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2.
Numerical Differentiation
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Numerical Differentiation using Newton's Forward, Backward Method
1. From the following table of values of x and y, obtain `(dy)/(dx)` and `(d^2y)/(dx^2)` for x = 1.2 .
| x |
1.0 |
1.2 |
1.4 |
1.6 |
1.8 |
2.0 |
2.2 |
| y |
2.7183 |
3.3201 |
4.0552 |
4.9530 |
6.0496 |
7.3891 |
9.0250 |
2. From the following table of values of x and y, obtain `(dy)/(dx)` and `(d^2y)/(dx^2)` for x = 2.2 .
| x |
1.0 |
1.2 |
1.4 |
1.6 |
1.8 |
2.0 |
2.2 |
| y |
2.7183 |
3.3201 |
4.0552 |
4.9530 |
6.0496 |
7.3891 |
9.0250 |
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3.
Numerical Integration
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Numerical Integration using Trapezoidal, Simpson's 1/3, Simpson's 3/8 Rule
1. From the following table, find the area bounded by the curve and x axis from
x=7.47 to x=7.52 using trapezodial, simplson's 1/3, simplson's 3/8 rule.
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x |
7.47 |
7.48 |
7.49 |
7.50 |
7.51 |
7.52 |
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f(x) |
1.93 |
1.95 |
1.98 |
2.01 |
2.03 |
2.06 |
2. Evaluate I = `int_0^1 (1)/(1+x) dx` by using simpson's rule with h=0.25 and h=0.5
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5. Solve numerical differential equation using Euler, Runge-kutta 2, Runge-kutta 3, Runge-kutta 4 methods
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1. Find y(0.1) for `y'=x-y^2`, y(0) = 1, with step length 0.1
2. Find y(0.5) for `y'=-2x-y`, y(0) = -1, with step length 0.1
3. Find y(2) for `y'=(x-y)/2`, y(0) = 1, with step length 0.2
4. Find y(0.3) for `y'=-(x*y^2+y)`, y(0) = 1, with step length 0.1
5. Find y(0.2) for `y'=-y`, y(0) = 1, with step length 0.1
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4.
Numerical Interpolation
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Numerical Interpolation using Forward, Backward Method
1. The population of a town in decimal census was as given below. Estimate population for the year 1895.
| Year |
1891 |
1901 |
1911 |
1921 |
1931 |
Population
(in Thousand) |
46 |
66 |
81 |
93 |
101 |
2. Let y(0) = 1, y(1) = 0, y(2) = 1 and y(3) = 10. Find y(4) using newtons's forward difference formula.
3. In the table below the values of y are consecutive terms of a series of which the number 21.6 is the 6th term. Find the 1st and 10th terms of the series.
| X |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
| Y |
2.7 |
6.4 |
12.5 |
21.6 |
34.3 |
51.2 |
72.9 |
4. The population of a town in decimal census was as given below. Estimate population for the year 1895.
| X |
0.10 |
0.15 |
0.20 |
0.25 |
0.30 |
| tan(X) |
0.1003 |
0.1511 |
0.2027 |
0.2553 |
0.3073 |
Find (1) tan 0.12 (2) tan 0.26
5. Certain values of x and log10x are (300,2.4771), (304,2.4829), (305,2.4843) and (307,2.4871). Find log10 301.
6. Find lagrange's Inerpolating polynomial of degree 2 approximating the function y = ln x defined by the following table of values. Hence find ln 2.7
| X |
2 |
2.5 |
3 |
| ln(X) |
0.69315 |
0.91629 |
1.09861 |
7. Using the following table find f(x) as polynomial in x
| x |
-1 |
0 |
3 |
6 |
7 |
| f(x) |
3 |
-6 |
39 |
822 |
1611 |
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