The task is to estimate the value of $\sqrt[4]{16.3}$ using differentials and compare it to the actual value calculated by a calculator. Here's a step-by-step breakdown to solve it using differentials:

Step 1: Approximation Using Differentials

We can estimate the value of $\sqrt[4]{16.3}$ by using the function: $f(x) = \sqrt[4]{x}$ This function can be approximated by using its tangent line near a value where we already know the result. Let's choose $x = 16$, since $\sqrt[4]{16} = 2$.

The differential approximation formula is: $f(x + \Delta x) \approx f(x) + f'(x) \cdot \Delta x$ Where $\Delta x = 16.3 - 16 = 0.3$.

Step 2: Compute the Derivative

The derivative of $f(x) = \sqrt[4]{x}$ is: $f'(x) = \frac{1}{4}x^{-\frac{3}{4}} = \frac{1}{4\sqrt[4]{x^3}}$ At $x = 16$: $f'(16) = \frac{1}{4 \cdot \sqrt[4]{16^3}} = \frac{1}{4 \cdot 8} = \frac{1}{32}$

Step 3: Estimate the Value

Now we can estimate $f(16.3)$ using the differential approximation: $f(16.3) \approx f(16) + f'(16) \cdot (16.3 - 16)$ $f(16.3) \approx 2 + \frac{1}{32} \cdot 0.3 = 2 + 0.009375 = 2.009375$

Step 4: Compare with the Actual Value

Using a calculator, we find the actual value of $\sqrt[4]{16.3}$.

$\sqrt[4]{16.3} \approx 2.00926$

Step 5: Comparison

The estimated value using differentials is $2.009375$, while the actual value from the calculator is $2.00926$. The differential method gives a very close approximation to the actual value.

Would you like more details on any part of the process?

Here are some related questions:

1. How do differentials help in estimating functions?
2. Can the same approach be applied to other roots or powers?
3. What role does the choice of $x$ (in this case, 16) play in the accuracy of the approximation?
4. How does the error in this approximation compare with larger values of $\Delta x$?
5. What other methods can be used to estimate the value of roots without a calculator?

Tip: When using differentials for approximation, choose a base value of $x$ for which you know the exact result, to minimize the error in estimation.