IGNOU MTE 10 2018 - Find by Newton’s method the roots of the following equations cor

Question

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

14 Feb 2018

Answer :

Word Count : 352

We will solve both equations using Newton’s Method. Recall the formula: $$ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} $$ --- ### i) $x \log_{10} x = 4.772393$, near $x = 6$ 1. Define function: $$ f(x) = x \log_{10} x - 4.772393 $$ Derivative: $$ f'(x) = \log_{10} x + \frac{x}{x \ln 10} = \log_{10} x + \frac{1}{\ln 10} $$ Since $\frac{1}{\ln 10} \approx 0.4343$, so $$ f'(x) = \log_{10} x + 0.4343 $$ 2. Initial guess: $x_0 = 6$ Compute $f(6)$ and $f'(6)$: $$ f(6) = 6 \cdot \log_{10} 6 - 4.772393 \approx 6 \cdot 0.778151 - 4.772393 \approx 4.668906 - 4.772393 = -0.103487 $$ $$ f'(6) = \log_{10} 6 + 0.4343 \approx 0.778151 + 0.4343 = 1.212451 $$ Next iteration: $$ x_1 = 6 - \frac{-0.103487}{1.212451} \approx 6 +

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x	− 3	− 2	− 1	0	1	2	3
f ( x)	13	7	3	1	1	3	7

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