Question

use a linear approximation to estimate the following quantity. choose a value of a that prod

√83

the linear approximation of √83 is
(type an exact answer.)

Explanation:

Step1: Select the function and \(a\)

Let \(f(x)=\sqrt{x}\), and choose \(a = 81\) (since \(\sqrt{81}=9\) is a perfect - square close to \(83\)).

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

The derivative of \(f(x)=\sqrt{x}=x^{\frac{1}{2}}\) using the power rule \(y = x^{n}\), \(y^\prime=nx^{n - 1}\). So \(f^\prime(x)=\frac{1}{2\sqrt{x}}\).

Step3: Evaluate \(f(a)\) and \(f^\prime(a)\)

When \(a = 81\), \(f(81)=\sqrt{81}=9\), and \(f^\prime(81)=\frac{1}{2\sqrt{81}}=\frac{1}{18}\).

Step4: Use the linear - approximation formula

The linear - approximation formula is \(L(x)=f(a)+f^\prime(a)(x - a)\). Here \(x = 83\), so \(L(83)=f(81)+f^\prime(81)(83 - 81)\).
Substitute \(f(81)=9\), \(f^\prime(81)=\frac{1}{18}\), and \(x - a=2\) into the formula:
\(L(83)=9+\frac{1}{18}\times2\)
\(L(83)=9+\frac{1}{9}\)
\(L(83)=\frac{81 + 1}{9}=\frac{82}{9}\approx9.11\)

Answer:

\(\frac{82}{9}\)