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Pre-Calc Homework Solutions 169

# Pre-Calc Homework Solutions 169 - (4 1 and has slope 1 2...

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Exploration 2 Using Newton’s Method on the Grapher 1–3. Here are the first 11 computations. 4. Answers will vary. Here is what happens for x 1 5 2 2. Quick Review 4.5 1. } d d y x } 5 cos ( x 2 1 1) ? } d d x } ( x 2 1 1) 5 2 x cos ( x 2 1 1) 2. } d d y x } 5 5 5 3. [ 2 2, 6] by [ 2 3, 3] x < 2 0.567 4. [ 2 4, 4] by [ 2 10, 10] x < 2 0.322 5. f 9 ( x ) 5 ( x )( 2 e 2 x ) 1 ( e 2 x )(1) 5 e 2 x 2 xe 2 x f 9 (0) 5 1 The lines passes through (0, 1) and has slope 1. Its equation is y 5 x 1 1. 6. f 9 ( x ) 5 ( x )( 2 e 2 x ) 1 ( 2 e 2 x )(1) 5 e 2 x 2 xe 2 x f 9 ( 2 1) 5 e 1 2 ( 2 e 1 ) 5 2 e The lines passes through ( 2 1, 2 e 1 1) and has slope 2 e . Its equation is y 5 2 e ( x 1 1) 1 ( 2 e 1 1), or y 5 2 ex 1 e 1 1. 7. (a) x 1 1 5 0 x 5 2 1 (b) 2 ex 1 e 1 1 5 0 2 ex 5 2 ( e 1 1) x 5 2 } e 2 1 e 1 } < 2 0.684 8. f 9 ( x ) 5 3 x 2 2 4 f 9 (1) 5 3(1) 2 2 4 5 2 1 Since f (1) 5 2 2 and f 9 (1) 5 2 1, the graph of g ( x ) passes through (1, 2 2) and has slope 2 1. Its equation is g ( x ) 5 2 1( x 2 1) 1 ( 2 2), or g ( x ) 5 2 x 2 1. 9. f 9 ( x ) 5 cos x f 9 (1.5) 5 cos 1.5 Since f (1.5) 5 sin 1.5 and f 9 (1.5) 5 cos 1.5, the tangent line passes through (1.5, sin 1.5) and has slope cos 1.5. Its equation is y 5 (cos 1.5)( x 2 1.5) 1 sin 1.5, or approximately y 5 0.071 x 1 0.891 [0, p ] by [ 2 0.2, 1.3] 10. For x . 3, f 9 ( x ) 5 } 2 ˇ x 1 w 2 w 3 w } , and so f 9 (4) 5 } 1 2 } . Since f (4) 5 1 and f 9 (4) 5 } 1 2 } , the tangent line passes through
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Unformatted text preview: (4, 1) and has slope } 1 2 } . Its equation is y 5 } 1 2 } ( x 2 4) 1 1, or y 5 } 1 2 } x 2 1. [ 2 1, 7] by [ 2 2, 2] Section 4.5 Exercises 1. (a) f 9 ( x ) 5 3 x 2 2 2 We have f (2) 5 7 and f 9 (2) 5 10. L ( x ) 5 f (2) 1 f 9 (2)( x 2 2) 5 7 1 10( x 2 2) 5 10 x 2 13 (b) Since f (2.1) 5 8.061 and L (2.1) 5 8, the approximation differs from the true value in absolute value by less than 10 2 1 . 1 2 cos x 2 ( x 1 1) sin x }}} ( x 1 1) 2 x 2 x sin x 1 1 2 sin x 2 x 2 cos x }}}} ( x 1 1) 2 ( x 1 1)(1 2 sin x ) 2 ( x 1 cos x )(1) }}}} ( x 1 1) 2 Section 4.5 169 x f ( x ) g ( x ) 0.7 2 1.457 2 1.7 0.8 2 1.688 2 1.8 0.9 2 1.871 2 1.9 1.0 2 2 2 2 1.1 2 2.069 2 2.1 1.2 2 2.072 2 2.2 1.3 2 2.003 2 2.3...
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