam type: Closed book
Aids: No calculators or other electronic aids
For examiners’ use only
Question Mark Possible marks
Rules governing formal examinations:
Each candidate must be prepared to produce, upon request, a
UBC card for identification.
No candidate shall be permitted to enter the examination room
after the expiration of one-half hour from the scheduled starting
time, or to leave during the first half hour of the examination.
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;
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18
2
6
3
4
4
5
5
5
6
15
7
5
8
6
9
6
Total
70
• 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.
Note that your answers must be in “calculator-ready” form, but they do not have to be
simplified.
In general, you may use any result proven in class or on assignments. However, in question
3, you must use the definition of derivative.
This page may be used for rough work. It will not be marked.
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1−x
√ .
x→1 1 −
x
1. (a) [2 marks] Evaluate lim
√
(b) [2 marks] Evaluate lim
x→0
x + 1 − ex
.
x
2
(c) [2 marks] Evaluate lim ecos(x )/x .
x→∞
3
(d) [2 marks] Find the domain of f (x) = log (2 − log(x)).
(e) [2 marks] Let f (x) = log (sin (log(x))). Find f 0 (x).
(f) [2 marks] Let f (x) = xlog(x) . Find f 0 (x).
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(g) [2 marks] Let f (x) = g (2 sin(x)), where g 0
√ √
2 = 2. Find f 0
π
4
.
(h) [2 marks] Find a function with horizontal tangent lines at exactly two values of x:
x = A and x = B.
(i) [2 marks] Find an equation of the line tangent to the curve y = esin(x) + x at (0, 1).
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2. (a) [3 marks] Find c such that
f (x) = sin(cx) − x2 + 2x + 3
has a critical point at x = 0.
(b) [3 marks] Find the coordinates of the global minimum of
f (x) = x3 − 6x + 4
on the interval [0, 3].
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3. [4 marks] Let f (x) =
x
. Find f 0 (x) using the definition of derivative.
2 + 3x
(You must use the definition of derivative; no credit will be given for proving the result
using the Power Rule, Product Rule, Quotient Rule or Chain Rule.)
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4. [5 marks] Let f (x) be a function for which f 0 (x) is strictly increasing (this means that
if a < b, then f 0 (a) < f 0 (b)). Prove that any line tangent to the graph of f (x) intersects
the graph of f (x) only once. (Hint: use the Mean Value Theorem.)
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5. Consider the point (c, 0) where − π2 < c < π2 .
(a) [2 marks] Prove that if a line through (c, 0) is tangent to the curve y = tan(x) at
x = b, then c = b − sin(b) cos(b).
(b) [1 mark] Find the derivative of f (x) = x − sin(x) cos(x).
(c) [2 marks] Explain carefully why you may conclude from parts (a) and (b) that there
is exactly one line through (c, 0) that is tangent to the curve y = tan(x).
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6. The Maxwell-Boltzmann distribution describes particle speeds in an ideal gas. It is a
probability distribution given by
√
2
2
2
f (v) = √ 3 v 2 e−v /(2a ) ,
πa
where a is a positive constant. f (v) is defined on the domain [0, ∞).
(a) [2 marks] Find the v-intercepts and y-intercepts of f (v), if any exist.
(b) [3 marks] Find the vertical and horizontal asymptotes of f (v), if any exist.
(c) [3 marks] Determine where f (v) is increasing and where f (v) is decreasing, given
that
√
2
2
2
f 0 (v) = √ 5 ve−v /(2a ) 2a2 − v 2 .
πa
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(d) [3 marks] Determine where f (v) is concave up and where f (v) is concave down, given
that
√
2
2
2
00
f (v) = √ 7 e−v /(2a ) v 4 − 5a2 v 2 + 2a4 .
πa
(e) [4 marks] Sketch a large graph of f (v), including the features determined in the
previous parts of this question.
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7. [5 marks] Recall that a curve is said to be nonsingular if there is exactly one line tangent
to the curve at every point on the curve. (For example, elliptic curves are nonsingular.)
The Kampyle of Eudoxus is the curve described by the equation
y 2 = x4 − x2
and excluding the point (0, 0). Prove that the Kampyle of Eudoxus is nonsingular.
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8. [6 marks] The curvature of a function at x indicates how much the function is “curving”
there. For example, a straight line has curvature 0 everywhere. The curvature of a
function f (x) at x is given by
κ(x) =
|f 00 (x)|
(1 + f 0 (x)2 )3/2
.
Determine where the curvature of f (x) = log(x) is maximal.
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9. Imagine two conical tanks of radius R and height H, one of which is inverted. Both tanks
are filled with water. Water drains out of the bottom of both tanks at a constant rate V 0 .
R
H
H
R
(a) [3 marks] In the case of the tank pictured on the left, find the rate at which the
height of the water is changing when the height of the water is equal to h0 . (Your
answer may involve the constants R, H, V 0 and h0 .)
(b) [3 marks] In the case of the tank pictured on the right, find the rate at which the
height of the water is changing when the height of the water is equal to h0 . (Your
answer may involve the constants R, H, V 0 and h0 .)
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