Consider the curve traced out by the parametric equations: { x =1 + cos(t) y = t + sin(t) for 0 ≤ t ≤ 4π.
1. Show that that dy dx = − 1 + cos(t)/sin(t) = − csc(t) −cot(t).
2. Make a Sign Diagram for dy dx to find the intervals of t overwhich C is increasing or decreasing.
• C is increasing on: • C is decreasing on:
3. Show that d2y/dx2 = − csc2 (t)(csc(t) + cot(t))
4. Make a Sign Diagram for d 2 y dx2 to find the intervals of tover which C is concave up or concave down.
• C is concave up on: • C is concave down on:
5. Using all of your work from numbers 1 through 4, sketch adetailed graph of C. Label the points (x, y), if any, where C hashorizontal tangents, vertical tangents, or where C is notsmooth.
