What is the local linearization of #y = (7+5x^2)^(-1/2)# at a=0?

Answer 1

It is #L(x)=7^(-1/2)# (or #L(x) = 1/sqrt7# if you prefer.)

We have #f(x) = (7+5x^2)^(-1/2)#, so
#f'(x) = (-1/2)(7+5x^2)^(-3/2)(10x)#.
At the chosen point, the derivative #f'(a)# is #0#.
So the liearization is horizontal: #L(x)=f(a) = 7^(-1/2)#
General Case The local linearization (aka linear approximation), of function #f# at point #x=a# is a form of the equation for the tangent line at #(a,f(a))#.
We have #y=f(x)#, and we note that the tangent at the point where #x=a# intersects the graph at #(a,f(a))# and has slope #m=f'(a)#.

Writing the equation of the tangent line in point-slope form, we get

#y-f(a)=f'(a)(x-a)#.
Adding #f(a)# to both sides get us the local linearization (linear approximation) of #f(x)# at #a#:
#L(x)~~f(a)+f'(a)(x-a)#,
Here is a picture of the graph of the function in this question. As you zoom in centered at #x=0#, you'll see that the graph looks very much like a horizontal line.

graph{y=(7+5x^2)^(-1/2) [-3.467, 3.465, -1.35, 2.117]}

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Answer 2

The local linearization of ( y = (7 + 5x^2)^{-1/2} ) at ( a = 0 ) is given by the equation: [ L(x) = y'(a)(x - a) + y(a) ] where ( y'(a) ) represents the derivative of ( y ) with respect to ( x ) evaluated at ( x = a ). The derivative of ( y ) with respect to ( x ) is: [ y'(x) = \frac{-5x}{2(7 + 5x^2)^{3/2}} ] Evaluating ( y'(0) ), we get: [ y'(0) = \frac{0}{2(7 + 5(0)^2)^{3/2}} = 0 ] Next, ( y(0) ) can be found by substituting ( x = 0 ) into the original function: [ y(0) = (7 + 5(0)^2)^{-1/2} = (7)^{-1/2} = \frac{1}{\sqrt{7}} ] Thus, the local linearization of ( y = (7 + 5x^2)^{-1/2} ) at ( a = 0 ) is: [ L(x) = 0 \cdot (x - 0) + \frac{1}{\sqrt{7}} = \frac{1}{\sqrt{7}} ]

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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.

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.

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.

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.

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