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MATH 10
8
(Applied Calculus)
Final Exam
Marks: 48
Please complete:
Last Name (capitals)
Full First Names
Student Number
For official use:
Question Marks
1-4 /1
6
5-8 /16
9
-12 /16
Total /48
Examiner: Stephen Benecke
Please read the following rules and instructions, and then sign the declaration below:
1. Communication with any other people is not allowed.
2. This test is open book, which means all resources are allowed, as long as it does not contradict with (1).
3. Latecomers are not allowed extra time.
4. Answers may be supplied in pencil.
5. Show sufficient work to justify your answer.
6. Any final answer must be rounded to two decimal places.
7. Students must scan and send their answers before the deadline, or be subject to a penalty at the discretion of
the instructor.
DECLARATION SIGNATURE
I hereby declare that I will abide by the above examination rules and
that the particulars supplied on this front cover are correct.
MATH 108 – Final Exam Page 2 of 1
4
1. State if the following limits exist or not. If it exists, state the limit value.
(a)
lim
x→−∞
−
7
1 − x
2
[1]
(b) lim
x→2
x2 − 7x + 10
x2 + 4
[1]
(c) lim
x→−1−
√
x −
1
x2 + 1
[1]
(d) lim
x→−∞
−2×5 + 4x [1]
MATH 108 – Final Exam Page 3 of 14
2. Consider the function f(x) = 2e−x + x
√
3x − 1. The derivative may be calculated as follows:
f ′(x) = Dx[2e
−x] + Dx[x
√
3x − 1]
= 2Dx[e
−x] + Dx[x
√
3x − 1]
= −2e−x + Dx[x
√
3x − 1]
= −2e−x +
√
3x − 1 + xDx[
√
3x − 1]
= −2e−x +
√
3x − 1 + x 1
2
√
3x−1Dx[3x − 1]
= −2e−x +
√
3x − 1 + x
3
2
√
3x−1
At each of the above steps, state all the derivative rule(s) used. Your options are:
[4]
• Derivative of a constant function
• Constant times a function property
• Sum property
• Difference property
• Power rule
• Product rule
• Quotient rule
• Chain rule
MATH 108 – Final Exam Page 4 of 14
3. (a) Determine the derivative of y = 2−2x
√
1 − x using the product rule first
. [2]
(b) Determine the derivative of y =
1 − x2
x + 1
using the quotient rule first. [2]
MATH 108 – Final Exam Page 5 of 14
4. Use the chain rule to find the derivatives of
(a) y = ex
4+1 [2]
(b) y = ln
(
x − e−x
)2 [2]
MATH 108 – Final Exam Page 6 of 14
5. Consider the function f(x) = x
2
3 − x
5
3 .
(a) Show that the derivative is f ′(x) = −
5x − 2
3 3
√
x
. [2]
(b) Find all intervals where the function f(x) is increasing or decreasing. [2]
MATH 108 – Final Exam Page 7 of 14
6. Sketch the graph of a function f(x) that clearly satisfies all of the following characteristics.
Your graph should clearly show the increasing, decreasing, and concave structure of f, as well
as other features such as local extrema, points of inflection, and asymptotes.
[4]
f(0) = 1
f(1) = 2
f(4) is
lim
x→∞
f(x) = 0
lim
x→−∞
f(x) = −∞
f ′(1) = 0
f ′(4) is
f ′′(4) is
Interval (−∞, 1] [1, 4) (4, ∞)
Sign of f ′(x) + – +
Interval (−∞, 4) (4, ∞)
Sign of f ′′(x) – –
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
MATH 108 – Final Exam Page 8 of 14
7. A piece of cardboard is 1 meter by 1/2 meter. A square is to be cut from each corner and the
sides folded up to make an open-top box.
(a) Define appropriate variables and state a function for the volume of the box. [2]
(b) What are the dimensions of the box with maximum possible volume? [2]
MATH 108 – Final Exam Page 9 of 14
8. A cylindrical tank standing upright (with one circular base on the ground) has radius 1 meter
and contains some water.
(a) Define appropriate variables and state a function for volume of water. [2]
(b) How fast does the water level in the tank drop when the water is being drained at 3 liters
per second? Use the fact that 1 cubic meter is equal to 1000 liters. [2]
MATH 108 – Final Exam Page 10 of 14
9. Determine the following integrals:
(a)
∫ (√
x − x3
3
√
x
)
dx [2]
(b)
∫ 2
1
(e3 − x−1)dx [2]
MATH 108 – Final Exam Page 11 of 14
10. (a) Define a substitution variable u and write
∫ 1
0
e2x−1 +
1
2x − 1
dx in terms of u. Do not
solve the integral. [2]
(b) Use the substitution u =
√
x to solve
∫
ln
√
x
x
dx. [2]
MATH 108 – Final Exam Page 12 of 14
11. (a) Setup the integral that represents the area enclosed by y = x4 and y = x. Do not solve
the integral. [2]
(b) Find the total area between y = (x2 − 1)2 and the x-axis on the interval [−3, 2]. [2]
MATH 108 – Final Exam Page 13 of 14
12. Let k = 0.0015 represent the growth constant for a certain bacteria, which grows at a rate
proportional to the amount at that time. Initially there are 500 bacteria present.
(a) Set up a differential equation and show why the number of bacteria is described by the
function y(t) = 500e0.0015t where t is measured in hours. [2]
(b) Determine how long it takes for the bacteria to grow to 2000. [2]
MATH 108 – Final Exam Page 14 of 14
Rough Work – Will not be graded