Question 7.4.1: Find the linear approximation to f (x) = √x about x = 1....

Essential Mathematics for Economic Analysis [1185377]

Find the linear approximation to $f (x) = \sqrt[3]{x}$ about x = 1.

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We have $f(x)=\ {\sqrt[3]{x}}=x^{1/3}$ , so f (1) = 1, and $f^{\prime}(x)={\textstyle\frac{1}{3}}x^{-2/3}$ , implying that $f^{\prime}(1)={\textstyle\frac{1}{3}}.$ Inserting these values into (7.4.1), when a = 1, yields

$f(x)\approx f(a)+f^{\prime}(a)(x-a)$ (7.4.1)

${{\sqrt[3]{{x}}}\approx f(1)+f^{\prime}(1)(x-1)=1+{\textstyle{\frac{1}{3}}}(x-1)}$ (x close to 1)

For example, ${\sqrt[3]{1.03}}\approx1+\textstyle{\frac{1}{3}}(1.03-1)=1.01.$ The correct value to four decimals is 1.0099.

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