Tarea

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CHAPTER 13 VECTOR FUNCTIONS

EXAMPLE 8 Find and graph the osculating circle of the parabola y � x 2 at the origin.
y

osculating circle

y=≈

SOLUTION From Example 5 the curvature of the parabola at the origin is ��0� � 2. So the

radius of the osculating circle at the origin is 1�� � 1 and its center is (0, 1 ). Its equation 2 2 is therefore 2 x2 � (y � 1) � 1 2 4 Forthe graph in Figure 9 we use parametric equations of this circle: x � 1 cos t 2 y � 1 � 1 sin t 2 2
M

1 2

0

1

x

FIGURE 9

We summarize here the formulas for unit tangent, unit normal and binormal vectors, and curvature. r��t� r��t� T��t� T��t�

TEC Visual 13.3C shows how the osculating circle changes as a point moves along a curve.

T�t� �

�

�

N�t� �

��

� �

��

B�t� � T�t� � N�t�

dT � ds

� T��t� � � � r��t� � r��t� � � r��t� � � r��t� �
3

13.3

EXERCISES
12. Find, correct to four decimal places, the length of the curve

1–6 Find the length of the curve. 1. r�t� � � 2 sin t, 5t, 2 cos t � , 2. r�t� � � 2t, t 2, 3 t 3 � ,
1

�10 � t � 10

of intersection of the cylinder 4x 2 � y 2 � 4 and the plane x � y � z � 2.
13–14Reparametrize the curve with respect to arc length mea-

0�t�1 0�t�1 0 � t � ��4

3. r�t� � s2 t i � e t j � e�t k,

sured from the point where t � 0 in the direction of increasing t.
13. r�t� � 2t i � �1 � 3t� j � �5 � 4t� k 14. r�t� � e 2t cos 2t i � 2 j � e 2t sin 2t k 15. Suppose you start at the point �0, 0, 3� and move 5 units

4. r�t� � cos t i � sin t j � ln cos t k, 5. r�t� � i � t 2 j� t 3 k,

0�t�1 0�t�1

6. r�t� � 12t i � 8t 3�2 j � 3t 2 k,

7–9 Find the length of the curve correct to four decimal places.

along the curve x � 3 sin t, y � 4t, z � 3 cos t in the positive direction. Where are you now?
16. Reparametrize the curve

(Use your calculator to approximate the integral.)
7. r�t� � � st , t, t 2 � ,

1�t�4 1�t�2 0 � t � ��4 r�t� �

8. r�t� � � t, ln t, tln t� ,

�

2 2t �1 i� 2 j t2 � 1 t �1

�

9. r�t� � � sin t, cos t, tan t� ,

; 10. Graph the curve with parametric equations x � sin t,
y � sin 2t, z � sin 3t. Find the total length of this curve correct to four decimal places.
11. Let C be the curve of intersection of the parabolic cylinder

with respect to arc length measured from the point (1, 0) in the direction of increasingt. Express the reparametrization in its simplest form. What can you conclude about the curve?
17–20

x � 2y and the surface 3z � xy. Find the exact length of C from the origin to the point �6, 18, 36�.

2

(a) Find the unit tangent and unit normal vectors T�t� and N�t�. (b) Use Formula 9 to find the curvature.
17. r�t� � �2 sin t, 5t, 2 cos t�

SECTION 13.3 ARC LENGTH AND CURVATURE||||

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18. r�t� � � t 2, sin t � t cos t, cos t � t sin t� , 19. r�t� � �s2 t, e , e
t �t

t�0

�

20. r�t� � � t, t , t 2 �
1 2 2

36 –37 Two graphs, a and b, are shown. One is a curve y � f �x� and the other is the graph of its curvature function y � ��x�. Identify each curve and explain your choices. 36.
y

37.
y

21–23 Use Theorem 10 to find the curvature. 21. r�t� � t 2 i� t k 22. r�t� � t i � t j � �1 � t 2 � k 23. r�t� � 3t i � 4 sin t j � 4 cos t k 24. Find the curvature of r�t� � � e t cos t, e t sin t, t� at the
CAS

a b
x

a b
x

38. (a) Graph the curve r�t� � �sin 3t, sin 2t, sin 3t� . At how

point (1, 0, 0).
25. Find the curvature of r�t� � � t, t , t � at the point (1, 1, 1).
2 3

; 26. Graph the curve with parametric equations
x�t y � 4t3�2 z � �t 2
CAS

many points on the curve does it appear that the curvature has a local or absolute maximum? (b) Use a CAS to find and graph the curvature function. Does this graph confirm your conclusion from part (a)?
39. The graph of r�t� � � t �

and find the curvature at the point �1, 4, �1�.
27–29 Use Formula 11 to find the curvature. 27. y � 2x � x 2 28. y � cos x 29. y � 4x 5�2...