Pre-Calc Homework Solutions 97

Pre-Calc Homework Solutions 97 - Section 3.6 dx dt dy dt dy...

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Unformatted text preview: Section 3.6 dx dt dy dt dy dx d t dt d dt dy dt dx dt 97 45. 1 t 2 1 t 1 2 t 1 4 1 1 , and has slope 4 2 1 4 1 , or 2 49. (a) dx dt dy dt dy dx d dy dt dx d 2 (t t) 2t dt d sin t cos t dt dy dt dx dt 1 1/(2 t) 1 cos t 2t 1 d cos t dt 2t 1 d (2t 1) (cos t) dt 1 The line passes through , 4 1 2 1 4 (b) (cos t) (2t 1) 2 d dt 1) 1. Its equation is y x 1 . 4 d (2t 2 dt d 4 (t ) dt dy dt dx dt 1 x (2t (2t y 46. dx dt dy dt dy dx 1)( sin t) (cos t)(2) (2t 1)2 1)(sin t) 2 cos t (2t 1)2 (2t 3) 4t 3 4t 3 4t 4t (c) Let u t2 Then d dy dx dx du dt dy . dx du dx du , so dx dt dx d dy dt dx (2t (2t du dt dx . Therefore, dt The line passes through (2( 1)2 3, ( 1)4) (5, 1) and has slope ( 1)2 = 1. Its equation is y 1(x 5) 1, or y x 4. 47. dx dt dy dt dy dx d (t dt d (1 dt dy dt dx dt dx dt 1)(sin t) 2 cos t (2t 1)2 1)(sin t) 2 cos t (2t 1)3 (2t 1) sin t) cos t) sin t cos t 1 cos t sin t (d) The expression in part (c). 50. Since the radius passes through (0, 0) and (2 cos t, 2 sin t), it has slope given by tan t. But the slope of the tangent is dy dx 3 2 3 1 , and has slope 2 dy dt dx dt 1 The line passes through 2 cos t 2 sin t cot t, which is the negative 3 sin 1 sin 3 3 ,1 cos 3 reciprocal of tan t. This means that the radius and the tangent are perpendicular. (The preceding argument breaks cos 3. Its equation is 3 down when t 3 3 2 1 , or 2 k , where k is an integer. At these values, 2 y 3 x either the radius is horizontal and the tangent is vertical or the radius is vertical and the tangent is horizontal, so the y 48. dx dt dy dt dy dx 3x 2 3 . result still holds.) 51. ds dt ds d d dt d d (cos ) dt d d dt 3 d and 2 dt d 2 (x dx dx dt dy dt 1 dy , 3 dt d cos t sin t dt d (1 sin t) cos t dt dy dt dx dt ( sin ) When 52. dy dt dy dx dx dt cos t sin t cot t 5, 7x ds dt sin dx dt 3 (5) 2 5. The line passes through cos has slope cot 2 ,1 sin 2 (0, 2) and 2. 5) 2 0. Its equation is y (2x When x 7) 1 and [2(1) 7] 1 3 3. ...
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