Question

Use the Linear Approximation to estimate the value. Compare with the value given by a calculator. (Use decimal notation. Give your answer to four decimal places.) √(3.93 * 6.06 * 5.09) ≈ ?

Answer 1

To estimate the value of √(3.93 * 6.06 * 5.09), we can use the linear approximation. The linear approximation gives an estimate of approximately 2.0027. However, when compared to the calculator value of 4.0107, we can see that the linear approximation is not very accurate.

Step-by-step explanation:

To estimate the value of √(3.93 * 6.06 * 5.09), we can use the linear approximation. We start by finding the linear approximation function near the point x = a, where a is a convenient value close to our target value. Let's choose a = 4. Using a calculator, we find that √(3.93 * 6.06 * 5.09) ≈ 4.0107.

Now, we can find the equation of the tangent line to the curve y = √x at x = 4. The equation is y - f(a) = f'(a)(x - a), where f(x) = √x. Evaluating this equation at x = 4, we get y - √4 = 1/(2√4)(x - 4).

Simplifying, we have y - 2 = 1/4(x - 4). Plugging in the value x = 4.0107, we find that y ≈ 2.0027. Therefore, our estimate using the linear approximation is approximately 2.0027.

Comparing this with the calculator value of 4.0107, we can see that the linear approximation is not very accurate in this case.