PTU Punjab Technical University B-Tech May 2019 Question Papers 5th Semester Automation Robotics Engineering (ECE-EIE)
Total No. of Pages : 03
Total No. of Questions : 09
B.Tech (Automation & Robotics) (2011 & Onwards) (Sem.?5)
NUMERICAL METHODS IN ENGINEERING
Subject Code : ME-309
M.Code : 70482
Time : 3 Hrs. Max. Marks : 60
INSTRUCTIONS TO CANDIDATES :
1.
SECTION-A is COMPULSORY consisting of TEN questions carrying T WO marks
each.
2.
SECTION-B contains FIVE questions carrying FIVE marks each and students
have to attempt ANY FOUR questions.
3.
SECTION-C contains T HREE questions carrying T EN marks each and students
have to attempt ANY T WO questions.
SECTION-A
1.
Write briefly :
i)
Write Newton's formula for interpolation.
ii)
Find the condition number of the function f (x) = sin x.
iii) Define a cubic spline interpolant with clampled boundary.
iv) Determine the Lagrange interpolating polynomial passing through the points (1, 1),
(2, 4) and (3, 9).
v)
Find the l norm of the vector (1, ?5, 9)t.
vi) Explain least square curve fitting.
2
vii) Compute
x
xe dx
using Simpson's rule.
0
viii) Use the forward-difference formula to approximate the derivative of f(x) = ln x at
x0 = 1.8 using h = 0.1.
1 | M -70482
(S2)-518
xi) What is the order of convergence when Newton Raphson's method is applied to the
equation x2 ? 6x + 9 = 0 to find its multiple root.
x)
Out of chopping of numbers and rounding off of numbers, which one introduce less
error?
SECTION-B
2.
Use forward-difference method with steps sizes h = 0.1 and k = 0.01 to approximate the
solution to the heat equation :
2
u
u
(x,t)
(x,t) 0,
0 x 1, t 0,
2
t
x
with boundary conditions
u (0, t) = v(1, t) = 0, t > 0,
and initial condition
u (x, 0) = sin( x), 0 x 1.
3.
Apply Taylor's method of order 2 with N = 10 to initial value problem
y1 = y ? t2 + 1, 0 t 2, y (0) = 0.5
4.
The following data is given
1.0
1.3
1.6
1.9
2.2
0.7651977
0.6200860
0.4554022
0.2818186
0.1103623
Use Lagrange's formula to approximate f (1,5).
5.
Use the data points (0, 1), (1, e), (2, e2) and (3, e3) to form a natural spline S(x) that
approximates f (x) = ex.
6.
Find the largest interval in which p* must lie to approximate p with relative error at most
1
10?4 for p =
3
(17) .
2 | M -70482
(S2)-518
SECTION-C
7.
Derive Secant's formula for solving the equation f (x) = 0 (specifying the assumptions
made). Use the secant method to solve the equation x = cos x starting with an initial
guesses 0.5 and
.
4
8.
Use Gauss elimination method with partial pivoting to solve the following linear system
of equations.
x
2x x x 0,
1
2
3
4
ex x x 2x 1,
1
2
3
4
x x 3x x 2,
1
2
3
4
x x x 5x 3.
1
2
3
4
9.
Determine the values of h that will ensure an approximation error of less than 10?4 when
2
approximating
2 x
e sin 3x dx
and employing :
0
a)
Composite trapezoidal rule.
b)
Composite Simpson's rule.
NOTE : Disclosure of identity by writing mobile number or making passing request on any
page of Answer sheet will lead to UMC case against the Student.
3 | M -70482
(S2)-518
This post was last modified on 04 November 2019