Logout Open In App
Open In App
Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 2: Q59E (page 77)

(a) Graph the function \(f\left( x \right) = {\rm{sin}}x - \frac{1}{{1000}}{\rm{sin}}\left( {1000x} \right)\) in the viewing rectangle \(\left( { - 2\pi ,2\pi } \right)\)by \(\left( { - 4,4} \right)\). What slopedoes the graph appear to have at the origin?

(b) Zoom into the viewing window \(\left( { - 0.4,0.4} \right)\)by \(\left( { - 0.25,0.25} \right)\)and estimate the value of\(f'\left( 0 \right)\). Doesthis agree with your answer from part (a)?

(c) Now zoom in to the viewing window \(\left( { - 0.008,0.008} \right)\)by \(\left( { - 0.005,0.005} \right)\). Do you wish to revise your estimate for \(f'\left( 0 \right)\)?

Short Answer

Expert verified

a. The slope at the origin is 1.


b. \(f'\left( 0 \right) = 1\) that is, the slope at the origin is 1. Yes, both the answers are same.


c. The slope at the origin is \(0\).


Step by step solution

01

 (a)                                                                                           Step 1: Plot the graph of the function.

Draw the graph of the function\(f\left( x \right) = \sin x - \frac{1}{{1000}}\sin \left( {1000x} \right)\) by using the graphing calculator as:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation \(\sin x - \frac{1}{{1000}}\sin \left( {1000x} \right)\) in the\({Y_1}\)tab.

2. Enter the “GRAPH” button in the graphing calculator.

The graph of the function \(f\left( x \right) = \sin x - \frac{1}{{1000}}\sin \left( {1000x} \right)\) is shown below.

From the graph, it is observed that the slope at the origin is 1.

02

(b)                                                                                           Step 2: Plot the graph of the function.

Draw the graph of the function\(f\left( x \right) = \sin x - \frac{1}{{1000}}\sin \left( {1000x} \right)\)by using the graphing calculator as:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(\sin x - \frac{1}{{1000}}\sin \left( {1000x} \right)\) in the\({Y_1}\)tab in the window\(\left( { - 0.4,0.4} \right)\) by\(\left( { - 0.25,0.25} \right)\).

2. Enter the “GRAPH” button in the graphing calculator.

The graph of the function \(f\left( x \right) = \sin x - \frac{1}{{1000}}\sin \left( {1000x} \right)\) is shown below.

From the graph, it is observed that \(f'\left( 0 \right) = 1\), that is, the slope at the origin is 1.

Yes, both the answers are same.

03

(c)                                                                                           Step 3: Plot the graph of the function. 

Draw the graph of the function\(f\left( x \right) = \sin x - \frac{1}{{1000}}\sin \left( {1000x} \right)\) by using the graphing calculator as:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(\sin x - \frac{1}{{1000}}\sin \left( {1000x} \right)\) in the\({Y_1}\) tab in the window\(\left( { - 0.008,0.008} \right)\)by\(\left( { - 0.005,0.005} \right)\).

2. Enter the “GRAPH” button in the graphing calculator.

The graph of the function \(f\left( x \right) = \sin x - \frac{1}{{1000}}\sin \left( {1000x} \right)\) is shown below.

From the graph, it is observed that \(f'\left( 0 \right) = 0\), that is, the slope at the origin is \(0\).

Yes, the slope is different.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

A particle moves along a straight line with the equation of motion\(s = f\left( t \right)\), where s is measured in meters and t in seconds.Find the velocity and speed when\(t = {\bf{4}}\).

\(f\left( t \right) = {\bf{80}}t - {\bf{6}}{t^{\bf{2}}}\)

If \(f\left( x \right) = 3{x^2} - {x^3}\), find \(f'\left( 1 \right)\) and use it to find an equation of the tangent line to the curve \(y = 3{x^2} - {x^3}\) at the point \(\left( {1,2} \right)\).

Calculate each of the limits

limh→012+h-12h

Find the values of a and b that make f continuous everywhere.

\(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}{\frac{{{x^{\bf{2}}} - {\bf{4}}}}{{{\bf{x}} - {\bf{2}}}}}&{{\bf{if}}\,\,\,x < {\bf{2}}}\\{a{x^{\bf{2}}} - bx + {\bf{3}}}&{{\bf{if}}\,\,\,{\bf{2}} \le x < {\bf{3}}}\\{{\bf{2}}x - a + b}&{{\bf{if}}\,\,\,x \ge {\bf{3}}}\end{array}} \right.\)

Sketch the graph of the function g for which \(g\left( {\bf{0}} \right) = g\left( {\bf{2}} \right) = g\left( {\bf{4}} \right) = {\bf{0}}\), \(g'\left( {\bf{1}} \right) = g'\left( {\bf{3}} \right) = {\bf{0}}\), \(g'\left( {\bf{0}} \right) = g'\left( {\bf{4}} \right) = {\bf{1}}\), \(g'\left( {\bf{2}} \right) = - {\bf{1}}\), \(\mathop {{\bf{lim}}}\limits_{x \to \infty } g\left( x \right) = \infty \), and \(\mathop {{\bf{lim}}}\limits_{x \to - \infty } g\left( x \right) = - \infty \).
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

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.

Sign-up for free