5. a) find the linear approximation to ( f(x)=sqrt3{x} ) near ( x = 27 ).\nb) use part (a) to approximate (…

Explanation:

Step1: Recall the formula for linear approximation

The formula for the linear approximation of a function (y = f(x)) at (x = a) is (L(x)=f(a)+f^{\prime}(a)(x - a)). Given (f(x)=\sqrt[3]{x}=x^{\frac{1}{3}}), then (f^{\prime}(x)=\frac{1}{3}x^{-\frac{2}{3}}=\frac{1}{3x^{\frac{2}{3}}}). When (a = 27), (f(27)=\sqrt[3]{27}=3), and (f^{\prime}(27)=\frac{1}{3\times27^{\frac{2}{3}}}). Since (27^{\frac{2}{3}}=(27^{\frac{1}{3}})^2 = 3^2=9), then (f^{\prime}(27)=\frac{1}{3\times9}=\frac{1}{27}).

Step2: Write the linear - approximation formula

Substitute (a = 27), (f(27) = 3), and (f^{\prime}(27)=\frac{1}{27}) into the linear - approximation formula (L(x)=f(a)+f^{\prime}(a)(x - a)). We get (L(x)=3+\frac{1}{27}(x - 27)). Simplify (L(x)=3+\frac{1}{27}x - 1=\frac{1}{27}x + 2).

Step3: Use the linear approximation to estimate (\sqrt[3]{26.46})

Let (x = 26.46). Then, using (L(x)=\frac{1}{27}x + 2), we substitute (x = 26.46) into (L(x)). (L(26.46)=\frac{1}{27}\times26.46+2). (\frac{26.46}{27}=0.98), so (L(26.46)=0.98 + 2=2.98).

Answer:

a) The linear approximation is (L(x)=\frac{1}{27}x + 2). b) (\sqrt[3]{26.46}\approx2.98)