Estimating Functions

Estimating Functions - Contents 7 Estimating Functions 115...

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Unformatted text preview: Contents 7 Estimating Functions 115 7.1 Estimating Functions by Simpler Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.2 A Strategy for Computing Bounds on | f ( x ) | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 2 CONTENTS Chapter 7 Estimating Functions 7.1 Estimating Functions by Simpler Functions Preliminaries You will also need the following results and the material given in your lectures. A. Let n be a positive integer. If | x | ≤ 1, then | x | ≥ | x | n . If | x | ≥ 1, then | x | ≤ | x | n . If | x | ≤ 1, then | x | ≤ | x | 1 n . If | x | ≥ 1, then | x | ≥ | x | 1 n . If | x | ≤ | y | and a > 0 then | x | a ≤ | y | a . B. If | x | ≥ 1, use | sin x | ≤ 1 and | cos x | ≤ 1. If | x | ≤ 1, use | sin x | ≤ | x | . C. | x | ≤ √ 1 + x 2 ≤ | x | + 1. Problems 1. Prove that for any real numbers x and y | x | - | y | ≤ | x + y | . 2. Using that the square root function is an increasing function show that p | x | ≤ p 1 + | x | ≤ 1 + | x | 2 . Hint: compute 1 + | x | 2 2 . 3. Show that if | x | ≤ 1 then 2 | x | ≤ | x 4 + 3 x | ≤ 4 | x | . 4. Show that if | x | ≤ 1 then 1- 6 | x | ≤ | x 4 +3 x 3- 2 x +1 | ≤ 6 | x | +1. Conclude that 0 ≤ | x 4 +3 x 3- 2 x +1 | ≤ 7 on the interval [- 1 , 1]. 116 Estimating Functions 5. Show that if | x | ≤ 1 then 0 ≤ | x 5- x 4 + x 3 + x- 1 | ≤ 4 | x | + 1. 6. Show that if | x | < 1 4 then 1- 4 | x | ≤ | x 5- x 4 + x 3 + x- 1 | ≤ 4 | x | + 1. 7. Show that if | x | ≤ 1 then 0 ≤ | 3 x 3- 2 x 2- x + sin x | ≤ 7 | x | . 8. Show that if | x | ≤ 1 then 2 | x | ≤ | 3 x 3- 2 x 2 + 8 x + sin x | ≤ 14 | x | . 9. Show that if | x | ≤ 1 10 then 0 < 5- 10 | x | ≤ | 2 x 2- 8 x + 5 | ≤ 10 | x | + 5. 10. Show that if | x | ≤ 1 then 0 ≤ | sin x + cos x | ≤ | x | + 1. 11. Show that if | x | ≤ 1 then 0 ≤ | 5sin x + 3cos x | ≤ 5 | x | + 3. 12. Show that if | x | ≤ 1 then 1- x 2 ≤ | x | + 1. 13. Show that √ x 2 + 2 x + 2 ≤ | x | + 2. 14. Show e x 2 ≤ q (sin 2 x ) + e x ≤ 1 + e x 2 . Conclude that in [0 , 1] 1 ≤ p (sin x ) 2 + e x ≤ 1 + √ e . 15. Show that p sin 2 x + 2cos 2 x ≤ √ 2. 16. On the interval [- 1 , 1] show that √ 1- x 2 ≤ cos x ≤ 1. Graph the three functions on a common coordinate system. 17. On the interval [- 1 , 1] show that 1- x 2 ≤ √ 1- x 2 ≤ 1. Graph the three functions on a common coordinate system. 18. On the interval [- 1 , 1] show that 1- | x | ≤ 1- x 2 . Graph both functions together on a common coordinate system. 19. If | f 1 ( x ) | ≤ 2 on [0 , 2] and | f 2 ( x ) | ≤ x on [1 , 3] what can you say about | f 1 f 2 | on [1 , 2]? 20. Show | x 2 + x + 2 | ≤ 2 | x | 2 if | x | ≥ 2. What can you say about | ( x 2 + x + 2)sin πx | on [2 , ∞ )?...
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This note was uploaded on 09/22/2011 for the course ECON 101 taught by Professor Mr.tull during the Spring '11 term at De La Salle University.

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Estimating Functions - Contents 7 Estimating Functions 115...

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