UNIVERSITY OF REGINA
Department of Mathematics and Statistics
Mathematics 110 (Calculus I)
Final Exam
8 December 2017
Time: 3 hours
Full Name:
Pages: 10
Student Number:
Instructors:
Alison Purdy (001)
Chun-Hua Guo (002)
Fernando Szechtman (003)
Gurmail Singh (004)
Vijayaparvathy Agasthian (C01,C02)
Shuchita Sharma (L01, L02, L03)
INSTRUCTIONS
1. Full credit is awarded only for well-presented, correct solutions in which all of your work
is shown. The marks allocated for each question are found to the left of the question.
2. Your solution and final answer should appear on the right side of the pages in this exam
booklet. If you require more space for your solution than the right side allows, then
use the left side (facing) page and indicate on your solution that there is work to be
found on the facing page.
3. The left side pages are to be used as scrap paper and are provided for rough work only;
these pages will not be graded unless you expressly indicate that there is work to be
found there.
4. The use of calculators, tablets, phones, formula sheets, or additional paper is not permitted.
For instructor use only:
Page:
2
3
4
5
6
7
8
9
10
Total
Marks:
12
10
12
12
7
10
12
16
9
100
Score:
MATH 110 Final Exam
Page 2
Marks
(12)
1. Evaluate the following limits.
x2 − x
.
x→1 x2 − 4x + 3
(a) lim
√
(b) lim ( x2 + x − x).
x→∞
(c) lim x2 cos
x→0
(d) lim+
t→0
1
. (Use the Squeeze Theorem)
x3
2t + sin t
.
t
Page 2
Fall 2017
MATH 110 Final Exam
(5)
Page 3
Fall 2017
2. Find all constants c such that the function
{ 2
3x + cx, x ≤ 1
f (x) =
x + c2 ,
x>1
is continuous on (−∞, ∞).
(5)
√
3. Find the derivative of f (x) = x + 3 at x = 1 using the limit definition of derivative.
(No marks will be given if you do not use the limit definition.)
Page 3
MATH 110 Final Exam
(12)
Page 4
4. Find f ′ (x) of the following: (Do not simplify)
√
9
x
9
(a) f (x) = x +
+ tan x.
2
(x + π)4
.
(b) f (x) = 4
x + π4
(c) f (x) = x cos x −
sin x
.
x
∫ x √
(d) f (x) =
5 1 + t2 dt.
4
Page 4
Fall 2017
MATH 110 Final Exam
Page 5
Fall 2017
(5)
5. Find the equation of the tangent line to the curve: (x2 + y 2 )2 = 2(x3 + y 2 ) at the point
(1, 1). (Use implicit differentiation)
(7)
6. Find the absolute maximum and minimum values of f (x) = x3 − 3x2 + 6 on the interval
[−2, 2].
Page 5
MATH 110 Final Exam
(7)
Page 6
Fall 2017
7. A ladder 10 m long is leaning against a vertical wall with its other end on the ground.
The top end of the ladder is sliding down the wall. When the top end is 6 m from the
ground it is sliding down at 2 m/sec. How fast is the bottom moving away from the wall
at this instant?
Page 6
MATH 110 Final Exam
(10)
Page 7
Fall 2017
8. A farmer wants to fence in a rectangular field beside a river. No fencing is required
along the river and the farmer’s neighbour will pay half of the cost of one of the sides
perpendicular to the river. If fencing costs $20 per linear meter and the field must have
an area of 600 m2 , what are the dimensions of the field that will minimize the cost to
the farmer?
Page 7
MATH 110 Final Exam
(12)
Page 8
x2 − 4
6x
6(3x2 + 1)
′
′′
.
Then
f
(x)
=
and
f
(x)
=
−
.
x2 − 1
(x2 − 1)2
(x2 − 1)3
Determine the following features of f
9. Let f (x) =
(a) domain;
(b) even, odd, neither;
(c) intercepts;
(d) asymptotes (show the relevant limits);
(e) intervals of increase and decrease;
(f) local maxima and minima;
(g) intervals of upward and downward concavity;
(h) points of inflection;
(i) use parts (a) through (h) to sketch the graph of f .
Page 8
Fall 2017
MATH 110 Final Exam
(16)
Page 9
Fall 2017
10. Integrals
(a) Use properties of integrals to verify the inequality without evaluating the integral.
∫ 1√
√
2≤
1 + x2 dx ≤ 2 2.
−1
∫ 4
(b) Find
√
(4 − x) x dx.
0
∫ π/6
(c) Find
0
sin x
dx.
(cos x)2
∫
(d) Find
(sec x)2 (tan x)2 dx.
Page 9
MATH 110 Final Exam
(9)
Page 10
Fall 2017
11. Find the area of the region enclosed between the curves y = 12 − x2 and y = x2 − 6.
Include a sketch of the relevant region as part of your solution.
Page 10