What is the equation of the line that is normal to #f(x)= x^3-4x# at # x=0 #?
A normal line is a straight line perpendicular to a tangent line. Since perpendicular lines have opposite reciprocal slopes, the way we find a normal line is to find the slope of a tangent line and then do the opposite reciprocal stuff.
But what is the slope of the tangent line at
That means the slope of the tangent line of
The slope of the normal line is perpendicular to this, so we take the opposite reciprocal of
Opposite reciprocal of
However, we're being asked for the equation of the normal line, not just the slope. As a result, we still have some more work to do. Recall that straight lines (the normal line is straight) take the form
Note that if we find a point
Our point is
Making substitutions in