12.3 Arc Length and Curvature by SGLee - HSKim - JHLee
1-5. Find the length of the curve.
1. 
, 
.
.
.
Calculus-Sec-12-3-Solution
12.3 Arc Length and Curvature by SGLee - HSKim - JHLee
1-5. Find the length of the curve.
1. ,
.
.
.
3. 
, 
.
.
.
5. 
, 
,
,
.
6-8. Reparametrize the curve with respect to arc length measured from the point where 
in the direction of increasing 
.
6. 
Since 
, 
.
Thus 
, this implies 
.
Substituting 
in 
, we have 
.
7. 
Since 
, 
.
Thus 
.
Substituting 
in 
, we have 
.
9-10. Find the unit tangent 
, unit normal vectors 
and the curvature 
.
9. 
, 
.
Hence 
.
, 
.
Hence 
.
.
10. 
, 
11-13. Use Theorem 7 to find the curvature.
11. 
,
,
.
13. 
,
.
.
14. Find the curvature of 
at the point 
.
,
,
,
,
,
, 
. <-- k(0)=sqrt(2)/5
15. Find the curvature of 
at the point 
.
,
,
,
.
16-18. Use Formula 6 to find the curvature.
16. 
,
. <-- 분자가 2
17. 
,
. <-- 분자가 abs(sin x + cos x)
18. 
,
.
19-20. Find the vectors 
, 
and 
at the given point, and plot at some point.
19. 
, 
http://matrix.skku.ac.kr/cal-lab/cal-13-3-19.html
21-22. Find equations of the normal plane and osculating plane of the curve at the given point, and plot the graphs.
21. 
http://matrix.skku.ac.kr/cal-lab/cal-13-3-21.html
23. At what point on the curve 
is the tangent plane parallel to the plane 
.
http://matrix.skku.ac.kr/cal-lab/cal-13-3-23.html
24. The curvature at a point 
of a curve is defined as 
,
when 
is the angle of inclination of the tangent line at 
, as shown in the figure.
Thus, the curvature is the absolute value of the rate of change of 
with respect to arc length.
It can be regarded as a measure of the rate of change of direction of the curve ar 
and will be studied in greater detail in Chapter 12.
(a) For a parametric curve 
, 
derive the formula
.
where the dots indicate derivatives with respect to 
, that is, 
. [Hint: Use 
and Equation 
to find 
. Then use the Chain Rule to find 
.]
(b) For a curve 
as the parametric curve 
, 
, show that the formula in part (a) becomes
.
(a) Here 
,
, and
.
Thus 
.
(b) Here 
.
By applying the result of (a) to this case, we obtain that
.
25. (a) Show that the curvature at each point of a straight line is 
.
(b) Show that the curvature at each point of a circle of radius 
is 
.
(a) For a straight line, we parametrize 
. Since 
is a straight line,
is a constant, and hence 
is zero. By #30-(b), we have 
.
(b) 
. Then,
and 
. By problem 30 (a), we have 
.
2. 
, 
4. 
, 
8. 
12. 
, 
