Name (print):
ID number:
Section (circle):
001
002
004
University of British Columbia
DECEMBER EXAMINATION for MATH 110
Date: December 21, 2009
Time: 3:30 p.m. to 6:00 p.m.
Number of pages: 15 (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
10
2
10
3
8
4
8
5
4
6
6
7
7
8
6
9
6
10
10
11
9
12
6
13
5 (bonus)
Total
90
1. State whether each of the following statements is true. If it is, explain why. If it is not,
explain why not (for example, by providing a counterexample).
1. (a) [2 marks]
d 3
e = 3e2 .
dx
1. (b) [2 marks] The graph of a function with a horizontal asymptote never crosses over
the asymptote.
1. (c) [2 marks]
d
d
tan2 x =
sec2 x
dx
dx
1. (d) [2 marks] If f and g are differentiable, then (f g)0 (x) = f 0 (x)g 0 (x).
1. (e) [2 marks] Continuous functions are differentiable.
2
2. Solve the following expressions for x.
2. (a) [6 marks] |x| = |x2 + x|
2. (b) [4 marks]
√
x+1<x+1
3
3. Let f (x) = x + c and g(x) = x2 .
3. (a) [3 marks] Find all values of c such that the equation
f (x) = g(x)
has no real solutions.
3. (b) [5 marks] Find all values of c such that the equation
(f ◦ g) (1) = (g ◦ f ) (1)
is satisfied.
4
4. Consider the curve y =
1
+ 2.
1−x
4. (a) [4 marks] Find the horizontal and vertical asymptotes of the curve.
4. (b) [4 marks] Sketch the curve.
5
5. Consider the following statement:
Since f (−1) < 0 and f (1) > 0, there exists a number c in the interval (−1, 1)
such that f (c) = 0.
5. (a) [2 marks] Give a function f such that the statement is true.
5. (b) [2 marks] Give a function f such that the statement is false.
6
6. Let f (x) =
5x − 3
if x < 1
.
3
−1
x + 2x − x
if x ≥ 1
6. (a) [3 marks] Determine if f is continuous on its domain.
6. (b) [3 marks] Determine if f is differentiable on its domain.
7
7. Let f (x) =
√
1 − x.
7. (a) [4 marks] Use the limit definition of derivative to find f 0 (x).
7. (b) [3 marks] Use the rules of differentiation to confirm your answer for part (a).
8
8. Find the number of solutions to each of the following equations. (Note: you are asked to
find the number of solutions, not the solutions themselves.)
8. (a) [3 marks] |x| = |x2 − 1|
8. (b) [3 marks]
d x
(e − 2x) = 0
dx
9
1
9. Let f (x) = x sin
.
x
3
9. (a) [4 marks] Evaluate lim f (x), or explain why it does not exist.
x→0
9. (b) [2 marks] Evaluate f 0 (0), or explain why it does not exist.
10
10. Find each of the following derivatives.
3
10. (a) [3 marks] f 0 (x), where f (x) = cos e2x
10. (b) [4 marks] f 00 (x), where f (x) =
x−1
x+1
10. (c) [3 marks] f 000 (x), where y = f (x) describes a parabola.
11
11. (a) [3 marks] Find the coordinates of the points on the curve
y = x4 − 2x2
with horizontal tangent lines.
11. (b) [6 marks] Find the coordinates of the points on the curve
x2 + xy + y 2 = 12
with tangent lines parallel to the line y = x.
12
12. Recall that curves are said to be orthogonal if their tangent lines are perpendicular at
every point of intersection. Consider the curves
x2
+ y 2 = b and y = cxa ,
a
where a, b and c are positive constants.
12. (a) [4 marks] Are the given curves orthogonal? (Justify your answer.)
12. (b) [2 marks] If a and b are positive and c = 0, are the given curves orthogonal? (Justify
your answer.)
13
13. [5 bonus marks] Let f (x) = g1 (x)g2 (x) · · · gn (x). Prove that
f 0 (x) = g10 (x)g2 (x) · · · gn (x) + g1 (x)g20 (x) · · · gn (x) + · · · + g1 (x)g2 (x) · · · gn0 (x) .
14
This page may be used for rough work. It will not be marked.
15