1. **Problem Statement:** Construct a forward difference table for the given values of $x$ and $y$, then find $\Delta^2 f(5)$ and $\Delta^3 f(10)$. 2. **Given Data:** $$\begin{array}{c|cccccc} x & 0 & 5 & 10 & 15 & 20 & 25 \\ y & 7 & 11 & 14 & 18 & 24 & 32 \\\end{array}$$ 3. **Forward Difference Table:** - The first forward difference $\Delta y_i = y_{i+1} - y_i$. - The second forward difference $\Delta^2 y_i = \Delta y_{i+1} - \Delta y_i$. - The third forward difference $\Delta^3 y_i = \Delta^2 y_{i+1} - \Delta^2 y_i$. Calculate step-by-step: | $x$ | $y$ | $\Delta y$ | $\Delta^2 y$ | $\Delta^3 y$ | |-----|-----|------------|--------------|--------------| | 0 | 7 | 11 - 7 = 4 | | | | 5 | 11 | 14 - 11 = 3| 3 - 4 = -1 | | | 10 | 14 | 18 - 14 = 4| 4 - 3 = 1 | 1 - (-1) = 2 | | 15 | 18 | 24 - 18 = 6| 6 - 4 = 2 | 2 - 1 = 1 | | 20 | 24 | 32 - 24 = 8| 8 - 6 = 2 | | | 25 | 32 | | | | 4. **Find $\Delta^2 f(5)$:** - $\Delta^2 f(5)$ corresponds to the second forward difference at $x=5$, which is $-1$. 5. **Find $\Delta^3 f(10)$:** - $\Delta^3 f(10)$ corresponds to the third forward difference at $x=10$, which is $2$. **Final answers:** $$\Delta^2 f(5) = -1$$ $$\Delta^3 f(10) = 2$$