| The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages. |
|
Home  Continuous Functions Properties of Continuous Functions Proof Of Theorem 1 on Curve Sketching |
|
 Search the VIAS Library | Index
|
  |
Proof Of Theorem 1 on Curve Sketching
Theorems 1 and 2 in Section 3.7 on curve sketching are consequences of the Mean Value Theorem. As an illustration, we prove part (ii) of Theorem 1: If f'(x) > 0 for all interior points x of I, then f is increasing on I. PROOF Let x1 < x2 where x1 and x2 are points in I. By the Mean Value Theorem there is a point c strictly between x1 and x2 such that
Since c is an interior point of I, f'(c) > 0. Because x1 < x2, x2 - x1 > 0. Thus
This shows that f is increasing on I.
|
|
| Home Continuous Functions Properties of Continuous Functions Proof Of Theorem 1 on Curve Sketching |
|
, f(x2) - f(x1) > 0, f(x2) > f(x1).
Last Update: 2006-11-05