Pre-Calc Homework Solutions 409

Pre-Calc Homework Solutions 409 - Section 10.3 51. u u1, u2...

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Unformatted text preview: Section 10.3 51. u u1, u2 , v (i) u (ii) v v1, v2 , w w1, w2 dy dx dy dx dy/dt dx/dt dy/dt dx/dt 5 cos t , which for t 4 sin t 5 cos t , which for t 4 sin t 409 3. 4. (iii) (iv) (v) (vi) (vii) (viii) (ix) u1 v1, u2 v2 v1 u1, v2 u2 v u (u v) w u1 v1, u2 v2 w1, w2 (u1 v1) w1, (u2 v2) w2 u1 (v1 w1), u2 (v2 w2) u (v w) u 0 u1, u2 0, 0 u1 0, u2 0 u1, u2 u u ( u) u1 , u 2 u1, u2 u1 u1, u2 u2 0, 0 0 0u 0 u1, u2 0u1, 0u2 0, 0 0 1u 1 u1, u2 1u1, 1u2 u1, u2 u a(bu) a(b u1, u2 ) a bu1, bu2 abu1, abu2 ab u1, u2 (ab)u a(u v) a u1 v1, u2 v2 au1 av1, au2 av2 au1, au2 av1, av2 a u1, u2 a v1, v2 au av (a b)u (a b) u1, u2 (a b)u1, (a b)u2 au1 bu1, au2 bu2 au1, au2 bu1, bu2 au bu 1 . Then their 3, a 1 1 , . 2 2 2 equals zero. is undefined: the tangent line is vertical. 5. dy dx dy/dt dx/dt 6 5 cos t , which for t 4 sin t 6 equals 5 3 . 4 Also, at t ,x 2 5 2 3 and y 5 4 3 5 . The equation for the 2 tangent line is y y 6. dy dx 5 3 x 4 dy/dt dx/dt 6 (x 2 3), or 10. 5 cos t , which for t 4 sin t 6 equals 5 3 . 4 Also, at t ,x 2 5 2 3 and y 4 5 3 5 . The equation for the 2 normal line is y y 7. lim 8. y x 4 15 2 4 3 (x 2 3), or x 9 . 10 2 x2 x x2 (x lim x 2 2)(x 2) 2 x2 x lim 1 2 1 4 3 2x, and Length 0 1 3.400. (3 2x)2 dx, which 52. Write the two vectors as a 1, 1 and b 1, sum is a a 7 ,b 2 using NINT evaluates to b, a b , so solve a 7 7 , 2 2 b b 4 to get 9. x t cos t 2 1 . So 3, 4 2 1 1 sin t, and y (t cos t sin t)2 t sin t cos t, and cos t)2 dt 53. (a) Slope y 1 1, so y 2) or y y1 x m(x 1. x1) becomes Length 0 2 0 ( t sin t (x t2 1 dt, 2.958. (b) Slope y 1 1, so y y1 m(x x 2 or y x 3. x1) becomes which using NINT evaluates to 10. y 2 y 54. The slopes of the lines are 3 and 1, which means that 4 xe x e x C (use integration by parts), so 0 e0 C and C 3: xe x e x 3 vectors 4, 3 and 1, 1 are parallel to the respective lines. cos 1 Section 10.3 Exercises 1. [5 ( 1)]i 3)i [0 (0 [0 (1 4)j 6i 3i 3j 4j 3i 4i i 7i 2j 6j 3(4)j] i 16j 2j and 3j. j 5j 2. (0 3. E AB CD E (a) [3 (b) [3 4. (a) (5 (b) (5 ( 4)]j 4 1 3 1 cos 1 7 5 2 2 10 8.130 . s Section 10.3 Vector-valued Functions (pp. 529539) Quick Review 10.3 1. f (x) 4 x x2 1 3 1 3 ( 3)]i (2 0)j 4)i ( 3 0)j ( 4)]i ( 4)]i 3)i 3)i [2 [2 ( 3)]j ( 3)]j 4]j 4]j 15i [3(3)i 8i 2i 6j [( 2) [( 2) 3( 2)j 2( 2)j] , so for x 3 1, f (x) 1 3 3 and (x 1) or (c) 3(5)i (d) [2(5)i f (x) y 2. f (x) f (x) . Then y x x 4 3 x2 . 1, f (x) 3 3(x 3 and 1) or y 3x. , so for x 1 4 3 . Then y ...
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