Chapter 3.6, Problem 77E

Investigation Let $P(x_0,y_0)$ be an arbitrary point on the graph of f that $f^{\text{'}}(x_0)\ne 0$, as shown in the figure. Verify each statement.

(a) The x-intercept of the tangent line is

$\left(x_0-\frac{f(x_0)}{f^{\text{'}}(x_0)},0\right)$

(b) The y-intercept of the tangent line is

$(0,f(x_0)-x_0{f}^{\text{'}}(x_0))$

(c) The x-intercept of the normal line is

$(x_0+f(x_0)f^{\text{'}}(x_0),0)$

(The normal line at a point is perpendicular to the tangent line at the point.)

(d) The y-intercept of the normal line is

$\left(0,y_0+\frac{x_0}{f^{\text{'}}(x_0)}\right)$

(e) $\left|BC\right|=\left|\frac{f(x_0)}{f^{\text{'}}(x_0)}\right|$

(f) $\left|PC\right|=\left|\frac{f(x_0)\sqrt{1+{[f^{\text{'}}(x_0)]}^2}}{f^{\text{'}}(x_0)}\right|$

(g) $\left|AB\right|=\left|f(x_0)f^{\text{'}}(x_0)\right|$

(h) $\left|AP\right|=\left|f(x_0)\right|\sqrt{1+{[f^{\text{'}}(x_0)]}^2}$