SECT. 9-B LAGRANGE ERROR OR REMAINDER
Lagrange or Taylor Polynomial Remainder
If a non-alternating series is approximated,the method for finding the error is called theLagrange Remainder or Taylor’s TheoremRemainder.
Taylor’s Theorem Remainder
If f has derivatives of all orders in an openinterval containing c then for each positiveinteger n and for each x in the interval
Taylor’s Theorem Remainder
n is the degree of the Taylor Polynomial
c is where it is centered
x is the value we are attempting to approximate
z is the x-value between x and c which makes
a maximum.
1. Use a fifth degree Maclaurin polynomial to
approximate then find the Lagrangeremainder
Case 1: Increasing function
Types of functions
Case 2: decreasing function
Types of functions
Case 3: increasing and decreasing function
Types of functions
Case 4: Sine and Cosine
Types of functions
You may know the maximum value forexample: (sine and cosine functions have amaximum value of 1).
2. If is a decreasing function, find theerror bound when a fifth degree TaylorPolynomial centered at x = 4 is used toapproximate f(4.1). (set up but do notevaluate)
3.Approximate
using a third degree Maclaurin polynomial
3.
b) then use the Lagrange error bound to showthat
4. Selected values of f and its first 4 derivativesare given in the table. The function f and itsderivatives are decreasing on the interval0<x<4
a)Write a third degree Taylor Polynomial for f about
x = 3 and use it to approximate f(3.1)
x
f(x)
f’(x)
f’’(x)
f’’’(x)
f(4)(x)
3
12
-18
-38
-67
-17
4.Continued
b) Use the Lagrange error bound to show thatthe third degree Taylor Polynomial for f aboutx = 3 approximates f(3.1) with an error lessthan 0.00008
5. The third degree Taylor Polynomial of f aboutx = -2 is given by:
a) Find
5.Continued
b) Does h have a relative max, relative min or neitherat x = -2 ?
5. continued
c) The fourth derivative of f satisfies the inequality
on the interval
Use the Lagrange error bound to show that
6. If and if x = 0.7 is theconvergence interval for the power seriescentered at x = 0, find an upper limit for theerror when the fourth-degree Taylorpolynomial is used to approximate f(0.7)
HOME WORK
Day 1 Worksheet 9-B
Day 2 Worksheet Error