How do you find parametric equations of a tangent line?

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Answer 1 · 2017-02-15T14:47:49

The parametric equations of the tangent line to the curve #y=f(x)# in the point #(x_0, f(x_0))# are:

#{(x=x_0+t),(y= f(x_0)+f'(x_0)t):}#

Given a curve #y=f(x)#, the slope intercept form of the equation of the tangent line to the point #(x_0, f(x_0))# is:

#y(x) = f(x_0) +f'(x_0)(x-x_0)#

So, if we pose:

#x= x_0+t#

we have:

#y = f(x_0) +f'(x_0)(x_0+t-x_0) = f(x_0)+f'(x_0)t#

The parametric equations are then:

#{(x=x_0+t),(y= f(x_0)+f'(x_0)t):}#