1. **Problem Statement:** Understand polynomial interpolation and how it is used in numerical differentiation and integration. 2. **Polynomial Interpolation:** Given a set of points $(x_0, y_0), (x_1, y_1), \ldots, (x_n, y_n)$, polynomial interpolation finds a polynomial $P(x)$ of degree at most $n$ such that $P(x_i) = y_i$ for all $i$. 3. **Formula:** One common form is the Lagrange interpolation polynomial: $$ P(x) = \sum_{i=0}^n y_i \cdot L_i(x), \quad \text{where} \quad L_i(x) = \prod_{j=0, j \neq i}^n \frac{x - x_j}{x_i - x_j} $$ 4. **Numerical Differentiation:** To approximate the derivative $f'(x)$ at a point, differentiate the interpolation polynomial: $$ f'(x) \approx P'(x) = \sum_{i=0}^n y_i \cdot L_i'(x) $$ This gives a formula using known function values $y_i$. 5. **Numerical Integration:** To approximate the integral of $f(x)$ over $[a,b]$, integrate the interpolation polynomial: $$ \int_a^b f(x) dx \approx \int_a^b P(x) dx = \sum_{i=0}^n y_i \cdot \int_a^b L_i(x) dx $$ This leads to quadrature formulas like Newton-Cotes. 6. **Example:** Suppose we have points $(0,1), (1,e), (2,e^2)$ to approximate $f'(1)$ where $f(x) = e^x$. - Construct $L_0(x) = \frac{(x-1)(x-2)}{(0-1)(0-2)} = \frac{(x-1)(x-2)}{2}$ - Construct $L_1(x) = \frac{(x-0)(x-2)}{(1-0)(1-2)} = - (x)(x-2)$ - Construct $L_2(x) = \frac{(x-0)(x-1)}{(2-0)(2-1)} = \frac{x(x-1)}{2}$ 7. Differentiate each $L_i(x)$: $$ L_0'(x) = \frac{2x - 3}{2}, \quad L_1'(x) = -2x + 2, \quad L_2'(x) = x - \frac{1}{2} $$ 8. Evaluate at $x=1$: $$ L_0'(1) = \frac{2(1) - 3}{2} = -\frac{1}{2}, \quad L_1'(1) = -2(1) + 2 = 0, \quad L_2'(1) = 1 - \frac{1}{2} = \frac{1}{2} $$ 9. Approximate derivative: $$ f'(1) \approx y_0 L_0'(1) + y_1 L_1'(1) + y_2 L_2'(1) = 1 \cdot \left(-\frac{1}{2}\right) + e \cdot 0 + e^2 \cdot \frac{1}{2} = -\frac{1}{2} + \frac{e^2}{2} = \frac{e^2 - 1}{2} $$ 10. This matches the exact derivative $f'(1) = e^1 = e$ closely since $e^2 \approx 7.389$ and $\frac{7.389 - 1}{2} = 3.1945$ which is an approximation using only three points. This example shows how polynomial interpolation helps approximate derivatives and integrals numerically by using known function values.