Pre-Calc Homework Solutions 467

Pre-Calc Homework Solutions 467 - Appendix A6 40. 180 x and...

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Unformatted text preview: Appendix A6 40. 180 x and y (a) (b) x a2 2 467 x cos 180 y sin 180 y x x sin 180 y b2 2 y cos 180 43. (a) B 2 4AC 42 4(1)(4) 0, so the discriminant indicates that this conic is a parabola. (b) The left-hand side of x 2 4xy 4y 2 6x 12y 9 0 factors as a perfect square: (x 2y 3)2 0 x 2y 3 0 2y x 3; thus the curve is a degenerate parabola (i.e., a straight line). 44. (a) B 2 4AC 62 4(9)(1) 0, so the discriminant indicates that this conic is a parabola. 1 1 a2 y b 1 y m( x ) y y m( x ) mx by mx b x2 a2 2 y2 b2 (c) x (d) y (e) y 41. (a) B y 2 mx mx 2 4AC 2x 2x x 1 1 4(0)(0) 0 y(x 1 hyperbola 1) 2x y and we want 2x x 1 (b) The left-hand side of 9x 2 6xy y 2 12x 4y 4 0 factors as a perfect square: (3x y 2)2 0 3x y 2 y 3x 2; thus the curve is a degenerate parabola (i.e., a straight line). 45. Assume the ellipse has been rotated to eliminate the xy-term the new equation is A x the semi-axes are 1 and A 2 AC 4A C 2 0 (b) xy (c) y dy dx 2 (x 1) 2 Cy 2 1 1 the area is C dy dx 2, the slope of y 2 (x 2 1)2 2x 1)2 4 3 (3, 3) is a 1 A 1 C . 4A C 2 4AC B2 x (x Since B2 (because B 4AC B 2 4A C 0) we find that the area is as 3 or x 1; x 3y claimed. point on the hyperbola where the line with slope m y x 3 2 is normal the line is 2(x 1y 3) or y 1 ( 1, 2x 3; 46. (a) A C (A cos2 B cos sin C sin2 ) 2 2 (A sin B cos sin C cos ) A(cos2 sin2 ) C(sin2 cos2 ) A C E sin )2 ( D sin (b) D 2 E 2 (D cos 2 cos ) D 2 cos2 2DE cos sin E2 sin2 D 2 sin2 2DE sin cos E 2 cos2 2 2 2 2 2 D (cos sin ) E (sin cos2 ) 2 2 D E E 1) is a point on the 2 is normal hyperbola where the line with slope m the line is y 1 2(x 1) or y s Appendix A6 2x 3 (pp. 618627) 1. sinh x 1 3 cosh x 1 sinh2 x 4 9 25 5 sinh x , tanh x 16 4 16 cosh x 3 , coth x 5 1 sin x 1 tanh x 4 3 1 3 2 4 [ 9.4, 9.4] by [ 6.1, 6.1] 42. (a) False: let A parabola C 1, B 2 B2 4AC 0 3 4 5 4 5 , sech x 3 1 cosh x (b) False: see part (a) above (c) True: AC 0 hyperbola 4AC 0 B2 4AC 0 4 , and csch x 5 ...
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