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CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Semester 2 Examinations 2009/
Module Title: Calculus 1 for Computing
Module Code: MATH 6003
School: School of Computing & Mathematics School of Science
Programme Title: Bachelor of Science (Honours) in Software Development – Year Bachelor of Science (Honours) in Software Development & Computer Networking – Year Bachelor of Science (Honours) in Instrument Engineering – Year
Programme Code: KSDEV_8_Y KDNET_8_Y SINEN_8_Y
External Examiner(s):Dr. P. Robinson Internal Examiner(s): Ms. M. Harley
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 are attempting the correct examination. If in doubt please contact an Invigilator.
1a Given the function ( ) 100(1 4 )
t y t e
, determine y (0), y (6). Draw a rough sketch of y t ( ) labeling axes appropriately Illustrate y (0)and y (6)on your sketch (7 Marks)
1b V 1 15sin(100 t )and V 2 12cos(100 t ),
State the amplitude and frequency of (^) V 2 and the frequency of ( (^) V 1 + (^) V 2 ). Draw vectors to represent (^) V 1 , (^) V 2 and ( (^) V 1 + (^) V 2 ) Hence express V 1 + V 2 in the form A sin( n t ) (7 Marks)
1c Differentiate from first principles f x ( ) 3 x^2 4 x 2
(5 Marks)
1d Evaluate x (0.3)given that x t ( ) 5cos(4 t )
(5 Marks)
1e Evaluate
1 2 0
x (^) dx x (5 Marks)
1f Evaluate the integral
2 0
dx x
(5 Marks)
1g Find the mean value of y ( x 1)(5 2 ) x in the interval [-1, 2.5]
(6 Marks)
(Total 40 Marks)
Q3a Evaluate the following integrals:
(i)
x x
dx
x
(ii)^5 1 3
dx
x
(iii)^32 1 4
dx
x
(iv) 2
^5^ xe^ x dx
(15 Marks)
Q3b Find the equation of the curve which passes through the origin if the slope of the
tangent line at any point ( , x y )is given by 2
(1 5 ) x (7 Marks)
Q3c The velocity v^ ms^1 of an object is given by 2
v t t
where t seconds is the time. Determine the distance moved by the object during the time interval t = 0.5 to t = 1.0s (8 Marks)
Q4a Show that the equation x e ^2 x 0 has a solution between x = 0 and x = 1. Taking x = 0.4 as a first approximation, use the Newton-Raphson approximation twice to determine the solution correct to 2 decimal places.
(8 Marks)
Q4b The graph below illustrates the graph of the function f x ( ) ( x 1)( x^2 6) Determine the total area enclosed between the graph and the x -axis.
(10 Marks)
Q4c For the function y x ( ) 8 x^2^ 0.5 2 x
determine y (0), lim( ( )) x 0 y x , (^) x lim ( ( )) y x , x lim ( ( )) y x (4 Marks) Determine the coordinates of the stationary points and the nature of these stationary points. (6 Marks)
Use your answers to draw a rough sketch of y x ( ). Do not attempt to plot the graph of y x ( ) accurately (2 Marks)
