Respuesta :

LammettHash
Let [tex]f(x)=x^{2/3}[/tex]. You have [tex]f(125)=(125)^{2/3}=(5^3)^{2/3}=5^2=25[/tex].
 
Taking the derivative, you get

[tex]f'(x)=\dfrac2{3x^{1/3}}\implies f'(125)=\dfrac2{3(125)^{1/3}}=\dfrac2{3\times5}=\dfrac2{15}[/tex]

The linear approximation to [tex]f(x)[/tex] near [tex]x=125[/tex] is given by

[tex]L(x)=f(125)+f'(125)(x-125)\approx f(x)[/tex]

So the value of [tex](125.07)^{2/3}[/tex] can be estimated as

[tex]L(125.07)=f(125)+f'(125)(125.07-125)[/tex]
[tex]L(125.07)=25+\dfrac2{15}(0.07)=25.009333\ldots[/tex]

The estimated value of the given number is 25.0093.

Given:

The given number is [tex](125.07)^{\frac{2}{3}}[/tex].

To find:

The estimated value of the given number by using the linear approximation formula.

Explanation:

Linear approximation formula:

[tex]f(x)=f(x_0)+f'(x_0)(x-x_0)[/tex]

Let

[tex]f(x)=x^{\frac{2}{3}}[/tex]

For [tex]x_0=125[/tex],

[tex]f(125)=(125)^{\frac{2}{3}}[/tex]

[tex]f(125)=5^2[/tex]

[tex]f(125)=25[/tex]

Differentiate [tex]f(x)[/tex] with respect to [tex]x[/tex].

[tex]f'(x)=\dfrac{2}{3}x^{\frac{2}{3}-1}[/tex]

[tex]f'(x)=\dfrac{2}{3}x^{-\frac{1}{3}}[/tex]

[tex]f'(x)=\dfrac{2}{3x^{\frac{1}{3}}}[/tex]

For [tex]x_0=125[/tex],

[tex]f'(125)=\dfrac{2}{3(125)^{\frac{1}{3}}}[/tex]

[tex]f'(125)=\dfrac{2}{3(5)}[/tex]

[tex]f'(125)=\dfrac{2}{15}[/tex]

Using the linear approximation formula, we get

[tex]f(125.07)=f(125)+f'(125)(125.07-125)[/tex]

[tex](125.07)^{\frac{2}{3}}=25+\dfrac{2}{3}(0.07)[/tex]

[tex](125.07)^{\frac{2}{3}}=25.0093333...[/tex]

[tex](125.07)^{\frac{2}{3}}\approx 25.0093333...[/tex]

Therefore, the estimated value of the given number is 25.0093.

Learn more:

https://brainly.com/question/2193112

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