Home > Calculus > Tangent Line to a Curve

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

Answer 1

Emilia Aguilar

There are two methods for doing this: the "normal" method and Newton's Linearization method.

COMMON WAY

#f'(x) = 2x + 2/(x^2)#

#f'(1) = 2 + 2 = 4#

The tangent line must pass through #(1, ?)#, with slope #4#. What is #y# when #x = 1?#

#f(1) = 1^2 - 2/1 = -1#

So the tangent line passes through #(1,-1)#. Therefore:

#(Deltay)/(Deltax) = (y_2 - (-1))/(0 - 1) = 4#

#-4 = y_2 + 1#

#y_2 = -5#

Thus, another point on the tangent line is #(0, -5)#. The y-intercept is therefore #y = -5#, and the equation overall is #color(blue)(y = 4x - 5)#.

Method of Linearization

#f_T(a) = f(a) + f'(a)(x-a)#

#color(blue)(f_T(1)) = f(1) + f'(1)(x-1)#

All you need is to plug in #1# to the function and its derivative, and plug it back into this formula.

#f(1) = 1^2 - 2/1 = -1#

#f'(1) = 2 + 2/1^2 = 4#

#f_T(1) = -1 + 4(x - 1)#

#= color(blue)(4x - 5)#

Try whichever method works for you; this one is a little more formulaic and may require less thought for some.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Axel Alvarez

The equation of the tangent line of f(x)=x^2-2/x at x=1 is y = 3x - 1.

By signing up, you agree to our Terms of Service and Privacy Policy

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.