12 SL Derivatives, Tangents and Normals

After today’s review of techniques for differentiation, try the questions listed below before our next class.

We’ve already seen how to find the equation of the tangent line to a curve that passes through a given point, and finding a normal through a given point on a curve is essentially the same. A normal is a line that is perpendicular to a tangent, and you will recall that if one line has slope $m$, any line perpendicular to it will have slope $-\frac{1}{m}$. For example, the equation of the line tangent to the function $f(x)=x^2$ that passes through the point $(1,1)$ has slope $2$ (found by taking the derivative), and so has equation $y=2x-1$. Similarly, the normal to the function $f(x)=x^2$ that passes through the point $(1,1)$ has equation $y=-\frac{1}{2}x-1$.