Equations of the normal line

Question

Find the equation of the normal line to the graph of the function f(x) at the point x_0:

_f(x)=tan(X), x_{0= pi}

Answer 1

To find the normal line, we must first find the tangent slope, then take its negative reciprocal. Taking the derivative, we have:

f'(π) = sec²(π) = 1

Since the tangent slope is 1, the normal slope is therefore -1. Now, we need to find f(π), which is:

f(π) = tan(π) = 0

Now, we use the point-slope form y = m(x - a) + b for a line with slope m passing through point (a,b). Our line has slope -1 and passes through (π,0), so the equation of the normal line is:

y = -(x-π) = π - x