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CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Semester 2 Examinations 2008/
Module Title: Calculus 1 for Computing
Module Code: MATH 6003
School: School of Computing & Mathematics
Programme Title:
Bachelor of Science (Honours) in Software Development – Year
Bachelor of Science (Honours) in Software Development & Computer Networking –
Year
Programme Code:
KSDEV_8_Y
KDNET_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: Summer 2008
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 exaination. If in doubt please contact an Invigilator.
1a Given the functions y 1 (^) ( ) t = 20 e −^300 t and y 2 (^) ( ) t = 12(1 − e −^300 t )where t is time.
State the initial and final values of y 1 and y 2. Evaluate y 2 (0.006 ) and determine the time t when y 1 (^) ( ) t = y 2 (^) ( ) t (6 Marks)
1b State the amplitude, period and frequency of the waveforms v 1 and v 2 below.
State the phase time of v 2 and calculate the phase angle. Hence determine expressions for v 1 and v 2 Determine v 2 when t = 0.004, using (i) the expression above and (ii) the graph below
(7 Marks)
1c Find the slope of the tangent line to the function y =
x
at the point
where x = 0.5. Hence find the equation of the tangent line.
(5 Marks)
1d Differentiate from first principles f ( ) x = x^2 + 4 x + 10
(5 Marks)
1e Given that y = 2.2cos(5 π t ) + 3.6sin(5 π t ), determine the rate if change of y
at t = 30 × 10 −^3
(6 Marks)
1f Evaluate
0 2
dt − t
∫
(5 Marks)
1g Determine the total area enclosed between the curve y = x^2^ + 3 x − 4
and the x -axis. Comment on your answer. (6 Marks)
(Total 40 Marks)
v 1
v 2
Q4a Show that the equation e −^2 x = 4 x has a solution in the interval [0, 1]
Determine x to 2 d.p. using the Newton-Raphson approximation twice
(10 Marks)
Q4b A rectangular sheet of metal having dimensions 16cm by 6cm 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^ − 44 x^2 + 96 x. Determine the maximum possible volume of the box. (10 Marks)
Q4c Determine where the function y = ( x^2 − 1)( x − 3)= x^3 − 3 x^2 − x + 3
crosses the x -axis. Draw a rough sketch of the graph of the function Determine the total area enclosed between the graph and the x -axis. (10 Marks)
