a. The graph y = (x) in the xy-plane automatically has the parametrization x = x, y = (x), and the vector formula r(x) =

a. The graph y = ƒ(x) in the xy-plane automatically has the parametrization x = x, y = ƒ(x), and the vector formula r(x) = xi + ƒ(x)j. Use this formula to show that if ƒ is a twice-differentiable function of x, then

b. Use the formula for k in part (a) to find the curvature of y = ln (cos x), -π/2

c. Show that the curvature is zero at a point of inflection.

Exercise 1

Find T, N, and κ for the plane curves.

r(t) = ti + (ln cos t)j, -π/2

. 7/ [z((x), f) + 1] |(x) uf| K(X) =