Approximate $f''(3)$ from Table of Values of $f(x)$

Question

Considering the table above, what is the best approximation for $f''(3)$?

How would I solve?

Answer 1

Using that $g'(x)=\lim_{h\to 0}\frac{g(x+h)-g(x)}{h}$ you can approximate $f''(3)$ using

$$f''(3)\approx \frac{f'(4)-f'(2)}{4-2}=\frac{f'(4)-f'(2)}{2}.$$

To approximate $f'(4)$ and $f'(2)$ use

$$f'(4)\approx \frac{f(5)-f(3)}{5-3}=\frac{f(5)-f(3)}{2}$$ and

$$f'(2)\approx \frac{f(3)-f(1)}{3-1}=\frac{f(3)-f(1)}{2}.$$

Thus you only have to substitute.

But this is not the only way you can follow. Using that (see http://en.wikipedia.org/wiki/Second_derivative)

$$g''(x)=\lim_{h\to 0}\frac{g(x+h)-2g(x)+g(x-h)}{h^2}$$ you can approximate

$$f''(3)\approx \frac{f(4)-2f(3)+f(2)}{1}=f(4)-2f(3)+f(2).$$

Answer 2

Hint:

$$ f''(x)\approx \frac{f(x+\epsilon)-2f(x)+f(x-\epsilon)}{\epsilon^2}$$

for small (smallness to be defined by context) $\epsilon$.

Answer 3

A different way: Using Lagrange Interpolation we have an approximation of $f(x)$ as $$f(x)=-\frac{x^5-13x^4+57x^3-131x^2+38x+72}{24}.$$

Taking the second derivative and evaluating at $x=3$ gives an approximation of $f''(3)$.