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
Here's how to solve question 5(b): Step 1: Define the function and its derivative. Let y = f(x) = x^(1)/(3). The derivative is: (dy)/(dx) = f'(x) = (1)/(3)x^(1)/(3)-1 = (1)/(3)x^-(2)/(3) = (1)/(3x^2)3 Step 2: Choose a suitable value for x and calculate x. To approximate [3]1003, we choose a value of x close to 1003 for which the cube root is easy to calculate. Let x = 1000. Then the change in x is x = 1003 - 1000 = 3. Step 3: Evaluate f(x) and f'(x) at x = 1000. f(1000) = 1000^(1)/(3) = (10^3)^(1)/(3) = 10 f'(1000) = (1)/(3(1000)^2)3 = (1)/(3((10^3)^1)3)^2 = (1)/(3(10)^2) = (1)/(3 × 100) = (1)/(300) Step 4: Use the approximation formula. The approximate value of f(x + x) is given by f(x) + f'(x) x. [3]1003 ≈ f(1000) + f'(1000) × x [3]1003 ≈ 10 + ((1)/(300)) × 3 [3]1003 ≈ 10 + (3)/(300) [3]1003 ≈ 10 + (1)/(100) [3]1003 ≈ 10 + 0.01 [3]1003 ≈ 10.01 That's 2 down. 3 left today — send the next one.