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CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Semester 1 Examinations 2008/
Module Title: Engineering Mathematics 201
Module Code: MATH
School: Mechanical & Process Engineering
Programme Title: Bachelor of Engineering (Honours) in Mechanical Engineering-Stage 2
Programme Code: EMECH_8_Y2 EBIOM_8_Y
External Examiner(s): Dr. P. Robinson Internal Examiner(s): Mr. T. O Leary
Instructions: Select any four questions. The questions carry equal marks.
Duration: 2 Hours
Sitting: Winter 2008
Requirements for this examination:
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.
- (a) State the Mean Value Theorem for derivatives. Verify that over the interval [0,2] the function f(x)=x^4 -6x 2 +8x satisfies the criteria of the theorem. Find correct to two places of decimal a value of x that satisfies the conclusion of the theorem. This value is close to x=1.50. (10 marks)
(b) Write down three terms of a Taylor Series expansion of f(x) about x=a. Include a remainder and show that f(a h) f(a - h) 2h f (a)^ O(h )
Briefly explain why the approximation above is more accurate than the approximation f(a + h)2h−f(a)=f′(a)+O(h).
Using the values below estimate the value of f ′(1). f(x) 1.45 1.47 1. x 0.99 1.00 1. (8 marks)
(c) In constructing a closed rectangular tank with a square base it costs €6m-2^ to construct the base and €2m-2^ to construct the other sides. By using a Lagrangian Multiplier find the dimensions of the tank of volume 16m^3 that can be constructed if the cost of constructing the box is to be minimised. (7 marks)
- In answering the following question you are required to use the Method of Undetermined Coefficients. No marks will be awarded if any other method is used.
A mass is attached to a spring and a dashpot. The displacement x of the mass at any instant t is found by solving the differential equation m ddtx 2 +cdxdt kx=f(t) x(0) x(0)= 0
2
(a) Solve this differential equation where m=1, c=6, k=8 and f(t)=24. (7 marks)
(b) Find the general solution of this differential equation where m=1, c=2, k=10, f(t)=50t (8 marks)
(c) Find the general solution of this differential equation where m=1, c=2, k=1, f(t)=10e -t^ (10 marks)
- (a) Show that (6x^2 -4y)dx+(2y-4x)dy is the total derivative of a function. Find this function and solve the exact differential equation. (6x^2 -4y)dx+(2y-4x)dy=0 y(1)=0 (5 marks)
(b) Select any two parts of the following:
(i) Find the Fourier Series for the periodic function below f(t) t if 0^ t^ π f(t+2π)=f(t) 2 π-t if π t 2 π
= ^ ≤^ ≤
^ ≤^ ≤
Note: ∫xsin ( nx)dx =-xcosn(^ nx)^ + sinn(^2 nx )^ ∫xcos( nx)dx =xsinn(^ nx)^ +cosn( 2 nx)
(9 marks)
(ii) The current i in a circuit at any instant t is found by solving the differential equation
dtdi^ +4i=^10 0cos3t i(0)=^0 Solve this differential equation. Express the steady state current as a single function of the form Rcos( 3 t-α). Write down the maximum and minimum values of this function. Find the smallest positive values of t for which these extreme values hold. (11 marks)
(iii) Solve for x and for y where dx =3x+y x(0)= dt dy =-x+y y(0)= dt (10 marks)
