13.1 Curves 5.

(a) This curve spirals up the cone x2 + y2 = Z2 from (0,1,1) to (0, e27r , e27r )

and has arc length V3(e27r - 1).

(b) This curve coincides with the graph of x = ±y3/2, ° ~ Y ~ 1 (looking like a stylized gull in flight) and has arc length 2( v'133 -1)/27.

(c) This curve is a straight-line segment from (0,0,0) to (4,4,4)

and has arc length 4V3.

(d) The arc length of the astroid is 6.

6.

(a) 27.

(b) ab(a2 + ab + b2 )/(3(a + b)).

(c) 127r.

(d) (5 + 3V5)/2.

7.

(b) Use Dini's Theorem.

9. Analyze what happens to (x,y) and dy/dx := (dy/dt)/(dx/dt) as t --> -00, t --> -1-, t --> -t+, t --> 0, and t --> 00. For example, prove that as t --> -1-, the trace of ¢(t) lies in the fourth quadrant and is asymptotic to the line y = -x.

11.

(b) Take the derivative of v' . v' using the Dot Product Rule.

(d) Observe that

¢(t) = v(f(t)) and use the Chain Rule to compute ¢'(t) and ¢"(t). Then calculate

¢' x ¢" directly.