Estimating Functions

# Estimating Functions - Contents 7 Estimating Functions 115...

This preview shows pages 1–5. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 , ∞ )?...
View Full Document

## 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.

### Page1 / 14

Estimating Functions - Contents 7 Estimating Functions 115...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online