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What is the slope of the line normal to the tangent line of #f(x) = e^(x-2)+x-2 # at # x= 0 #?

Answer 1

Paisley Johnson

#e^2/(2-e^2)#

Differentiating,

#f'(x)=(x-2)e^(x-2)+1#

#f'(0)=(-2)e^(-2)+1=1-2/e^2=(e^2-2)/e^2#

Slope of the tangent at #x=0# is #(e^2-2)/e^2#

Slope of the normal at #x=0# is #e^2/(2-e^2)#

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Answer 2

Isabella Ackerman

To find the slope of the line normal to the tangent line of ( f(x) = e^{x-2} + x - 2 ) at ( x = 0 ), follow these steps:

Find the derivative of ( f(x) ) using the rules of differentiation.
Evaluate the derivative at ( x = 0 ) to find the slope of the tangent line.
Calculate the negative reciprocal of the slope found in step 2 to find the slope of the line normal to the tangent line.

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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.