   Chapter 4, Problem 1PS

Chapter
Section
Textbook Problem

Using a Function Let L ( x ) = ∫ 1 x 1 t d t ,   x > 0 (a) Find L (1).(b) Find L'(x) and L'(1).(c) Use a graphing utility to approximate the value of x (to three decimal places) for which L ( x ) = 1 .(d) Prove that L ( x 1 x 2 ) = L ( x 1 ) + L ( x 2 ) for all positive values of x 1 , and x 2 .

(a)

To determine

To calculate: The value of L(1) for the function, L(x)=1x1tdt,x>0.

Explanation

Given:

L(x)=1x1tdt,x>0

Formula used:

Integration formula:

ab1tdt=[lnt]ab

Calculation:

Consider the provided expression:

L(x)=1x1tdt,x>0

Substituting

(b)

To determine

To calculate: The value of L(x),L(1) for the function, L(x)=1x1tdt,x>0.

(c)

To determine

To calculate: The value of x by the use of graphic utility such that L(x)=1 where, L(x)=1x1tdt,x>0.

(d)

To determine

To prove: That L(x1x2)=L(x1)+L(x2), where L(x)=1x1tdt,x>0.

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