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.

21-22. Find equations of the normal plane and osculating plane of the curve at the given point, and plot the graphs.

21.

23. At what point on the curve  is the tangent plane parallel to the plane .

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 .