MA211: AdvancedCalculus
Faculty of Science, Technology and Environment
School of Computing, Information and Mathematical Sciences
Final Examination
Semester I, 2017
Mode: Face-to-Face and Online
DurationofExam: 3hours+10minutes
Reading Time: 10 minutes
Writing Time: 3 hours
Total Marks: 100
INSTRUCTIONS:
1. There are 6 questions and ALL are compulsory.
2. Start each question on a new page.
3. There are seven (7) pages (including this cover page) in this exam paper.
4. Show all necessary working. Partial marks will be awarded for partially
correct answers. Write legibly.
5. Only non-programmable calculators are allowed.
6. A list of formulae is attached at the end.
7. This exam is worth 50% of your overall mark. To be considered for a clear
pass in this course, you must achieve at least 40 out of 100 in this final ex-
amination.

MA211 Final Examination Semester I, 2017
Question1: Startonanewpage. (
4
+
6
+
3
=
13
marks)
(a) Show that if
A
and
B
are not both zero, then the graph of the polar equation
r
=
A
sin
θ
+
B
cos
θ
is a circle. Find its radius.
[4 marks]
(b) Find the area of the region outside the cardioid
r
=
2
−
2 cos
θ
and inside
the circle
r
=
4.
[6 marks]
(c) Find parametric equations for the line through the point
(
x
1
,
y
1
,
z
1
)
and
parallel to the line
x
=
x
0
+
at
,
y
=
y
0
+
bt
,
z
=
z
0
+
ct
.
[3 marks]
Question2: Startonanewpage. (
3
+
4
+
6
=
13
marks)
(a) Sketch the graph of
f
(
x
,
y
)=
4
−
x
2
−
y
2
.
[3 marks]
(b) Evaluate
lim
(
x
,
y
)
→
(
0,0
)
sin
(
x
2
+
y
2
)
x
2
+
y
2
.
[4 marks]
(c) Given the following information use the Chain Rule to sketch the tree dia-
gram and find
∂
z
∂
u
and
∂
z
∂
v
.
z
=
8
x
2
y
−
2
x
+
3
y
,
x
=
uv
,
y
=
u
−
v
[6 marks]
Page 2 of 7

MA211 Final Examination Semester I, 2017
Question3: Startonanewpage. (
6
+
6
+
6
=
19
marks)