this is a b.tech assinment

1. INDERPRASTHA ENGINEERING COLLEGE, GHAZIABAD
Assignments No. 4
Subject Name ENGINEERING MATHEMATICS-
III
Subject Code NAS-301
Q1 (a) Find
𝑑𝑦
𝑑𝑥
at x = 6 from following table:
x 0 2 3 4 7 8
y 4 26 58 112 466 922
(b) Find
𝑑𝑦
𝑑𝑥
and
𝑑2
𝑦
𝑑 𝑥2 at x = 1.1, 1 , 1.2 ,1.4, 1.8 , 2.0
x 1.0 1.2 1.4 1.6 1.8 2.0
y 0 0.128 0.554 1.296 2.432 4.0
(c) For what value of x , y is minimum from the following table
x 3 4 5 6 7 8
y 0.205 0.240 0.259 0.262 0.250 0.224
Q2 (a) Estimate the error term in trapezoidal Rule and Simpson
1
3
Rule.
(b) Explain about Numerical Differentiation and Numerical integration (Newton cotes
formulas for integration: Trapezoidal rule Simpson 1/3 rule, Simpson 3/8 rule)
(c) Evaluate ∫ sin 𝑥 𝑑𝑥
𝜋
2
0
using simpson
3
8
Rule take h =
𝜋
18
(d) Evaluate ∫ x . sin 𝑥 𝑑𝑥
𝜋
0
using Trapezoidal Rule
(e) Evaluate ∫ √1 − 0.162 𝑠𝑖𝑛2 𝑥
𝜋
2
0
𝑑𝑥 using simpson
1
3
Rule by taking no. of intervals = 6
Q3 (a) Use R-K 4th order method to find y(0.2) from differential equation
𝑑𝑦
𝑑𝑥
= x + y2 ,given
y(0) = 1 ,take h = 0.1
(b) Solve
𝑑𝑦
𝑑𝑥
= x + z ,
𝑑𝑧
𝑑𝑥
= x - y2 to find y(0.1) and z(0.1) by Runge –Kutta Method ,given
that y(0) =2, z(0) = 1
………………………………………………………………………………………………………
Answers
………………………………………………………………………………………………………
1(a) 125.4744 (b) y ' (1.1) = 0.66724 , y″(1.1) = 8.13125 (c) 5.6875,0.2628
2(c) 0.974353 (d) 3.14 (e) 1.505103
3(a) y(0.1) = 1.116492, y(0.2) = 1.273562 (b) y(0.1) = 2.0851667
z(0.1) = 0.583745