/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q. 18 Consider the function聽F(x)=聽聽... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Q. 18 (page 362)

Consider the function

F(x)=lnx,ifx<0lnx+4,ifx>0

Show that the derivative of this function is the function f(x)=1x. Compare the graphs of F(x) and lnx, and discuss how this exercise relates to the second part of Theorem 4.16.

Short Answer

Expert verified

Proved that鈭�f(x)g(x)dx鈮�鈭�f(x)dx鈭�g(x)dx

Step by step solution

01

Step 1. Given information 

The given functionF(x)=lnx,ifx<0lnx+4,ifx>0

02

Step 2. Prove ∫f(x)g(x)dx≠∫f(x)dx∫g(x) dx

Let

f(x)=xg(x)=1x

Now,

=鈭�f(x)g(x)dx=鈭�x1xdx=鈭�1dx=x+C

And

=(鈭�f(x)dx)(鈭�g(x)dx)=(鈭�xdx)(鈭�1xdx)=x22+Clnx+C

Therefore,鈭�f(x)g(x)dx鈮�鈭�f(x)dx鈭�g(x)dx

Hence Proved.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use integration formulas to solve each integral in Exercises 21鈥�62. You may have to use algebra, educated guess- and- check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating. (Hint for Exercise 54: tanx=sinxcosx).

鈭�x2x3+15dx.

Calculate the exact value of each definite integral in Exercises 47鈥�52 by using properties of definite integrals and the formulas in Theorem 4.13.

鈭�613(1-2x)2+4xdx

Use integration formulas to solve each integral in Exercises 21鈥�62. You may have to use algebra, educated guess and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating .

鈭�x2-1(3x+5)dx

Approximations and limits: Describe in your own words how the slope of a tangent line can be approximated by the slope of a nearby secant line. Then describe how the derivative of a function at a point is defined as a limit of slopes of secant lines. What is the approximation/limit situation described in this section?

Find the sum or quantity without completely expanding or calculating any sums.

Given鈭�k=025k=325and鈭�k=328k-32=14,910, find the value of鈭�k=325k2-5k+9.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.