Find the number of positive and negative roots of the polynomial
P( x) = x3 − 3x2 + 4x − 5 . Find P(2) and P′(2) using synthetic division method.
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A table of values is to be constructed for the function f (x) given by in the interval [1, 4] with equal step length. Determine the spacing h such that quadratic interpolation gives result with accuracy 1x10-6.
Derive a suitable numerical differentiation formula of 0(h2 ) to find f'(2,4) with h = 0.1 given the table
| x | 0.1 | 1.2 | 2.4 | 3.9 |
| f(x) | 3.41 | 2.68 | 1.37 | -1.48 |
Estimate the eigenvalues of the matrix
using the Gershgorin bounds. Draw a rough sketch of the region where the eigenvalues lie.
The equation x2 + ax + b = 0 has two real roots p and q such that | p | < | q |. If we use the fixed point iteration , to find a root then to which root does it converge?
The position f (x) of a particle moving in a line at various times xk is given in the following table. Estimate the velocity and acceleration of the particle at x = 1.2.
| x | 1.0 | 1.2 | 1.4 | 1.6 | 1.8 | 2.0 | 2.2 |
| f(x) | 2.72 | 3.32 | 4.06 | 4.96 | 6.05 | 7.39 | 9.02 |
Use the polynomial in part (i) to approximate
.
Find by Newton’s method the roots of the following equations correct to three places of decimals
i) x log10 x = 4.772393 near x = 6
ii) f (x) = x − 2 sin x near x = 2