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# ODD11 - C H A P T E R 11 Vector-Valued Functions Section...

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C H A P T E R 1 1 Vector-Valued Functions Section 11.1 Vector-Valued Functions . . . . . . . . . . . . . . . . . . . . . . 39 Section 11.2 Differentiation and Integration of Vector-Valued Functions . . . . 44 Section 11.3 Velocity and Acceleration . . . . . . . . . . . . . . . . . . . . . 48 Section 11.4 Tangent Vectors and Normal Vectors . . . . . . . . . . . . . . . 54 Section 11.5 Arc Length and Curvature . . . . . . . . . . . . . . . . . . . . . 60 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

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39 C H A P T E R 1 1 Vector-Valued Functions Section 11.1 Vector-Valued Functions Solutions to Odd-Numbered Exercises 1. Component functions: Domain: , 0 0, h t 1 t g t 4 t f t 5 t r t 5 t i 4 t j 1 t k 3. Component functions: Domain: 0, h t t g t e t f t ln t r t ln t i e t j t k 5. Domain: 0, r t F t G t cos t i sin t j t k cos t i sin t j 2 cos t i t k 7. Domain: , r t F t G t i sin t 0 j cos t sin t k 0 cos t cos 2 t i sin t cos t j sin 2 t k 9. (a) (b) (c) (d) 2 t 1 2 t 2 i t j 2 2 t 1 2 t 2 i 1 t j 2 i j r 2 t r 2 1 2 2 t 2 i 2 t 1 j 2 i j r s 1 1 2 s 1 2 i s 1 1 j 1 2 s 1 2 i s j r 0 j r 1 1 2 i r t 1 2 t 2 i t 1 j 11. (a) (b) is not defined. does not exist. (c) (d) ln 1 t i 1 1 t 1 j 3 t k r 1 t r 1 ln 1 t i 1 1 t j 3 1 t k 0 i j 3 k r t 4 ln t 4 i 1 t 4 j 3 t 4 k ln 3 r 3 r 2 ln 2 i 1 2 j 6 k r t ln t i 1 t j 3 t k
13. r t sin 3 t 2 cos 3 t 2 t 2 1 t 2 r t sin 3 t i cos 3 t j t k 15. a scalar. The dot product is a scalar-valued function. 5 t 3 t 2 , 3 t 3 t 2 2 t 3 4 t 3 r t u t 3 t 1 t 2 1 4 t 3 8 4 t 3 17. Thus, Matches (b) z x 2 . z t 2 y 2 t , x t , 2 t 2 r t t i 2 t j t 2 k , 19. Thus, Matches (d) y x 2 . z e 0.75 t y t 2 , x t , 2 t 2 r t t i t 2 j e 0.75 t k , 21. (a) View from the negative x -axis: (c) View from the z -axis: 0, 0, 20 20, 0, 0 (b) View from above the first octant: (d) View from the positive x -axis: 20, 0, 0 10, 20, 10 23. x 6 4 2 2 4 y y x 3 1 y t 1 x 3 t 25. x 3 4 5 3 2 4 5 1 4 3 2 y 2 1 2 5 3 6 7 y x 2 3 x t 3 , y t 2 27. Ellipse 2 3 2 3 1 2 x y x 2 y 2 9 1 x cos , y 3 sin 29. Hyperbola x 12 9 6 6 6 3 9 12 9 12 12 9 6 3 y x 2 9 y 2 4 1 x 3 sec , y 2 tan 31. Line passing through the points: x y (0, 6, 5) (1, 2, 3) (2, 2, 1) 4 3 5 6 4 3 5 1 3 z 1, 2, 3 0, 6, 5 , z 2 t 3 y 4 t 2 x t 1 33. Circular helix x y 3 3 3 7 z z t x 2 4 y 2 4 1 z t y 2 sin t , x 2 cos t , 35. z e t x y 3 3 3 6 z x 2 y 2 4 z e t y 2 cos t , x 2 sin t , 37. , z 2 3 x 3 y x 2 , x y 5 2 6 4 2 2 4 6 2, 4, 2, 4, ( ( ) ) 16 16 3 3 z z 2 3 t 3 y t 2 x t , t 0 1 2 x 0 1 2 y 4 1 0 1 4 z 0 16 3 2 3 2 3 16 3 1 2 1 2 40 Chapter 11 Vector-Valued Functions

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39. Parabola y x 2 3 2 3 1 2 3 2 3 4 5 1 z r t 1 2 t 2 i t j 3 2 t 2 k 41. Helix y x 2 3 4 1 2 1 2 2 z r t sin t i 3 2 cos t 1 2 t j 1 2 cos t 3 2 k 43. y x π 2 π 2 2 2 2 z (a) The helix is translated 2 units back on the x -axis. y x π 2 π 1 3 1 2 2 2 z (b) The height of the helix increases at a faster rate. y x π 8 π 4 2 2 2 2 z (d) The axis of the helix is the x -axis.
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