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INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Semester 2 Examinations 2009/
Module Title: Technological Mathematics 312
Module Code: MATH 7021
School: Building & Civil Engineering
Programme Title: Bachelor of Engineering in Civil Engineering – Year
Programme Code: CCIVL-7-Y
External Examiner(s): Internal Examiner(s): Dr.P.RobinsonMr. T. O Leary
Instructions: Select any four questions. These questions carry equal marks.
Duration: 2 Hours Sitting: Autumn 2010
Requirements for this examination:
Note to Candidates: correct examination paper. Please check the Programme Title and the Module Title to ensure that you have received the If in doubt please contact an Invigilator.
- (a) Corresponding values of x and y are given in a table attached to this examination paper. There is an error in one of the values of y. Form a forward difference table up as far as and including second differences for these values. Include the completed table with your examination script. (i) Locate and correct the error in the values of y. (ii) Extend the table to calculate the values of y at x=0 and at x=6. (iii) By using the Newton-Gregory interpolation formula estimate the value of y at x=4. (iv) By using linear interpolation estimate the value of y at x=3. (v) Estimate the value of the slope of y at x=3.4. (15 marks) (b) Variables R and T are related by a formula of the type R=aT^2 +b where a and b are constants. For the data below by using the Least Squares Method find the best values of the constants a and b. R 2.0 2.1 2.4 2.9 3. T 0 1 2 3 4 (10 marks)
- (a) Find the Inverse Laplace Transform of the expressions
(i) (^) s (^2) - 4s6s 8 (ii) (^) s (^2) - 6s^12 13 (9 marks)
(b) By using Laplace Transformations solve the differential equations (i) ddt^2 x 2 4x10et x(0)x(0) 0
(ii) ddt^2 x 2 4 dxdt 4x10et x(0)x(0) 0 (16 marks)
- There is a choice in part (b) of this question.
(a) By using the Least Squares method fit a parabola to the set of points (-2,2), (-1,1), (0,2), (1,0), (2,4) (9 marks) (b) By using Gaussian Elimination with partial pivoting solve the set of simultaneous equations (I) below. All calculations should be correct to two places of decimal. or By using Gaussian Elimination without partial pivoting solve the set of simultaneous equations (II) below
(I)
z
y
x 5 10 6
(II)
1 2 3 1 x 6 2 7 7 3 y 0 3 9 8 6 z 0 4 5 3 7 w 0
(8 marks)
(c) Consider the set of simultaneous equations 10x+y= x+10y= The solution of this set of equations is close to x=0.1, y=0.2. Use two iteration of Jacobis Method and two iterations of the Gauss Siedel Method to find correct to two places of decimal further approximations to the solution of this set of equations. (8 marks)
DERIVATIVES
f(x) a=constant n f (x) lnx^ x^ nx^ n^1 x
eax a eax cosx^ sinx^ -sinxcosx uv dx vdu dx u dv v
u v^2 dx
udv dx v du
INTEGRALS
f(x) a=constant f(x)dx
xn xn+1 (^) if n - n+1 x
1 lnx eax^1 sinx a-cosx^ a^ eax cosx sinx INTERPOLATION (^1 20 0 2 1 0 ) 0 1 0 2 1 0 1 2 2 0 2 1 f(x) = (x - x )(x - x ) f(x ) + (x - x )(x - x )^ f(x ) + (x - x )(x - x ) f(x ) (x - x )(x - x ) (x - x )(x - x ) (x - x )(x - x ) f(x 0 rh)f(x 0 )rf 0 r(r2!-1)^2 f 0 ...
y (x 0 rh) (^1) h^ ^ y 0 2r-12! ^2 y 0 f (x^0 rh)h^1 ^ rf 0 (2r2!-1)^2 f 0 ...
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