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 the Lagrange Remainder or Taylor’s Theorem Remainder.

Taylor’s Theorem Remainder

If f has derivatives of all orders in an open interval containing c then for each positive integer n and for each x in the interval )()()( xRxPxf n

)()()( xfxPxR n

Taylor’s Theorem Remainder

)()(!

)(...)(

!2

)("))((')()( 2 xRcx

n

cfcx

cfcxcfcfxf n

nn

! 1

)()( where

11

n

cxzfxR

nn

n

n is the degree of the Taylor Polynomialc is where it is centeredx is the value we are attempting to approximatez is the x-value between x and c which makes

a maximum. 1nf z

1. Use a fifth degree Maclaurin polynomial to approximate then find the Lagrange remainder

6.e

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 for example: (sine and cosine functions have a maximum value of 1).

2. If is a decreasing function, find the error bound when a fifth degree Taylor Polynomial centered at x = 4 is used to approximate f(4.1). (set up but do not evaluate)

)()6( xf

!1

)( 11

n

cxzfLR

nn

3. Approximate using a third degree Maclaurin polynomial

35sin)(

xxf

3. b) then use the Lagrange error bound to show that

35sin)(

xxf

1200

1

15

1

15

1

Pf

4. Selected values of f and its first 4 derivatives are given in the table. The function f and its derivatives are decreasing on the interval 0<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 that the

third degree Taylor Polynomial for f about x = 3 approximates f(3.1) with an error less than 0.00008

5. The third degree Taylor Polynomial of f about x = -2 is given by:

a) Find 32

3 26

52

3

1)2(94)( xxxxP

)2(''' h

5.Continued

b) Does h have a relative max, relative min or neither at x = -2 ?

323 2

6

52

3

1)2(94)( xxxxP

5. continued

c) The fourth derivative of f satisfies the inequality on the interval

Use the Lagrange error bound to show that

323 2

6

52

3

1)2(94)( xxxxP

10)(4 xf 0,2

7)0()0(3 fP

6. If and if x = 0.7 is the convergence interval for the power series centered at x = 0, find an upper limit for the error when the fourth-degree Taylor polynomial is used to approximate f(0.7)

xxf sin700)()5(

! 1

)( 11

n

cxzfLR

nn

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