MA102 Mock Final Exam
Name:
Time Allowed: 150 minutes
Total Value: 85 marks
Number of Pages: 10
**** Mock tests are meant as a means of providing an extra set of practice questions and basis for a review
class. Do not study for the exam based solely on the topics covered by the mock test! Go back through
notes/assignments/homework to ensure you have reviewed all concepts discussed in the course.
Instructions:
optional, but recommended – create one page of study notes ahead of time, to use while completing the mock test
download a copy of the mock test – either print a paper copy of the test, or use a pdf annotator (or MS OneNote)
and a tablet/iPad
complete the problems under "test-like" conditions:
– restrict yourself to a 2.0 to 2.5 hour period to work on the problems
– attempt the problems without using any reference material, other than your page of study notes (of course,
this "cheat sheet" is not allowed for your actual final exam, but is allowed for mock tests to aid in your study
process)
– use only any additional aids that will be allowed for the actual exam (such as specific calculator,
instructor-provided formula sheet, etc.)
provide complete solutions to the mock test questions; all numerical answers should be given as exact values,
and not as calculator approximations – regardless of the format for your actual exam, we use traditional types of
problems for the mocks so that you can test your understanding of the concepts/methods involved
check your work against the answer key, which will be posted on the MaSt site in MyLS before the scheduled
takeup/review session
1
1. [5 marks] For each of the following limits, write your answer in the blank provided. Your answer should
be one of: a finite number, the symbol +1 or 1, or d.n.e. (for "does not exist"). No justifications are
necessary.
p
(a) lim 25 x
x!25
Answer:
(b) lim 3
1=x
x!1
Answer:
(c)
(d)
(e)
lim
e
x
x! 1
lim
x!0
lim
x!1+
Answer:
jxj + x
jxj x
Answer:
1
ln x
Answer:
2. [5 marks] For each of the following, indicate whether the given statement is true (T ) or false (F ) by writing T
or F in the space provided.
(a) If f is discontinuous at x = a, then lim f (x) does not exist.
Answer:
x!a
(b) The graph of a function can have at most two horizontal asymptotes.
(c) The graph of a function can have at most two vertical asymptotes.
Answer:
Answer:
(d) For all positive values of a and b, ln (a + b) = ln a + ln b.
(e) The domain of the inverse sine function, f (x) = sin
2
1
x, is
Answer:
h
;
2 2
i
Answer:
3. [10 marks] Evaluate each of the following limits.
p
x2 + 5 3
(a) lim
x!2
x2 2x
ln 5e3x
x!1 ln (3e5x )
(b) lim
(c)
lim
!0
3 sin
cos2
1
3
4. [4 marks] Suppose f is a function such that:
1+x+e
x
f (x)
ex + ln (x + e) for all x > 0.
Evaluate lim f (x) and justify your answer. [Hint: Use the Squeeze Theorem.]
x!0+
p
1 + 4x2
.
2x
(a) Determine all, if any, vertical asymptotes for the graph of y = f (x). Justify each of your answers by
evaluating an appropriate limit.
5. [8 marks] Let f (x) =
3x
(b) Determine all, if any, horizontal asymptotes for the graph of y = f (x). Justify each of your answers by
evaluating an appropriate limit.
4
6. [6 marks] Consider the piecewise-defined function
8
>
if x <
< kx + 5
f (x) =
l
if x =
>
: 2k 4x2
if x >
1
1 , where k and l are constants.
1
(a) Determine the value of k and the value of l for which f is continuous at x =
1.
[Hint: Justify your answer using appropriate limits!]
(b) If f is continuous at x = 1, using only that fact, can we then conclude that f is differentiable at
x = 1? Explain your answer.
5
7. [10 marks]
(a) Use the limit definition to find the slope of the line tangent to the curve y = f (x) = p
point P =
3;
1
.
4
(b) Find an equation of the normal line to y = f (x) at P .
6
1
at the
5x + 1
8. [5 marks] Solve the following inequality for x. State your answer using interval notation.
2
x
9. [5 marks] Solve the equation:
10. [3 marks] Evaluate sin
1
cos
3
x
2
log2 (x + 1) + log2 (x)
5
3
log2 x2
1
2
= 2.
, expressing your answer as an exact value.
7
11. [8 marks] Consider the function:
f (x) = 3 ln (x − 1).
(a) Find f −1 , the inverse function of f .
(b) State the domain and range of f −1 .
(c) Sketch the curves y = f (x) and y = f −1 (x) on the axis provided below.
features.
6
5
4
3
2
1
−6 −5 −4 −3 −2 −1
−1
1
2
−2
−3
−4
−5
−6
8
3
4
5
6
Label any key
12. [4 marks] Determine k 0 (2) if k (x) = f (g (x)) + ex h (x) where f , g and h are all differentiable functions,
given that:
f (2) = 5
f 0 (2) = 3
f (3) = 1
f 0 (3) = 10
g (2) = 3
g 0 (2) = 6
13. [8 marks] Determine the derivative of the each given function.
(a) f (x) =
x2
2x + 5
p
x
3
(b) g (x) =
(x 6)
2x5 + 1
9
h (2) = 1
h0 (2) = 4
13. continued... Determine the derivative of the each given function.
(c) h (x) =
x 6
2x5 + 1
9
14. [4 marks] Determine y 00 given that: y = 32x+1 + 2x3
10
1
1=3