Question

1. (a) We need to calculate accurate values of the function for very large values of x. However, it is found that just programming this formula into a computer gives very poor accuracy for large x Explain why this happens, and show how to re-write the function so that it can be used reliably, even when x is large. [6 points] (b) In diffraction theory, it is sometimes necessary to evaluate the function sin θ f(x) for small to moderate (positive) values of the variable x. One way to do this is to make use of the Taylor-MacLaurin series 03 0s sino-e-31+5!-... + (-1)"-1 with remainder term o2n (2n) llere, ξ is some number in the interval 0 < ξ < θ. Derive a Taylor-series expression for fe) , and give an upper bound for the error when the series is terminated after the (2n-1)-th order term. [10 points] (e) Prove that the polynomial function p(x) x + 2x 1 must have at least one (real) root in the interval x e [0.1] [4 points)

Answer 1

IF YOU HAVE ANY DOUBTS COMMENT BELOW I WILL BE TTHERE TO HELP YOU..ALL THE BEST..

AS FOR GIVEN DATA.

The term $\sqrt{x^2+3}-\sqrt{x^2+2}$ is the main reason for such poor accuracy because if x is large then $\sqrt{x^2+3}$ and $\sqrt{x^2+2}$ both are large too and subtraction of such a large numbers doesn't make sense. This is the same thing as $\infty -\infty$. However if we follow the steps :

$f(x)=x^2\left[ \sqrt{x^2+3}-\sqrt{x^2+2}\right ]$

f (x)

f (x)-

$f(x)=\frac{x^2\left[x^2+3-x^2-2\right ]}{\left[ \sqrt{x^2+3}+\sqrt{x^2+2}\right ]}$

f(x) =

This is the form where we can get more accurate value than the previous one.

I HOPE YOU UNDERSTAND..

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