1. **Problem Statement:** Given the values of $x$ and $y$, construct the forward difference table and find the values of $\Delta^2 f(5)$ and $\Delta^3 f(10)$.\n\n2. **Given Data:**\n$$\begin{array}{c|cccccc} x & 0 & 5 & 10 & 15 & 20 & 25 \\ y & 7 & 11 & 14 & 18 & 24 & 32 \\end{array}$$\n\n3. **Forward Difference Table Construction:**\n- The first forward difference is defined as $\Delta y_i = y_{i+1} - y_i$.\n- The second forward difference is $\Delta^2 y_i = \Delta y_{i+1} - \Delta y_i$.\n- The third forward difference is $\Delta^3 y_i = \Delta^2 y_{i+1} - \Delta^2 y_i$.\n\nCalculate each step:\n| $x$ | $y$ | $\Delta y$ | $\Delta^2 y$ | $\Delta^3 y$ |\n|-----|-----|------------|--------------|--------------|\n| 0 | 7 | 11 - 7 = 4 | | |\n| 5 | 11 | 14 - 11 = 3| 3 - 4 = -1 | |\n| 10 | 14 | 18 - 14 = 4| 4 - 3 = 1 | 1 - (-1) = 2 |\n| 15 | 18 | 24 - 18 = 6| 6 - 4 = 2 | 2 - 1 = 1 |\n| 20 | 24 | 32 - 24 = 8| 8 - 6 = 2 | |\n| 25 | 32 | | | |\n\n4. **Find $\Delta^2 f(5)$:**\n- This is the second forward difference at $x=5$, which is $-1$.\n\n5. **Find $\Delta^3 f(10)$:**\n- This is the third forward difference at $x=10$, which is $2$.\n\n**Final answers:**\n$$\Delta^2 f(5) = -1$$\n$$\Delta^3 f(10) = 2$$