The University of the South Pacific
Serving the Cook Islands, Fiji, Kiribati, Marshall Islands, Nauru, Niue, Samoa, Solomon Islands, Tokelau, Tonga, and Vanuatu
School of Computing, Information and Mathematical Sciences
Division of Mathematics
MA112 Calculus II
Mode: OC Face–to–Face
Final Examination
Semester 2, 2010
Time Allowed:
Total Marks:
3 Hours + 10 Minutes Reading
100
Instructions:
1. Attempt ALL FOUR (4) Questions.
2. There are 5 pages (including this cover page).
3. Start each question on a new page.
4. Marks for each question are shown.
5. Show all working as partly correct answers may be rewarded. Write legibly.
6. Approved NON–PROGRAMMABLE calculators may be used.
7. This exam is worth 55% of your overall mark. To be considered for a clear pass in this
course, you must achieve at least 22 out of 55 in this final examination.
Turn Over/...☞
Semester 2, 2010
MA112 Final Examination
Page 2 of 5
QUESTION 1.
Start this question on a new page
[25 Marks]
(a) A 13 ft ladder is leaning against a wall. If the top of the ladder is slips along the wall at a
rate of 2 ft/s, how fast will the foot of the ladder be moving away from the wall when the
top is 5 ft above the ground?
[5 marks]
(b) Use an appropriate local linear approximation to estimate the value of tan−1 (0.99).
[4 marks]
(c) Use the right endpoint approximation method to find the area under the curve f (x) = 9−x2
over the interval [0, 3].
Note:
n
X
k2 =
k=1
n(2n + 1)(n + 1)
6
[7 marks]
(d) Evaluate the following.
tet
(i) lim
t→0 1 − et
x
(ii) lim (e + x)
x→0
1/x
d
(iii)
dx
Z x
sin−1 t dt
0.5
[3+4+2=9 marks]
Turn Over/...☞
Semester 2, 2010
MA112 Final Examination
Page 3 of 5
QUESTION 2.
Start this question on a new page
[20 Marks]
(a) Find the area of the region enclosed by the curves x2 = y and x = y − 2.
[2 marks]
(b) Use cylindrical shells to find the volume of the solid that results when the region enclosed
by y = 2x − 1, y = −2x + 3, and x = 2 is revolved about the y–axis.
[5 marks]
(c) Find the volume of the solid generated when the region between the graphs of the equations
y = x2 and x = y 2 is revolved about the y–axis.
[4 marks]
1
(d) Find the exact arc length of the curve x = (y 2 + 2)3/2 from y = 0 to y = 1.
3
[5 marks]
(e) Find the area of the surface generated by revolving the curve x =
−2 ≤ y ≤ 2 about the y–axis.
√
9 − x2 over the interval
[4 marks]
Turn Over/...☞
Semester 2, 2010
MA112 Final Examination
Page 4 of 5
QUESTION 3.
Start this question on a new page
[29 Marks]
(a) Let f be a function whose second derivative is continuous on [−1, 1]. Show that
Z 1
xf ′′ (x) dx = f ′ (1) + f ′ (−1) − f (1) + f (−1).
−1
[3 marks]
(b) Evaluate the following integrals.
Z
(i)
tan5 x sec x dx
[Hint: tan5 x = tan4 x tan x]
(ii)
Z
dx
(4 + x2 )2
(iii)
Z
dx
x2 − a2
,
a 6= 0
[4+4+4=12 marks]
(c) (i) Describe the domain and graph of f (x, y) =
(ii) Show that
x2 − y 2
does not exist.
(x,y)→(0,0) x2 + y 2
p
9 − x2 − y 2 .
lim
[3+3=6 marks]
(d) How large should we take n in order to guarantee that the Simpson’s Rule approximation
for
Z 3
1
dx
1 x
is accurate to within 0.0001?
[4 marks]
(e) Show that
Z ∞
1
dx = π.
2
−∞ 1 + x
[4 marks]
Turn Over/...☞
Semester 2, 2010
MA112 Final Examination
Page 5 of 5
QUESTION 4.
Start this question on a new page
[26 Marks]
(a) (i) Calculate the limit of the sequence
ln n
n
∞
.
n=1
(ii) Consider the sequence {an }∞
n=1 . Prove that if lim |an | = 0, then lim an = 0.
n→∞
n→∞
[2+3=5 marks]
(b) Show that the sequence
n
2
n +1
∞
is decreasing.
n=1
[5 marks]
(c) Determine whether each of the following series is convergent or divergent. You must clearly
state the test and any assumption(s) being used in each case.
√
∞
∞
∞
3
X
X
X
5
5−2 n
nn
(ii)
(iii)
(i)
(−1) n
3
2n2 + 4n + 3
n3
n=1
n=1
n=1
[3+3+4=10 marks]
(d) Find the radius of convergence and interval of convergence of the power series
∞
X
n(x + 2)n
n=0
3n+1
.
[6 marks]
***End of Final Examination Questions***
.../The End ! ©