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How to find the equation of the tangent to y=√x−1 at the point where x=t ?

(a) Find the equation of the tangent to #y=sqrtx−1# at the point where #x = t#.

(b) Hence find t and the equation of the tangent if the tangent passes through the origin.

(c) Draw a sketch.

Answer 1

Scarlett Allison

#y=1/(2*sqrt(t))*x+sqrt(t)/2-1#

We get the slope by differentiating with respect to #x# #y'=1/(2sqrt(x))# for #x=t# we get

#y'=1/(2sqrt(t))# For #x=t# we get #y=sqrt(t)-1# so we get the searched equation as

#y=1/(2sqrt(t))x+n#

with #y=sqrt(t)-1# and #x=t# we get

#y=1/(2sqrt(t))x+sqrt(t)/2-1#

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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.

How to find the equation of the tangent to y=√x−1 at the point where x=t ?

(a) Find the equation of the tangent to #y=sqrtx−1# at the point where #x = t#. (b) Hence find t and the equation of the tangent if the tangent passes through the origin. (c) Draw a sketch.

(a) Find the equation of the tangent to #y=sqrtx−1# at the point where #x = t#.

(b) Hence find t and the equation of the tangent if the tangent passes through the origin.

(c) Draw a sketch.