Download MATH6015 Exam: Technological Maths 2, Cork Institute of Technology, Autumn 2010 and more Exams Mathematics in PDF only on Docsity!
CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Semester 1 Examinations 2009/
Module Title: Technological Maths 2
Module Code: MATH
School: School of Mechanical & Process Engineering
School of Manufacturing, Biomedical & Facilities Engineering
School of Manufacturing, Building & Civil Engineering
Programme Title:
Bachelor of Engineering in Mechanical Engineering – Year
Bachelor of Engineering in Biomedical Engineering – Year
Bachelor of Engineering in Building Services Engineering – Year
Bachelor of Engineering (Honours) in Sustainable Energy – Year
Bachelor of Engineering in Building Civil Engineering – Year
HC in industrial Measurement and Control - Year
Certificate in Process Control and Automation - Year
Programme Code:
EBIMI_7_Y
EBSEN_7_Y
EMECH_7_Y
ESENT-8-Y
CCIVL-7-Y
SIMCT-6-Y
SPRCA-6-Y
External Examiner(s): Dr Padraig Kirwan
Internal Examiner(s): Ms M Harley, Ms H Lordan , Dr J O Donovan,
Mr T O Leary, Dr S O Rourke
Instructions:
Answer Q1 (compulsory – 40 Marks) and 2 other questions
(30 Marks each)
Duration: 2 Hours
Sitting: Autumn 2010
Requirements for this examination: Mathematical Tables
Note to Candidates: Please check the Programme Title and the Module Title to ensure
that you have received the correct examination paper.
Q1(a) Differentiate from first principles ( ) 2 3 4
2 f x x x
(5 marks)
Q1(b) The displacement s meters of a mass from its starting point at any instant t
seconds is given by
2 s 12 t 2 t
Find the velocity and the acceleration of the mass after 2 seconds. (5 marks)
Q1(c) Find the slope of the tangent to the curve
x y e
2 at the point where x = 0.
Also find the equation of the tangent line. (5 marks)
Q1(d) If V 3 sin( 2 t ) 4 cos( 2 t )verify that 2 4 0
2 V dt
dV (5 marks)
Q1(e) Determine the integral ^ dx
x
x )
(^4 (5 marks)
Q1(f) Evaluate 12 cos( 3 x ) 6 sin( 3 x ) dx
6
0
^
(5 marks)
Q1(g) Evaluate .the integral
4
3
2 25
x
dx (5 marks)
Q1(h) Find the area between the curve y ( x 1 )( x 3 )and the x axis. (5 marks)
Q3(a) Evaluate the following integrals.
(i)
2
1
2
6 dx x
x
(ii) dx x x
x
5
4 (^1 )(^3 )
`
(iii)
1
0
2 4
x
xdx
(22 marks)
Q3(b) Find the mean value of the function f ( t ) 3 t 4 t
2 over the interval [0,2].
(8 marks)
Q4(a) Show that the equation
3 2 x x x
has a root in the interval [1,2].
Use x =1.0 as a first approximation and use the Newton-Raphson Method
twice to find further approximations to the solution.
(10 marks)
Q4(b) A sheet of metal is 30cm by 16cm. Equal squares of side x units
are removed from each corner. The sheet is then bent to form an
open box. Show that the volume of this box is given by
2 3 V 480 x 92 x 4 x
Find the maximum value of V.
(10 marks)
Q4(c) A body moves in a straight line and the acceleration of the body is given
by
a 12 t 14
2 m s
Initially the body is at the point P and the initial velocity of the
body was
1 8
ms.
(i) Derive expressions for the velocity and the displacement of the
body from the point P at any instant t.
(ii) Determine the distance covered by the object in the first 2
seconds and the velocity at t = 4s.
(iii) At what instants is the velocity equal to zero?
(10 marks)
