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How do you find the linearization of #f(x) = 1/sqrt(2+x)# at the point a=0?

Answer 1

Emilia Aguilar

Linearization is finding the linear approximation to a function at a given point #y=f(a)+f'(a)(x-a)#

#f'(x)=-1/2(2+x)^(-3/2)# #f'(0)=-1/2(2+0)^(-3/2)=-0.18# #f(0)=1/sqrt(2)=0.71# #f(0)~=f(0)+f'(0)*(x-0)=0.71-0.18*x#

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Answer 2

Emma Aldridge

To find the linearization of ( f(x) = \frac{1}{\sqrt{2+x}} ) at the point ( a = 0 ), follow these steps:

Find the first derivative ( f'(x) ) of the function.
Evaluate ( f'(0) ) to find the slope of the tangent line at ( x = 0 ).
Use the point-slope form of the equation for a line, ( y - y_1 = m(x - x_1) ), where ( m ) is the slope and ( (x_1, y_1) ) is the point of tangency (in this case, ( x_1 = 0 ) and ( y_1 = f(0) )).
Simplify the equation obtained in step 3 to get the linearization of ( f(x) ) at ( a = 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.