King’s College London
University Of London
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Midsessional Examination
4CCM111A Calculus I
January 2012
Time Allowed: Two Hours
This paper consists of two sections, Section A and Section B.
Section A contributes half the total marks for the paper.
All questions in Section B carry equal marks.
Answer all questions.
NO calculators are permitted.
DO NOT REMOVE THIS PAPER
FROM THE EXAMINATION ROOM
TURN OVER WHEN INSTRUCTED
c
2012 King’s
College London
-2-
4CCM111A
In section A there are 5 marks for a correct answer, -1 for a wrong
answer, 0 for a blank answer. Put the correct letter in the box.
A 1.
A 2.
The complex number 1/(3 − 2i) is equal to:
√
A. (3 + 2i)/ 13
B. (3 + 2i)/13
√
D. (3 − 2i)/13
E. 3 + 2i/ 13
A 6.
A 7.
E. 2π − π/5
The ratio (e2x−y − 1)/(e2x−y + 1) is equal to:
A. sinh(2x − y)
B. cosh(x − y)
D. cosh(2x − y)
A 5.
E. iπ/5
F. None of these
Which one of the following angles equals arccos(cos(−π/5)) ?
A. −π/5
B. π/5
C. π − π/5
D. −π/5 − π
A 4.
F. None of these
The argument arg(z) of the complex number z = 2eiπ/5 equals:
A. −π/5
B. π/5
C. 2
D. 2e
A 3.
√
C. (3 − 2i)/ 13
E. sech(x − 21 y)
F. None of these
C. tanh(x − 21 y)
F. None of these
The quantity sin(ix) is equal to:
A. i sinh(ix)
B. i sinh(x)
C. −i sinh(ix)
D. sinh(ix)
F. None of these
E. sinh(x)
The limit limh→0 (3h − 1)/h is equal to:
A. 1
B. 3
C. ln(1)
D. ln(3)
F. None of these
E. 0
The limit limx→∞ tanh( 32 x)/x is equal to:
A. 3/2
B. 0
C. limit does not exist
D. 2/3
F. None of these
E. 3π/2
See Next Page
-3-
A 8.
1 2
If f (x) = e 2 x , then f 0 (x) is equal to:
A. ex
B. 12 x2 ex
1 2
E. e 2 x
R
1
3
tanh(3)
The integral
A. 1
The integral
A. (2n)!/n!
R √2π
0
x sin(x2 )dx is equal to:
B. 0
R ∞ n −x
x e dx
0
F. None of these
is equal to:
B. (2n)!
C. n!
E. (n + 1)!
F. None of these
E. 2π
The radius of convergence of the series
A. 3/2
B. e
D. 1
A 14.
C. integral does not exist
The volume enclosed by rotating the segment of the curve y = x(2 − x)
between x = 0 and x = 2 about the x-axis is equal to:
A. π
B. 2π/9
C. π/2
D. 2π/7
A 13.
F. None of these
E. 2π
D. (n − 1)!
A 12.
C. tanh(3)
E. sinh(3)
D. π
A 11.
F. None of these
The integral 01 cosh(3x)dx is equal to:
A. 16 (e3 + e−3 ) − 13
B. 61 (e3 − e−3 )
D.
A 10.
1 2
C. xe 2 x
1 2
D. 12 x2 e 2 x
A 9.
4CCM111A
F. None of these
P
n
xn /n! is equal to:
C. 1/e
E. 23/2
F. None of these
The first three terms in the Taylor expansion of (1 + x)1/3 about x = 0 are:
A. 1 + 3x − 19 x2
B. 1 + 13 x + 29 x2
C. 1 − 31 x + 29 x2
D. 1 + 31 x − 92 x2
E. 1 − 3x + 19 x2
F. None of these
See Next Page
-4-
4CCM111A
In section B you must provide FULL DERIVATIONS of your solutions to gain marks. Each question carries a maximum of 14 marks.
B 15.
Use the induction method to prove the following identity:
n
X
1
(2k − 1)2 = n(4n2 − 1)
3
k=1
(∀n ∈ ZZ, n > 0) :
B 16.
Give the series representations of the two trigonometric functions sin(x) and
cos(x). Use these to derive the following properties for x ∈ IR (you are allowed
to interchange differentiation and summation without justification):
d
sin(x) = cos(x),
dx
d
cos(x) = − sin(x),
dx
B 17.
sin(0) = 0
cos(0) = 1
Calculate the following three limits:
sin(3x)
(a) lim
x→0 tan(2x)
cos(x)
(b) lim
x→∞
x2
ax −bx
(c) lim
(a > b > 0)
x→0
x
You may use (if you wish), without proof, the following three elementary limits:
sin(h)
=1
h→0
h
lim
B 18.
cos(h) − 1
=0
h→0
h
lim
Each of the following is an equation which determines y as an implicit function
of x. Find in all cases an expression for dy/dx.
(a) y 4 + x4 = 1
B 19.
eh − 1
=1
h→0
h
lim
(b) 3xy − ex+y = 2
(c) 2y 2 + x(x − 1)(x + 1) = 0
Find the radii of convergence for the following three powers series:
X
X
X
√
(a)
xn /n n
(b)
(−1)n xn / ln(1+ n)
(c)
xn n2 e−n
n
n
n
Final Page