Question

In Evercises \(2.5 .19-2.5 .22, \Delta\) is the forwand difference operator with spacing \(h>0\). Suppose that \(f^{(n+1)}\) exists on \((a, b), x_{0}, \ldots, x_{n}\) are in \((a, b),\) and \(p\) is the polynomial of degree \(\leq n\) such that \(p\left(x_{i}\right)=f\left(x_{i}\right), 0 \leq i \leq n .\) Prove: If \(x \in(a, b),\) then $$ f(x)=p(x)+\frac{f^{(n+1)}(c)}{(n+1) !}\left(x-x_{0}\right)\left(x-x_{1}\right) \cdots\left(x-x_{n}\right) $$ where \(c,\) which depends on \(x,\) is in \((a, b) .\) HiNT: Let \(x\) be fixed, distinct from \(x_{0}\) \(x_{1}, \ldots . x_{n},\) and consider the function $$ g(y)=f(y)-p(y)-\frac{K}{(n+1) !}\left(y-x_{0}\right)\left(y-x_{1}\right) \cdots\left(y-x_{n}\right) $$ where \(K\) is chosen so that \(g(x)=0 .\) Use Rolle's theorem to show that \(K=\) \(f^{(n+1)}(c)\) for some c in \((a, b)\).

Short answer

Answer: The formula for f(x) is: $$ f(x) = p(x) + \frac{f^{(n+1)}(c)}{(n+1)!}(x-x_0)(x-x_1)\cdots(x-x_n) $$ where c depends on x and lies in the interval (a, b).

Step-by-step solution

Define g(y)

Let's define the function g(y) as: $$ g(y) = f(y) - p(y) - \frac{K}{(n+1)!} (y-x_0)(y-x_1)\cdots(y-x_n) $$ where K is a constant such that g(x) = 0.

Apply Rolle's theorem

According to the given hint, we can use Rolle's theorem. For the function g(y) to satisfy Rolle's theorem, it must be continuous on the closed interval [x_0, x], differentiable on the open interval (x_0, x), and g(x_0) = g(x) = 0. Since both f(y) and p(y) are differentiable on the given interval (a, b) and are equal for \(0\leq i \leq n\), the function g(y) will also be differentiable. The conditions for Rolle's theorem are satisfied.

Find g'(y)

Now, we need to find the derivative of g(y) with respect to y. Let's find g'(y): $$ g'(y) = f'(y) - p'(y) - \frac{K}{n!}\sum_{i=0}^n\prod_{\substack{j=0\\j\neq i}}^n(y-x_j) $$

Apply Rolle's theorem multiple times

By applying Rolle's theorem n times to g(y), we get that there exists numbers \(c_1, c_2, \ldots, c_n\) in the interval (a, b) such that $$ g^{(n+1)}(c) = (n+1)!\,\frac{K}{(n+1)!} = K = f^{(n+1)}(c) $$

Find f(x) using the derived formula

Now, we have the correct formula for K, we can substitute it back in the formula for f(x) using g(x) = 0: $$ f(x) = p(x) + \frac{f^{(n+1)}(c)}{(n+1)!}(x-x_0)(x-x_1)\cdots(x-x_n) $$ This is the formula for f(x) that we needed to prove. The constant c depends on x and lies in the interval (a, b).

Key concepts

Polynomial Interpolation

Polynomial Interpolation is a method used to approximate functions by fitting a polynomial that passes through a given set of data points. Imagine you have several points plotted on a graph, and you want to find a smooth curve that not only follows these points but also is defined across the entire domain that covers these points.

Interpolation is useful for making predictions or when you need to fill in gaps within your data. When we use a polynomial of degree \(n\) to approximate \(f(x)\), it ensures that the polynomial will pass through \(n+1\) specific points. Polynomial Interpolation relies heavily on choosing the right degree for the polynomial because it can significantly affect the accuracy and efficiency of the approximation.

For example, if you have three data points, a quadratic polynomial (degree 2) is needed. Here, the interpolating polynomial doesn't just simply connect the dots; it smoothens the transition and maintains consistency between known points. The beauty of Polynomial Interpolation is in the balance it strikes between simplicity and accuracy in various applications from science to economics.

Rolle's Theorem

Rolle's Theorem is a foundational result in calculus that connects the behavior of a function to the existence of derivatives. The theorem states that if a function \(f\) is continuous on the closed interval \([a, b]\), differentiable on the open interval \((a, b)\), and if \(f(a) = f(b)\), then there exists at least one \(c\) in \((a, b)\) such that the derivative \(f'(c) = 0\). Essentially, this theorem is saying that if you have a smooth mountain range that starts and ends at the same height, there has to be at least one peak or trough along the way where the slope is zero.

In our context, by applying this theorem repeatedly, we prove that there exists some point \(c\) where the polynomial and the function have the same \(n\)-th derivative. This conclusion is crucial for showing that the polynomial approximates the function very closely, especially when you're working with interpolations characterized by repeated applications of derivatives.

Differentiability

Differentiability is a characteristic of functions that allows them to have derivatives, meaning that one can compute the slope of the function at any point over its domain. For a function to be differentiable at a point \(x\), it must also be continuous at \(x\). However, continuity alone does not ensure differentiability. Differentiable functions are smooth and have no breaks, bends, or corners at the point where a derivative is considered.

In the exercise, differentiability ensures that not only can we apply Rolle's theorem, but also allows us to trust that the polynomial interpolation is valid. It means the transition between the known data points is smooth, and we can reliably compute high-order derivatives like \(f^{(n+1)}(y)\) used in the proof. Being differentiable means that both \(f(y)\) and \(p(y)\) provide a reliable canvas over which calculus can be enforced without losing integrity.