Question

a table of values is to be constructed for the function f(x) =1 /1+x in the interval [1,4] with equal step length. determine the spacing h such that quadratic interpolation gives result with accuracy 1×10 - 6

Answer 1

Answer to Question #74945, Math / Quantitative Methods

A table of values is to be constructed for the function $f(x) = 1 / (1 + x)$ in the interval [1,4] with equal step length. Determine the spacing $h$ such that quadratic interpolation gives result with accuracy $1 \times 10^{-6}$

Solution.

The error is

$$\left| R _ {n} (x) \right| \leq \frac {h ^ {3}}{9 \sqrt {3}} \max _ {x \in [ a, b ]} \left| f ^ {\prime \prime \prime} (x) \right|$$

We have:

$$f ^ {\prime} (x) = - \frac {1}{(1 + x) ^ {2}}$$ $$f ^ {\prime \prime} (x) = \frac {2}{(1 + x) ^ {3}}$$ $$f ^ {\prime \prime \prime} (x) = - \frac {6}{(1 + x) ^ {4}}$$ $$\max _ {x \in [ 1, 4 ]} | f ^ {\prime \prime \prime} (x) | = | f ^ {\prime \prime \prime} (1) | = \left| - \frac {6}{(1 + 1) ^ {4}} \right| = \frac {6}{1 6} = \frac {3}{8}$$

Then:

$$\left| R _ {n} (x) \right| \leq \frac {3}{8} \cdot \frac {h ^ {3}}{9 \sqrt {3}}$$ $$\frac {3}{8} \cdot \frac {h ^ {3}}{9 \sqrt {3}} = 1 0 ^ {- 6}$$

Answer:

$$h = \sqrt [ 3 ]{\frac {8 \cdot 9 \sqrt {3}}{3} \cdot 1 0 ^ {- 6}} = \frac {2}{1 0 0} \sqrt [ 3 ]{3 \sqrt {3}} = 0. 0 3 4 6 4$$

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