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Description
SESSION | april 2024 |
PROGRAM | Bachelorof CoMPUTER APPLICATIONS (BCA) |
SEMESTER | III |
course CODE & NAME | DCA2101 – Computer Oriented Numerical Methods |
SET-I
- Show that
(a) δμ= 1/2(∆+∇)
(b) ∆-∇=∆∇
Ans 1.
- a) To show that δμ=1/2(∆+∇), we can start with the definition of the Laplacian (δ) and the gradient (∇):
δf = ∇²f = ∇•∇f
Now, let’s consider the Laplacian of a function μ:
δμ = ∇²μ = ∇•∇μ
Next, we can use the identity that relates the Laplacian and the gradient:
∇•(∇μ) = ∇²μ Its Half solved only
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- Find Lagrange’s interpolation polynomial fitting the pointsy(1) = -3,y(3)= 0,y(4)= 30, y(6) = 132 Hence find y(5).
Ans 2.
Step 1: Find the Lagrange basis polynomials
For each point, we need to create a Lagrange basis polynomial that has a value of 1 at that point and a value of 0 at all other points. We’ll use these polynomials to build the final interpolation polynomial.
For example, for the point (1, -3), the Lagrange basis polynomial is:
L1(x) = (x – 3)(x – 4)(x – 6) / ((1 – 3)(1 – 4)(1 – 6))
Simplifying this expression, we get:
L1(x) = (x – 3)(x – 4)(x
- Evaluate , given the following table of values:
10 | 20 | 30 | 40 | 50 | |
46 | 66 | 81 | 93 | 101 |
Ans 3.
To evaluate f(15) given the table of values, we need to interpolate the value of y corresponding to x = 15.
Looking at the table, we can observe that the values of x are evenly spaced by 10 units. To interpolate, we can use linear interpolation.
First, let’s find the interval in which 15 lies:
– x = 10 corresponds to y = 46
– x = 20
SET-II
- Find the equation of the best fitting straight line for the data:
X | 1 | 3 | 4 | 6 | 8 | 9 | 11 | 14 |
Y | 1 | 2 | 4 | 4 | 5 | 7 | 8 | 9 |
Ans 4.
Linear Regression is a method to model the relationship between two continuous variables, X and Y. In this case, we have a set of data points (X, Y) and we want to find the best straight line that fits these data points.
To find the
- Calculate the intercept (b): Finally, calculate the intercept (b) using the formula:
b = μY – m * μX
Substituting the values, we get:
b = 5 – m * 7
= 5 – approximately m * 7
= approximately b ≈ 5 – approximately m * 7
= approximately b ≈ 5 – approximately (32.00 * 7)
= approximately b ≈ -15
- For what value of λ & μ the following system of equations:
x + y + z = 6
x +2y+3z =10
x+2y +λz =μ may have
(i) Unique solution
(ii) Infinite number of solutions
(iii) No solution
Ans 5.
To determine the values of λ and μ that result in each type of solution, we can use the properties of systems of equations.
(i) For a unique solution:
- The rank of the coefficient matrix should be equal to the rank of the augmented matrix.
- The
- Find the solution for x=0.2 taking interval length 0.1 using Euler’s method to solve: dy/dx=1-y given y(0)=0.
Ans 6.
Step 1: Understand the problem
We have the differential equation:
dy/dx = 1 – y
with the initial condition:
y(0) = 0