Name (print):
ID number:
Section (circle):
001
002
004
University of British Columbia
MIDTERM TEST 1 for MATH 110
Date: October 21, 2009
Time: 6:00 p.m. to 7:30 p.m.
Number of pages: 11 (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
8
3
8
4
8
5
8
6
6
7
6
8
8
9
3 (bonus)
Total
60
1. Solve the following equations for x.
2
x+2
3x − 2x − 1
=2
1. (a) [5 marks]
x2 − 4
3x + 1
1. (b) [3 marks]
√
x − 2 = 2x
2
2. Solve the following inequalities for x.
2. (a) [4 marks] −x2 + 4x + 5 < 0
2. (b) [4 marks] −x2 + 4x − 5 < 0
3
3. (a) [4 marks] Solve the inequality 2|x − 2| ≤ 10.
3. (b) [4 marks] Let f (x) = 2|x − 2| and g(x) = 10. Find the area of the triangle enclosed
by f (x) and g(x).
4
4. Let f (x) = x + 1 and g(x) = x2 .
4. (a) [4 marks] Sketch the graphs of f ◦ g(x) and g ◦ f (x) on the same axes.
4. (b) [4 marks] How many solutions, if any, does
f (x) = f ◦ g(x) − 2
have? Justify your answer.
5
5. Evaluate the following limits, or explain why they do not exist.
√
3− x
(a) [3 marks] lim
x→9 9 − x
(b) [5 marks] lim f (x), where f (x) is the slope of the line from (0, −1) to (x, 0), as shown
x→0
below:
y
(x,0)
-1
6
1
6. (a) [2 marks] Determine the domain and range of the function f (x) = sin
.
x
6. (b) [4 marks] Evaluate the limit
1 √ 3
lim sin
x + x2 ,
x→0
x
or explain why it does not exist.
7
7. (a) [3 marks] Define what it means for a function f to be continuous at x = a.
7. (b) [3 marks] Let f be the function with the following graph:
y
1
-1
x
1
-1
Where is f discontinuous? (Justify your answer.)
8
8. For each of the following functions, find c such that f is continuous everywhere.
2
cx + 2x if x < 2
8. (a) [5 marks] f (x) =
.
x3 − cx if x ≥ 2
8. (b) [3 marks] f (x) =
x
if x < c
.
1/x if x ≥ c
9
9. [3 bonus marks] Let f and g be two functions. Let g be continuous at a, and f be
continuous at g(a). Prove that f ◦ g is continuous at a.
10
This page may be used for rough work. It will not be marked.
11