Allen – Math 265 B2 Homework #5 (20 points) Date Due: Monday, February 12, 2007
Instructions: Please work the following problems on other pieces of paper. You must show all work
and calculations in order to receive full credit. I would prefer to collect your assignments in class on the day
it is due, but I will accept it as late as 5:00 PM on the due date. Assignments submitted later will receive
50% credit.
1. Section 13.5 # 2. Use the formula K(t) = |~v(t)×~a(t)|
|~v(t)|3. Treat ~r(t) like a 3-D vector, but set the third
component to zero.
2. Section 13.5 # 22. Just to clarify, y= cosh (x/2).
3. Section 13.5 # 29.
4. Let Cbe the plane curve defined by the vector equation ~r(t) = (2t+ 1)
~
i+ (t2−2)~
j.
(a) Find aT(t) and aN(t).
(b) Find ~a(−1), aT(−1), aN(−1), and P(−1) (the point on the curve at t=−1).
(c) Sketch a graph of the curve for −2≤t≤0. Then draw ~a(−1) emanating from the point P(−1).
Finally, sketch the vectors aT(−1)~
T(−1) and aN(−1) ~
N(−1) emanating from the point P(−1). You
do not need to find ~
T(−1) and ~
N(−1) to do this.
5. Section 14.5 # 16. Hint: You will need the following integral formula.
Zpu2±a2du =u
2pu2±a2+a2
2ln
u+pu2±a2
+C.
6. Let Cbe the space curve defined by the parametric equations x=t,y=1
3t3,z=1
t;t > 0.
(a) Find ~v(1), ~a(1), and ~
T(1).
(b) Find K(1).
(c) Find aT(1) and aN(1).
(d) Find ~
N(1).
(e) Find ~
B(1).
(f) Torsion: Let τ(t) denote the torsion of a curve at time t. Its formula is given by
τ(t) = D(t)
|~v(t)×~a(t)|2
where D(t) =
x0(t)y0(t)z0(t)
x00(t)y00(t)z00 (t)
x000(t)y000 (t)z000 (t)
. Find τ(1).
1