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CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Semester 1 Examinations 2008/
Module Title: Technological Maths 2
Module Code: MATH
School: School of Mechanical & Process Engineering
School of Manufacturing, Biomedical & Facilities 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
Programme Code:
EBIME_7_Y
EBSEN_7_Y
EMECH_7_Y
External Examiner(s): Dr. Brendan O’Regan
Internal Examiner(s): Ms M Harley, Mr Gerard O Driscoll
Instructions: Answer Q1 (compulsory – 40 Marks) and 2 other questions
(30 Marks each)
Duration: 2 Hours
Sitting: Winter 2008
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.
Q1a Differentiate from first principles f ( ) x = 2 x^2 − 5 x + 8.
(5 marks)
Q1b If y = 5 e −^4 t^ − 3 e −^2 t , show that
2 2 6 8 0
d y dy y dt dt
(5 marks)
Q1c Find the slope of the tangent line to the function y = 3 x^2 − 8 x + 1 at the point
where x = 2. Hence find the equation of the tangent line. (5 marks)
Q1d The surface area A of a closed cylindrical container is given
by 2
A 2 r r
= π + where r is the radius of the tank.
Determine the radius which will minimise this surface area. (5 marks)
Q1e Evaluate ∫ +
5
2
10 sin( 3 t ) 4 cos( 2 t ) dt.
(5 marks)
Q1f Evaluate dx x x
x
1
3
2
( 3 )
(5 marks)
Q1g Determine the total shaded area in the diagram below given that the equation of the
graph is y = x^2 − 4 x + 3.
(5 marks)
Q1h The acceleration of an object is given by 2 t − 5 ms -2where t (s) is the time. Initially the object is 2 m from a point P and has a velocity of 4 ms -1. Find expressions for the object’s velocity and displacement from P at any time t.
Q4a Show that the equation x^2 − e −^3 x = 0 has a solution between x = 0 and x = 1. Use the Newton-Raphson method twice to solve the equation correct to 2 decimal places. (10 marks) Q4b A rectangular sheet of metal having dimensions 18 cm by 8 cm has squares of side x cm removed from each of the four corners and the sides bent upwards to form an open box. Show that the volume of the box is given by V = 4 x^3 − 52 x^2 + 144 x. Determine the maximum possible volume of the box. (10 marks)
Q4c The force F (kN) acting on an object is given by (^2)
F
x
where x cm is the
displacement of the object from a point P. A and B are points 1.5 cm and 5 cm from P respectively_._
Determine (i) the mean force between P and A (ii) the work done by the force in moving the object from A to B stating the units.
