Estimate a value knowing the values of: the function, the derivative and the second derivative in 0

Question

Please suppose you have an unknown function r(x).

This function r(x) is defined in the range: [-5; 5]

You know that:

r(0) = 1;

r'(0) = -1;

r"(0) = 1.

Please estimate the value of r(x) in the point 1/10 (= 0.1), explaining the method or the theorems you've used to estimate.

Thank you in advance for your kind help.

Please suppose that in a certain classroom, Taylor Series concept has not yet been explained, so you aren't authorized to use Taylor Series to solve this problem.

Answer 1

using the concept of taylor series $$ P(x) = \frac{r(0)}{0!} + x\frac{r'(0)}{1!} + x^2\frac{r''(0)}{2!} + \ldots $$ as you are requiring an approximation stop at degree $ x^2$ then $$r(x) \approx 1 - x + \frac{x^2}{2!} $$ at $x =0.1$ $$ r(0.1) \approx \frac{181}{200} = 0.905$$

Answer 2

If you have no idea of Taylor series, you need to approximate your function $r$ by another one, with 3 parameters (since you have 3 informations). Without more informations, the most reasonable choice is to approximate it using a parabola $f(x) = ax^2 + bx + c$.

To determine the values of $a$,$b$ and $c$, set $f(0)= r(0), f'(0) = r'(0), f''(0) = r''(0)$. You get immediatly the values of the parameters: $c=1$, $b=-1$ and $a=0.5$.

From there, $r(0.1) \approx f(0.1) = 0.5 (0.1^2) - 0.1 + 1 = 0.905$.

Without surprise, this yields the same answer as the Taylor serie.