(a) Explanation Given: The value of a 1 = a , a 2 = f ( a ) , a 3 = f ( a 2 ) and this is equal to , then the nth sequence is a n + 1 = f ( a n ) , here f is the continuous function. Calculation: From the given data lim n → ∞ a n = L then, lim n → ∞ a n + 1 = lim n → ∞ a n lim n → ∞ a n + 1 = L Then, a n + 1 = f ( a n ) lim n → ∞ a n + 1 = lim n → ∞ a n lim n → ∞ a n + 1 = f { lim n → ∞ a n } lim n → ∞ a n + 1 = f ( L ) Then, L = f ( L ) . Hence, proved. (b) To determine To Prove: The term f ( L ) = L . Explanation Given: The value of a 1 = a , a 2 = f ( a ) , a 3 = f ( a 2 ) and this is equal to , then the nth sequence is a n + 1 = f ( a n ) , here f is the continuous function. Calculation: Consider that f ( x ) = cos that is equal to a = 1 . Since, cos x is trigonometric function is continuous that f ( x ) is continuous. Then, f ( L ) = L cos ( L ) = L L = 0.73909 Hence, proved.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition

4th Edition

ISBN: 9781337687805

Author: James Stewart

Publisher: Cengage Learning

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Question

Chapter 8.1, Problem 58E

(a)

To determine

To Prove: The term $f(L)=L$ .

(a)

Expert Solution

Explanation of Solution

Given:

The value of $a1=a$ , $a2=f(a)$ , $a3=f(a2)$ and this is equal to , then the nth sequence is $an+1=f(an)$ , here $f$ is the continuous function.

Calculation:

From the given data $limn→∞an=L$ then,

$limn→∞an+1=limn→∞anlimn→∞an+1=L$

Then,

$an+1=f(an)limn→∞an+1=limn→∞anlimn→∞an+1=f{limn→∞an}limn→∞an+1=f(L)$

Then,

$L=f(L)$ .

Hence, proved.

(b)

To determine

To Prove: The term $f(L)=L$ .

(b)

Expert Solution

Explanation of Solution

Given:

The value of $a1=a$ , $a2=f(a)$ , $a3=f(a2)$ and this is equal to , then the nth sequence is $an+1=f(an)$ , here $f$ is the continuous function.

Calculation:

Consider that $f(x)=cos$ that is equal to $a=1$ .

Since, $cosx$ is trigonometric function is continuous that $f(x)$ is continuous.

Then,

$f(L)=Lcos(L)=LL=0.73909$

Hence, proved.

Chapter 8.1, Problem 57E

Chapter 8.1, Problem 59E

Chapter 8 Solutions

Single Variable Calculus: Concepts and Contexts, Enhanced Edition

Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E Ch. 8.1 - Prob. 8E Ch. 8.1 - Prob. 9E Ch. 8.1 - Prob. 10E

Ch. 8.1 - Prob. 11E Ch. 8.1 - Prob. 12E Ch. 8.1 - Prob. 13E Ch. 8.1 - Prob. 14E Ch. 8.1 - Prob. 15E Ch. 8.1 - Prob. 16E Ch. 8.1 - Prob. 17E Ch. 8.1 - Prob. 18E Ch. 8.1 - Prob. 19E Ch. 8.1 - Prob. 20E Ch. 8.1 - Prob. 21E Ch. 8.1 - Prob. 22E Ch. 8.1 - Prob. 23E Ch. 8.1 - Prob. 24E Ch. 8.1 - Prob. 25E Ch. 8.1 - Prob. 26E Ch. 8.1 - Prob. 27E Ch. 8.1 - Prob. 28E Ch. 8.1 - Prob. 29E Ch. 8.1 - Prob. 30E Ch. 8.1 - Prob. 31E Ch. 8.1 - Prob. 32E Ch. 8.1 - Prob. 33E Ch. 8.1 - Prob. 34E Ch. 8.1 - Prob. 35E Ch. 8.1 - Prob. 36E Ch. 8.1 - Prob. 37E Ch. 8.1 - Prob. 38E Ch. 8.1 - Prob. 39E Ch. 8.1 - Prob. 40E Ch. 8.1 - Prob. 41E Ch. 8.1 - Prob. 42E Ch. 8.1 - Prob. 43E Ch. 8.1 - Prob. 44E Ch. 8.1 - Prob. 45E Ch. 8.1 - Prob. 46E Ch. 8.1 - Prob. 47E Ch. 8.1 - Prob. 48E Ch. 8.1 - Prob. 49E Ch. 8.1 - Prob. 50E Ch. 8.1 - Prob. 51E Ch. 8.1 - Prob. 52E Ch. 8.1 - Prob. 53E Ch. 8.1 - Prob. 54E Ch. 8.1 - Prob. 55E Ch. 8.1 - Prob. 56E Ch. 8.1 - Prob. 57E Ch. 8.1 - Prob. 58E Ch. 8.1 - Prob. 59E Ch. 8.1 - Prob. 60E Ch. 8.2 - 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To Prove: The term f ( L ) = L .

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To Prove: The term f(L)=L<m:math><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mi>L</m:mi><m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:mi>L</m:mi></m:mrow></m:math> .

Explanation of Solution

To Prove: The term f(L)=L<m:math><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mi>L</m:mi><m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:mi>L</m:mi></m:mrow></m:math> .

Explanation of Solution

Chapter 8 Solutions

To Prove: The term $f(L)=L$ .

To Prove: The term $f(L)=L$ .