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جستجوهای پرتکرار

میتونی لایو بذاری!

The curve $y = \frac{{ln{\text{ }}x}}{{x + 1}}$ has one stationary point.

a) Show that the x-coordinate of this point satisfies the equation

$x = \frac{{x + 1}}{{ln{\text{ }}x}}$,

and that this x-coordinate lies between 3 and 4.

b) Use the iterative formula

${x_{n + 1}} = \frac{{{x_n} + 1}}{{ln{\text{ }}{x_n}}}$

to determine the x-coordinate correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

پاسخ تشریحی :
نمایش پاسخ

a) Use correct quotient or product rule

Obtain correct derivative in any form, e.g. $\frac{1}{{x\left( {x + 1} \right)}} - \frac{{\ln x}}{{{{\left( {x + 1} \right)}^2}}}$

Equate derivative to zero and obtain the given equation correctly

Consider the sign of $x - \frac{{\left( {x + 1} \right)}}{{\ln x}}$ at $x = 3$ and $x = 4$, or equivalent

Complete the argument with correct calculated values

b) Use the iterative formula correctly at least once, using or reaching a value in the interval (3, 4)

Obtain final answer 3.59

Show sufficient iterations to at least 4 d.p. to justify its accuracy to 2 d.p.,

or show there is a sign change in the interval (3.585, 3.595)

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