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CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Autumn 2009
Module Title:Technological Mathematics 2
Module Code: Math 6015
School: Building and Civil Engineering
Mechanical & Process Engineering
Manufacturing, Biomedical & Facilities Engineering
Programme Title: Bachelor of Engineering in Civil Engineering –Year 1
Bachelor of Engineering in Mechanical Engineering –Year 2
Bachelor of Engineering in Biomedical Engineering –Year 2
Bachelor of Engineering in Building Service Engineering –Year 2
Programme Code: CCIVL_7_Y
EBIME_7_Y
EBSEN_7_Y
EMECH_7_Y
External Examiner(s): Dr. P. Robinson
Internal Examiner(s): D.Cremin, M. Harley, H.Lordan, G. O’Driscoll
Instructions: Answer Question 1 and Two other questions.
Duration: 2 HOURS
Sitting: Autumn 2009
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.
If in doubt please contact an Invigilator.
Q1. Answer each of the following:
(i) Differentiate by rule:
1 5 5 2 3
y 5 x 3 x 3 x 12 x
− = − + − + (5 marks)
(ii) Differentiate by rule:
2 3 tan 2 10 ln 3 cos 2
x x y x x e x
− = + − − + (5 marks)
(iii) The distance s metres moved by an object in a time t seconds is given by the equation:
3 2 s = 2 t + 5 t + 3 t + 8
Write down expressions for the velocity and acceleration.
Find the velocity after 4 seconds and the acceleration after 2 seconds. (5 marks)
(iv) Given y = 2sin 3 x + 3cos 3 x write down values for
2
2 and
dy d y
dx dx
If
2
2
d y ky dx
= find the value of k. (5 marks)
(v) Evaluate the integral:
3 2
2
( x − 4 x +6) dx
(5 marks)
(vi) Evaluate the integral:
2
1
(2sin x −3cos x dx )
(5 marks)
(vii) A vehicle has an acceleration a of 2
m t s
If the vehicle starts from rest find its velocity after 8 seconds. (5 marks)
(viii) Find the particular solution of the following differential equation :
2 3cos 0
dr
d
θ = (5 marks)
- (a) Solve the differential equation
2 (1 ) 2
dy x x dx
given that x = 0 when y = 2. (10 marks)
(b) A rectangular sheet of metal having dimensions 30cm by 22cm has equal squares
of side x cm removed from each of the four corners and the sides bent upward to form
an open box. Show that the volume of the box is given by
3 2 4 x − 104 x + 660 x
and use differentiation to find the maximum volume of the box. (12 marks)
(c) Calculate the mean value of the function
2 y = 2 x + 5 x + 8
in the region x = 4 to x = 7. (8 marks)
