This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To estimate the missing term in the table, we assume that the fourth-order differences are zero, as there are 5 data points (including the missing one). This implies that the underlying function is a polynomial of degree 3. The given data points are: f(0) = 4 f(1) = 3 f(2) = 4 f(3) = ? (Let this be y_3) f(4) = 12 Step 1: State the assumption for estimating the missing term. Since there are 5 data points, we assume that the fourth-order forward difference is zero, i.e., ^4 f(x_0) = 0. Step 2: Write the formula for the fourth-order forward difference. The formula for ^4 f(x_0) is given by: ^4 f(x_0) = f(x_4) - 4f(x_3) + 6f(x_2) - 4f(x_1) + f(x_0) Step 3: Substitute the known values into the formula. Let f(x_0)=4, f(x_1)=3, f(x_2)=4, f(x_3)=y_3, and f(x_4)=12. Setting ^4 f(x_0) = 0: 12 - 4(y_3) + 6(4) - 4(3) + 4 = 0 Step 4: Simplify the equation and solve for y_3. 12 - 4y_3 + 24 - 12 + 4 = 0 (12 + 24 - 12 + 4) - 4y_3 = 0 28 - 4y_3 = 0 4y_3 = 28 y_3 = (28)/(4) y_3 = 7 The missing term in the table is 7. Send me the next one 📸