/*! 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} Problem 11 (a) Over what open interval does... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 11

(a) Over what open interval does the formula \(F(x)=\int_{1}^{x} \frac{d t}{t}\) represent an antiderivative of \(\mathrm{f}(\mathrm{x})=1 / \mathrm{x} ?\) (b) Find a point where the graph of F crosses the \(\mathrm{x}\)-axis.

Short Answer

Expert verified
Question: Determine the open interval where the function F(x) represents an antiderivative of f(x) = 1/x and the point where the graph of F crosses the x-axis. Answer: The open interval for which F(x) represents an antiderivative of f(x) is (-鈭�,0)鈭�(0,鈭�), and the graph of F crosses the x-axis at the point (1,0).

Step by step solution

01

Check the interval of the integral and check continuity

Since \(\frac{1}{t}\) is continuous for all nonzero values of t, we can find the antiderivative of \(\frac{1}{t}\). The integral is from 1 to x, meaning that the function is continuous for all x except x=0. Thus, the open interval for which the formula represents an antiderivative is \((-\infty,0)\cup(0,\infty)\).
02

Check if the function F is an antiderivative of f

To check if F(x) is an antiderivative, we will find the derivative of F(x) and see if it matches the function f(x). Using the Fundamental Theorem of Calculus, we have: \(F'(x)=\frac{d}{dx}(\int_{1}^{x} \frac{d t}{t})=\frac{1}{x}\) Since \(F'(x)=\frac{1}{x}\) and we are given that \(f(x)=\frac{1}{x}\), we can conclude that F(x) is indeed an antiderivative of f(x) over the interval \((-\infty,0)\cup(0,\infty)\).
03

Find the point where the graph of F crosses the x-axis

To do this, we must find the value of x for which F(x) = 0. We are given the formula for F(x): \(F(x)=\int_{1}^{x} \frac{d t}{t}\) We need to find x such that: \(\int_{1}^{x} \frac{d t}{t}=0\) The value of the integral is zero when the lower limit is equal to the upper limit. Therefore, \(F(x)=0\) when \(x=1\). This means that the graph of F crosses the x-axis at the point (1,0).

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

Prove that \(\int_{0}^{2 \lambda} \frac{\sin x}{x} d x=\int_{0}^{i} \frac{\sin 2 y}{y} d y=\frac{\sin ^{2} \lambda}{\lambda}+\int_{0}^{i} \frac{\sin ^{2} x}{x^{2}} d x .\) Deduce that \(\int_{0}^{\infty} \frac{\sin x}{x} d x=\int_{0}^{\infty} \frac{\sin ^{2} x}{x^{2}} d x\) (It may be assumed that the integrals are convergent)

Show that \(\int_{0}^{\infty} x^{2} e^{-x^{2}} d x=\frac{1}{2} \int_{0}^{\infty} e^{-x^{2}} d x\)

Evaluate the following integrals: (i) \(\int_{-1}^{0} \frac{e^{\frac{1}{x}}}{x^{3}} d x\) (ii) \(\int_{-\infty}^{\infty} \frac{1}{e^{x}+e^{-x}} d x\) (iii) \(\int_{3}^{5} \frac{x^{2} d x}{\sqrt{(x-3)(5-x)}}\) (iv) \(\int_{-1}^{1} \frac{d x}{(2-x) \sqrt{1-x^{2}}}\)

Show that \(\int_{0}^{1} \frac{\ell n\left(1-a^{2} x^{2}\right)}{x^{2} \sqrt{\left(1-x^{2}\right)}} d x\) \(=\pi\left[\sqrt{1-a^{2}}-1\right],\left(a^{2}<1\right)\)

Let \(\mathrm{f}\) be twice continuously differentiable in \([0,2 \pi]\) and concave up. Prove that \(\int_{0}^{2 \pi} f(x) \cos x d x \geq 0\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Geometry

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.