Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Sent to:
Search glass icon
  • Login
  • Textbooks
  • Ask our Educators
  • Study Tools
    Study Groups Bootcamps Quizzes AI Tutor iOS Student App Android Student App StudyParty
  • For Educators
    Become an educator Educator app for iPad Our educators
  • For Schools

Problem

Show that if $ a \not= 0 $ and $ b \not= 0, $ the…

00:31

Question

Answered step-by-step

Problem 58 Easy Difficulty

Investigate the family of functions
$ f_n(x) = \tanh (n \sin x) $
where $ n $ is a positive integer. Describe what happens to the graph of $ f_n $ when $ n $ becomes large.


Video Answer

Solved by verified expert

preview
Numerade Logo

This problem has been solved!

Try Numerade free for 7 days

Yuki Hotta
Numerade Educator

Like

Report

Textbook Answer

Official textbook answer

Video by Yuki Hotta

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

More Answers

00:08

Amrita Bhasin

Related Courses

Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 3

Differentiation Rules

Section 11

Hyperbolic Functions

Related Topics

Derivatives

Differentiation

Discussion

You must be signed in to discuss.
Top Calculus 1 / AB Educators
Grace He
Catherine Ross

Missouri State University

Heather Zimmers

Oregon State University

Caleb Elmore

Baylor University

Calculus 1 / AB Courses

Lectures

Video Thumbnail

04:40

Derivatives - Intro

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

Video Thumbnail

44:57

Differentiation Rules - Overview

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

Join Course
Recommended Videos

05:05

Investigate the family of …

03:03

Graph several members of t…

10:04

$$ \begin{array}{l}{\text …

01:37

Investigate the family of …

00:49

Graph several members of t…

01:56

Use a graphing utility to …

03:44

A family of curves is give…

02:12

Give a short written answe…

Watch More Solved Questions in Chapter 3

Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
Problem 43
Problem 44
Problem 45
Problem 46
Problem 47
Problem 48
Problem 49
Problem 50
Problem 51
Problem 52
Problem 53
Problem 54
Problem 55
Problem 56
Problem 57
Problem 58
Problem 59

Video Transcript

All right, let's answer the question. What is the behavior off? Why equals two hyperbolic tangent? Some people call it Van. I like to call it Tancho, so I'm going to read it that way. The can't off and sign effects. How does this thing behave? Okay, let's go ahead and graphic. If n is equal to one, you can see that it's kind of going to resemble a sine function, as you can see right here. So before we actually observe this, we can take a look at one Tansi. Looks like if you recall the tant function resembles that of a inverse tangent function. Very well. It's just that instead, off negative power to two pi over to it actually goes from negative. Want to positive one? Okay. And if you remember what sign functions looks like that's actually the end side effects know that it's simply going to be one of these shapes. And as an increases or decreases, you can see that the amplitude changes like that. Mhm. All right. So as you can imagine, it's going to be some sort of a mixture of thes to where you can plug in, uh, Ensign X into can church and let's see what's gonna happen. Why equals two engine age off n sign of X. We already did the one place. Let's see what happens if it's equal to two. Uh, it kind of looks a little bit block here, right? How about a much bigger number? Like 17? It's getting even Mawr block here. So what's important to remember about tans? Uh, hyperbolic tangent is the fact that the upper and the lower bound of it it's always going to be between negative 1 to 1. So no matter how large the input is going to be, um, tension is going to be between negative 1 to 1. However, because the input is periodic, you see a periodic, um, shape in tange off and sign effects, and that's basically what's happening here. Let's see what happens if we use a very large number. For example, 10 factorial. You can see that the blocks are going to be almost perpendicular right here, and of course it's not. But I can imagine that these kind of functions can be very useful for things such as engineering's and stuff. So I come from a mathematics or statistics backgrounds, but so I've never really worked with this particularly, but I think it's a really cool function. So this is how can of Ensign of X behaves.

Get More Help with this Textbook
James Stewart

Calculus: Early Transcendentals

View More Answers From This Book

Find Another Textbook

Study Groups
Study with other students and unlock Numerade solutions for free.
Math (Geometry, Algebra I and II) with Nancy
Arrow icon
Participants icon
95
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
54
Hosted by: Alonso M
See More

Related Topics

Derivatives

Differentiation

Top Calculus 1 / AB Educators
Grace He

Numerade Educator

Catherine Ross

Missouri State University

Heather Zimmers

Oregon State University

Caleb Elmore

Baylor University

Calculus 1 / AB Courses

Lectures

Video Thumbnail

04:40

Derivatives - Intro

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

Video Thumbnail

44:57

Differentiation Rules - Overview

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

Join Course
Recommended Videos

05:05

Investigate the family of polar curves $$ r = 1 + \cos^n \theta $$ where $ n …

03:03

Graph several members of the family of curves $ x = \sin t + \sin nt $, $ y = \…

10:04

$$ \begin{array}{l}{\text { Graph several members of the family of curves }} \\…

01:37

Investigate the family of curves given by $f(x)=x^{n} e^{-x}$ where $n$ is a p…

00:49

Graph several members of the family of curves $x=\sin t+\sin n t, y=\cos t+\co…

01:56

Use a graphing utility to investigate how the family of polar curves $r=1+a \co…

03:44

A family of curves is given by the equations $r=1+c \sin n \theta,$ where $c$ …

02:12

Give a short written answer. The graphs of $f(x)=x^{n}$ for $n=3,5,7, \ldots$ r…
Additional Mathematics Questions

02:41

The diagram below shows a triangle inscribed in a rectangle:
8
5
Wh…

02:40

Name the" vertex and sides of each angle:
1 )
3 )
2 )
Name …

02:52



please do it on paper work thank yoh

The radioactive isoto…

01:28

If the triangle pictured is reflected across the X-axis, the coordi…

02:00

Not lo scale
The distance AB is 8 units (a) Write down the equation of th…

03:46

PQ has endpoints at P(-5,4) and Q (7,-5). what are the coordinates of the po…

04:07

how to find the perimeter for this triangle...
4. Triangle RST
48 36

03:23

Please answer 1,2, & 3
If a shoelace represents the Nile river; and t…

03:52

Supply the missing coordinates for the vertices using as few
variables as…

02:13

Jasmine has designed her own flag on rectangle that is 8 inches tall and 12 …

Add To Playlist

Hmmm, doesn't seem like you have any playlists. Please add your first playlist.

Create a New Playlist

`

Share Question

Copy Link

OR

Enter Friends' Emails

Report Question

Get 24/7 study help with our app

 

Available on iOS and Android

About
  • Our Story
  • Careers
  • Our Educators
  • Numerade Blog
Browse
  • Bootcamps
  • Books
  • Notes & Exams NEW
  • Topics
  • Test Prep
  • Ask Directory
  • Online Tutors
  • Tutors Near Me
Support
  • Help
  • Privacy Policy
  • Terms of Service
Get started