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:

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