alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

terms with higher powers of x a l 3 limit step now

Info iconThis preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: -- f(a) b -- a INSTANT ANTANEOUS RATE 14. INSTANTANEOUS RATE OF CHANGE of xn Let us consider how to derive the general expression for I N S TA N TA N E O U S R AT E O F C H A N G E of the general polynomial : a n x n + a n -- 1 x n -- 1 + . . . + a 1 x + a 0 Obser ve that the ter ms of this function are of the for m a n x n for n = 0, 1, 2, 3, . . . To DIFFERENTIATE the general polynomial we should be able to find the INSTANTANEOUS RATE OF CHANGE of the general variable term x n . We can think of x n a s the function x n. In other words, we should be able to DIFFERENTIATE the function f(x) = x n. As usual we will first find the INSTANTANEOUS RATE OF CHANGE of the function f(x) = xn at par ticular point a . Then we will find the general expression of the particular par any INSTANTANEOUS RATE OF CHANGE f(x) = xn at any point x. an We note that : Value x=a f(x) = an We can differentiate f(x) = xn at x = a in two ways : 1. The Vedic way - without the use of δ x . 2. The Western way - using the infinitesimal δ x . 65 wa Vedic w ay : f ’ (a) = (a) Limit x→a AVERAGE RATE OF CHANGE close to a . = x n -- a n x -- a CHANGE in function CHANGE in variable So : f(x) − f(a) xn − an = x−a x−a Properly speaking we have to compute both : L imit f(x) − f(a) L imit f(x) − f(a) and x → a+ x−a x−a But in the Vedic way the Algebraic notation is lacking. For n = 1 and n = 2 the reader may wish to review example 4 (page 28) and example 5 (page 29). x → a-- For n = 3 : Limit x 3--a 3 = Limit ( x--a) (x 2 +xa+a 2 ) Limit = x → a (x 2 +xa+a 2 ) x → a x--a x→a (x--a) Substitute for x = a to get this Limit = 3a 2 Let us do this in three steps for general n. VERAGE step 1. AVERAGE step : f(x) − f(a) xn − an = x−a x−a 2. TENDS TO step : x n--a n = (x − a)(x n--1 + x n--2 a + x n--3 a 2 + . . . + xa n-- 2 + a n-- 1 ) x--a x--a = (x n--1 + x n--2 a + x n--3 a 2 + . . . + xa n-- 2 + a n-- 1 ) We may simplify because (x − a) ≠ 0 . 3. LIMIT step : Limit (x n--1 + x n--2 a + x n--3 a 2 + . . . + xa n-- 2...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online