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What is the equation of the tangent line of #f(x) =sqrt(x^2-2x+1)-x+1# at # x = 1#?

Answer 1

Isabella Ackerman

The tangent line at #x=1# is undefined as #f(x)# is continuous but not differentiable at that point.

#f(x) = sqrt(x^2-2x+1)-x+1#

#=sqrt((x-1)^2)-(x-1)#

#=abs(x-1)-(x-1)#

Hence:

#f(x) = { (-2x+2, "if x <= 1"), (0, "if x >= 1") :}#

When #x < 1# the derivative #f'(x) = -2#

When #x > 1# the derivative #f'(x) = 0#

So the left and right limits of the slope disagree at #x=1#. So the derivative is undefined at #x=1# and the tangent is undefined.

x^2-2x+1)-x+1 graph{sqrt[-4.333, 5.667, -0.7, 4.3]}

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Answer 2

Axel Alvarez

The equation of the tangent line of f(x) = sqrt(x^2-2x+1)-x+1 at x = 1 is y = -x+2.

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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.