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How do you find the linear approximation of the function #g(x)=root5(1+x)# at a=0?

Answer 1

Ezekiel Alexander

#root(5)(1+x) ~= 1+x/5#

The best linear approximation of #g(x)# around #x=0# is its tangent line:

#y(x) = g(0)+g'(0)x#

Given that:

#g(x) = root(5)(1+x) = (1+x)^(1/5)#

#g'(x) = 1/5(1+x)^(-4/5)#

we have:

#g(0) = 1#

#g'(0) = 1/5#

so:

#root(5)(1+x) ~= 1+x/5#

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Answer 2

Luke Alvarez

To find the linear approximation of the function ( g(x) = \sqrt{5(1+x)} ) at ( a = 0 ), you can use the formula for linear approximation:

[ L(x) = f(a) + f'(a)(x-a) ]

First, find ( f(a) ) and ( f'(a) ) at ( a = 0 ):

[ f(a) = \sqrt{5(1+a)} ] [ f'(a) = \frac{d}{dx} \sqrt{5(1+x)} ]

Evaluate these at ( a = 0 ) to get ( f(0) ) and ( f'(0) ). Then plug these values into the linear approximation formula:

[ L(x) = f(0) + f'(0)(x-0) ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.