UNIVERSITY OF DUBLIN
TRINITY COLLEGE
Faculty of Science
school of mathematics
JF Engineering
JF MEMS
JF MSISS
Annual Examination
Trinity Term 2006
Course 1E1
Dr. M. Kurth
Answer ALL questions.
Log tables are available from the invigilators, if required.
Non-programmable calculators are permitted for this examination,—please indicate the make
and model of your calculator on each answer book used.
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1. Calculate the following limits, provided they exist. Otherwise explain why they fail to exist.
(a)
lim x3 − x2 + x − 1
x→−2
(b)
√
lim
x→2
(c)
√
2x − x + 2
x−2
(x − 1)2 + |x − 1|
x→1
x−1
lim
(d)
1
lim x cos
x→0
x
2
(e)
sin2 x
x→0 x2
lim
2. Calculate the following derivatives.
(a)
d
[cos(2x) sinh(3x)]
dx
(b)
1
d
√
dx cos x
(c)
d cos x
dx cosh x
(d)
√ d
ln x
dx
(e)
d
sin (sinh x)
dx
3. Calculate the following integrals, using substitution, integration by parts, or partial fractions
as appropriate.
(a)
Z
cosh x esinh x dx
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(b)
Z
π
sin2 x dx
0
(c)
Z
(d)
x+4
dx
(x + 1)(x − 2)
Z
x sin x dx
Z
x3
dx
x4 + 1
(e)
4. Consider the curve defined by
x2 + y 2 = 1.
(a) Use implicit differentiation to find
dy
.
dx
(b) Find the point-slope equation of the line tangent to the curve at the point ( √12 , √12 ).
(c) Find the slope-intercept equation of the line orthogonal to the curve at the point ( √12 , √12 ).
5. Hyperbolic functions
(a) Use the definition of hyperbolic functions in terms of exponential functions to show
cosh(x + y) = cosh x cosh y + sinh x sinh y.
(b) Use implicit differentiation to find
d
cosh−1 x.
dx
(c) Calculate the length of the graph of the function f defined by
f (x) = cosh x
between the points x = 0 and x = ln 5.
6. Calculate the limits of the following series, provided they exist.
(a)
∞
X
2n−1
1
3n
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(b)
∞
X
1
1
1+
n
n
1
(c)
∞
X
1
2n
3n+1 n!
7. Solids of revolution.
(a) Consider two functions f and g defined by
f (x) = x2 − 1,
g(x) = −1 − x.
Find the volume of the solid of revolution generated by revolving the area bounded by the
graphs of f and g about the x-axis.
(b) Find a function h, such that for the area A bounded by the graph of h, the y-axis, the
x-axis and the line x = 1
A=
Z
0
1
h(x) dx = ∞,
but the volume of the solid of revolution obtained by revolving that area about the y-axis is
finite.
8. Taylor series
(a) Write down the definition of the Taylor series of a function f generated at a point x0 .
(b) Find the Taylor polynomial of degree 3 for the function f defined by
f (x) = ln(x + 1).
c UNIVERSITY OF DUBLIN 2006