Name (print):
ID number:
Section (circle):
001
002
003
University of British Columbia
DECEMBER EXAMINATION for MATH 110
Date: December 15, 2010
Time: 3:30 p.m. to 6:00 p.m.
Number of pages: 13 (including cover page)
Exam type: Closed book
Aids: No calculators or other electronic aids
For examiners’ use only
Question Mark Possible marks
1
6
2
8
3
6
4
5
5
10
6
8
7
6
8
7
9
14
10
4 (bonus)
Total
70
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;
• 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.
1. (a) [2 marks] Given polynomials f (x) and g(x), lim
x→∞
f (x)
is equal to 0, ∞ or −∞.
g(x)
1. (b) [2 marks] Not all continuous functions are differentiable.
1. (c) [2 marks] If f 0 (x) = g 0 (x), then f (x) = g(x).
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2. Find the derivatives of each of the following functions.
2. (a) [2 marks] f (x) =
x2 x3 x 4 x5 x6
x
√
2. (b) [3 marks] f (x) =
x
cos x
2. (c) [3 marks] f (x) = tan (tan (tan (x2 )))
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3. (a) [2 marks] Complete the statement of the Intermediate Value Theorem by filling in
the blanks:
and
Suppose the function f is continuous on the interval
L is a number between f (a) and f (b). Then there is at least one number
in [a, b] satisfying f (c) = L.
3. (b) [2 marks] Sketch the graph of a continuous function illustrating the Intermediate
Value Theorem as stated in part (a). Make sure to include the labels a, b, c, f (a),
f (b) and L.
3. (c) [2 marks] Give an example of a function f defined on the interval [−1, 1], satisfying
f (−1) = −1 and f (1) = 1, such that f (x) 6= 0 for any x in [−1, 1].
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4. Given a differentiable function f , recall that there are two definitions for f 0 (a):
f (x) − f (a)
f (a + h) − f (a)
and f 0 (a) = lim
.
x→a
h→0
x−a
h
f 0 (a) = lim
4. (a) [2 marks] Pick one of these definitions, and draw a picture to illustrate the fact that
the limit on the right-hand side is the limit of slopes of secant lines. Make sure to
include the labels a and f (a).
4. (b) [3 marks] Use one of the definitions given to find the derivative of the function
f (x) = 2x + 1 at x = 0. (No credit will be given for differentiating directly.)
5
5. Two curves are said to intersect orthogonally at x = a if (i ) they intersect at x = a and
(ii ) their tangent lines are perpendicular at x = a.
√
intersect at
5. (a) [2 marks] Verify that the curves y = 16 2x2 + 7 and y = −2x2 + 7x − 13
6
x = 3.
5. (b) [4 marks] Are the tangent lines of the curves in part (a) perpendicular at x = 3?
Justify your answer.
5. (c) [4 marks] Consider the curve y = sin x. Find constants b and c such that the curve
y = x2 + bx + c intersects it orthogonally at x = 0.
6
6. A circle of radius 5 centred on the origin has the equation
x2 + y 2 = 25.
(1)
6. (a) [1 mark] This circle is not a function. Why not?
6. (b) [4 marks] Find the equation of the line tangent to the circle at the point (3, −4). (It
may help first to find an expression for the bottom half of the circle by solving the
equation (1) for y.)
6. (c) [3 marks] Prove that the line through the origin and the point (3, −4) is perpendicular
to the tangent line in part (b).
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7. [6 marks] A function f is defined piecewise by
−x2
if x < 1
f (x) =
.
3
ax + bx − 2 if x ≥ 1
Find values of a and b so that f is continuous and differentiable everywhere.
8
8. (a) [4 marks] Let f (x) = sin x and g(x) = cos x. Find f
π
3
,g
π
3
, f0
π
3
and g 0
π
3
.
8. (b) [3 marks] Find the equation of the line tangent to the curve y = 4 sin x cos x at
x = π3 .
9
9. Let f (x) =
x+1
.
x−1
9. (a) [1 mark] State the domain of f .
9. (a) [3 marks] Find all vertical and horizontal asymptotes of f , if they exist.
9. (b) [2 marks] Find all x- and y-intercepts of f , if they exist.
9. (b) [2 marks] Find all critical points of f , if they exist.
10
9. (c) [2 marks] Determine where f is increasing and where it is decreasing.
9. (d) [4 marks] Sketch the graph of f .
11
10. [4 bonus marks] Prove that the curve
y=
cos x − 1
x
has an infinite number of horizontal tangent lines.
12
This page may be used for rough work. It will not be marked.
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