Question

use a linear approximation to estimate the following quantity choose a value of a that produces a small error and does not require the use of a calculator

√66

the linear approximation of √66 is □
(type an exact answer)

Explanation:

Step1: Choose the function and \(a\)

Let \(f(x)=\sqrt{x}\), and we choose \(a = 64\) since \(\sqrt{64}=8\) and \(66\) is close to \(64\).

Step2: Find the derivative of \(f(x)\)

The derivative of \(f(x)=\sqrt{x}=x^{\frac{1}{2}}\) is \(f^{\prime}(x)=\frac{1}{2\sqrt{x}}\).

Step3: Calculate \(f(a)\) and \(f^{\prime}(a)\)

When \(a = 64\), \(f(64)=\sqrt{64}=8\), and \(f^{\prime}(64)=\frac{1}{2\sqrt{64}}=\frac{1}{16}\).

Step4: Use the linear approximation formula \(L(x)=f(a)+f^{\prime}(a)(x - a)\)

Here \(x = 66\), so \(L(66)=f(64)+f^{\prime}(64)(66 - 64)\)
Substitute the values: \(L(66)=8+\frac{1}{16}\times(66 - 64)\)
\(L(66)=8+\frac{2}{16}=8+\frac{1}{8}=\frac{64 + 1}{8}=\frac{65}{8}\)

Answer:

\(\frac{65}{8}\)