We are tasked with approximating $\sqrt{4.1}$ correctly to three decimal places, and using an error estimate theorem to ensure the accuracy of the approximation.

Step 1: Use a binomial approximation for small deviations

We start by noting that $4.1$ is close to $4$, so we can use the following approximation for square roots:

$\sqrt{4.1} = \sqrt{4(1 + 0.025)} = 2 \cdot \sqrt{1 + 0.025}.$

Step 2: Apply the binomial expansion

For small $x$, the binomial expansion $\sqrt{1 + x}$ is approximately given by:

$\sqrt{1 + x} \approx 1 + \frac{x}{2} - \frac{x^2}{8} + O(x^3).$

In our case, $x = 0.025$. Substituting into the expansion:

$\sqrt{1 + 0.025} \approx 1 + \frac{0.025}{2} - \frac{(0.025)^2}{8}.$

Step 3: Compute the terms

Let's calculate each term:

1. $\frac{0.025}{2} = 0.0125$,
2. $\frac{(0.025)^2}{8} = \frac{0.000625}{8} = 0.000078125$.

Now, adding the terms:

$\sqrt{1 + 0.025} \approx 1 + 0.0125 - 0.000078125 = 1.012421875.$

Step 4: Multiply by 2 to get the approximation for $\sqrt{4.1}$

Now, multiply by 2 to find $\sqrt{4.1}$:

$\sqrt{4.1} \approx 2 \times 1.012421875 = 2.02484375.$

Step 5: Round to three decimal places

Rounding this to three decimal places gives:

$\sqrt{4.1} \approx 2.025.$

Step 6: Error estimate

To ensure that this approximation is correct to three decimal places, we use the error term from the binomial expansion:

$\text{Error term} \approx \frac{x^3}{16} = \frac{(0.025)^3}{16} = \frac{0.000015625}{16} = 0.0000009765625.$

Since the error is much smaller than $0.001$, our approximation of $2.025$ is correct to three decimal places.

Final Answer:

$\boxed{2.025}.$

Would you like more details on the binomial expansion or error estimates?

Here are 5 related questions for further exploration:

1. How can you approximate cube roots using a similar method?
2. What is the general formula for binomial expansion?
3. How do error estimates change when using more terms in the expansion?
4. Can we use Newton's method for a similar approximation?
5. How would you approximate $\sqrt{4.01}$ to three decimal places?

Tip: Binomial expansions are great for approximating square roots when the number you're working with is close to a perfect square.