Final answer:
To estimate (32.03)4/5 using linear approximation, calculate the derivative of f(x) = x4/5 at a nearby point where the value is known easily, such as 32. Use the derivative and the change in x to find the approximate value.
Step-by-step explanation:
To estimate the number (32.03)4/5 using linear approximation (or differentials), we can consider a function f(x) = x4/5 and use a point near 32.03, where we can easily compute the exact value. For simple computation, we choose 32 as this nearby point because 324/5 is a value that can be calculated without difficulty.
First, let's find the derivative of f(x) which is f'(x) = (4/5)x-1/5. Now we evaluate it at x = 32 to get f'(32) = (4/5)32-1/5. The exact value of f(32) is 324/5 = 24 = 16.
Now, let's calculate the change in x, which is dx = 32.03 - 32 = 0.03. The differential, dy, is f'(32) × dx = (4/5)(32-1/5)(0.03). Next, we calculate the estimated value for f(32.03) as f(32) + dy = 16 + (4/5)(32-1/5)(0.03).
After performing the calculations and applying the rounding rules to two decimal places, we arrive at the linear approximation for (32.03)4/5.