Chapter 4 Conceptual Project:

Consider a function $f\left(x\right)$ that is at least twice differentiable. In this project, you will show that the second derivative of $f\left(x\right)$ at $x=c$ can be found as the limit of so-called second-order differences, as follows.

$f^{″}\left(c\right)=\underset{h\to 0}{\lim }{\displaystyle \frac{f\left(c+h\right)-2f\left(c\right)+f\left(c-h\right)}{h^2}}$

1. Instead of working with a secant line through the points $\left(c,f\left(c\right)\right)$ and $\left(c+h,f\left(c+h\right)\right)$ like we did when approximating the first derivative, suppose that

   $y=a_1{x}^2+a_{2}x+a_3$

   is the parabola through the following three points on the graph of f: $\left(c-h,f\left(c-h\right)\right)$, $\left(c,f\left(c\right)\right)$ and $\left(c+h,f\left(c+h\right)\right)$. Do you expect to always be able to find coefficients $a_1,a_2,a_3\in ℝ$ such that the resulting parabola satisfies the desired conditions? Why or why not? Why would you expect $2a_1$ to be "close” to $f^{″}\left(c\right)$ if h is "small”? What will happen to $2a_1$ as $h\to 0$? Write a short paragraph answering the above questions.

2. By substituting the points $\left(c-h,f\left(c-h\right)\right)$, $\left(c,f\left(c\right)\right)$, and $\left(c+h,f\left(c+h\right)\right)$ into $y=a_1{x}^2+a_{2}x+a_3$, obtain a system of linear equations in unknowns $a_1$, $a_2$, and $a_3$. Solve the system for the unknown $a_1$.

3. Use Questions 1 and 2 to argue that $f^{″}\left(c\right)$ is the following limit of the second‑order differences.

   $f^{″}\left(c\right)=\underset{h\to 0}{\lim }{\displaystyle \frac{f\left(c+h\right)-2f\left(c\right)+f\left(c-h\right)}{h^2}}$

4. Use L'Hôpital's Rule to verify the result you found in Question 3.