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Cork Institute of Technology
Bachelor of Engineering in Biomedical Engineering-Stage 2
Autumn 2009
(EBIME_7_Y2)
Mathematics & Computing -Legacy
(Time: 3 Hours)
Instructions Answer FOUR questions from Section A and ONE question from Section B. All questions carry equal marks.
Examiners: Dr. P. Robinson Ms.H.Lordan. Ms. G. O’Callaghan.
Section A
1 (a) The parametric equations are defined as x = t^2 and y = t^5. Find the value of dx
dy and (^2)
2
dx
d y
at the point (4, -32). (6 marks)
(b) Determine dx
dy : 2 y^3 − y + 3 x − 2 = 0.
(6 marks)
(c) Show that the function f ( x )= x^3 − 4 x − 2 has a root between x = 2 and x = 3. Use Newton’s method twice to get a closer approximation to this root. (8 marks)
2 (a) Investigate the function 3
( ) x x
f x = + for maximum and minimum values. Sketch the
curve. What happens as x →∞? (8 marks)
(b) Given z = 4 x^2 y^3 − 2 x^3 + 7 y^2 , find (^2)
2 , , x
z y
z x
z ∂
(6 marks) (c) The resonant frequency f (^) 0 of a series connected circuit containing inductance L and
capacitance C is given by LC
f
0 =^. Determine the approximate percentage error in
measuring f (^) 0 if errors of ± 1 % and ± 2 %are possible in the measurement of L , C respectively. (6 marks)
3 Determine any three of the following:
(a) dx
∫ x −
2
1
( 2 1 )^3
(7 marks)
(b) ∫ x xdx
6
0
sin^3 cos
π
(7 marks)
(c) ∫
3
0 9 2
dx x (7 marks)
(d) ∫ 5 xe −^4 xdx
(7 marks)
4 (a) A solid of revolution is formed by rotating the curve with the equation x
y x
= + between
the limits x = 2 and x = 4 about the x −axis. Find (i) the volume of revolution formed and (ii) the co-ordinates of the centre of gravity. (14 marks)
(b) The velocity, v , (in metres per second) of an object varies with time t, according to the equation: v^ =^40 (^1 − e −^0.^05 t )(metres per second). Calculate the distance travelled by the object between t = 0 and t = 30 seconds. (6 marks)
5 (a) Sketch the curve f^ (^ x )=^ x^2 +^1 and g (^^ x )=^ x +^3. Determine the area enclosed by them.
(7 marks)
(b) Determine the root mean square of the function x x between the ordinates x = 1 and x = 4. (6 marks)
(c) Calculate the centroid of the area enclosed by the curve y = e − x , the x - axis and the ordinates x = -1 and x = 1. (7 marks)
(b) Write a C++ function that receives the array and the size of the array as parameters and calculates the lowest result entered along with the highest result entered. You should also display your results to the screen. (9 marks)
(c) Write a C++ function that receives the array and the size of the array as parameters and calculates the average result achieved by all students. You should also display your results to the screen. (6 marks)
ABC Haulage Ltd is concerned with the mileage obtained by their trucks. They have asked one driver to keep track of all tanks of diesel used by recording miles driven and litres used for each tank full. Develop a C++ program that will input miles driven and litres used for each tank of diesel. The program should calculate and display the miles per litre obtained for each tank full of diesel. After processing all input information, the program should calculate and print the average miles per litre obtained for all tanks of diesel.
Sample Output: Enter the litres used (-1 to quit): 28. Enter the miles driven: 120 The miles/litres for this tank was 4.
Enter the litres used (-1 to quit): 40. Enter the miles driven: 300 The miles/litres for this tank was 7.
Enter the litres used (-1 to quit): -
The overall average miles/litres was 5. (20 marks)
