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aev [14]
3 years ago
13

Use a linear approximation to estimate the number (125.07)2/3

Mathematics
1 answer:
Ivahew [28]3 years ago
8 0
Let f(x)=x^{2/3}. You have f(125)=(125)^{2/3}=(5^3)^{2/3}=5^2=25.
 
Taking the derivative, you get

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

The linear approximation to f(x) near x=125 is given by

L(x)=f(125)+f'(125)(x-125)\approx f(x)

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

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