   Chapter 3.R, Problem 51E

Chapter
Section
Textbook Problem

# Use Newton’s method to find the absolute maximum value of the function f ( t ) = cos t + t − t 2 correct to eight decimal places.

To determine

To find:

The absolute maximum value of given function correct to eight decimal places

Explanation

1) Concept:

Working rule of Newton’s method is

To find a root of y=f(x) start with an initial approximation x1.  After the first iteration of Newton’s method, we will get x2 which is actually the x-intercept of the tangent at that point (x1,f(x1)) . Now we draw a tangent at that point (x2,f(x2)), and the x-intercept at that tangent will be x3. Continue to do this till the value of xn tends to converge

2) Formula:

i. Newton’s formula for nth approximation is xn+1=xn-fxnf'xn for n=1,2,3,

ii. Power rule of differentiation ddxxn=nxn-1

iii.

ddxconstant=0

3) Given:

ft=cost+t-t2

4) Calculation:

Here, ft=cost+t-t2

Differentiate f(t) with respect to t, using power rule,

f't=-sint+1-2t

Differentiate f't with respect to t,

f''t=-cost-2

Draw the graph of f(x) to maximum root,

From the graph, take t1=0.5

So Newton’s formula for nth approximation becomes

tn+1=tn-ftnf'tn

To find the absolute maximum value, use Newton’s formula for f' and f'' means

tn+1=tn-f'(tn )f''tn

For t1=0

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