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

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

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.

Problem

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

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

More Answers

Related Courses

Calculus: Early Transcendentals

Related Topics

Discussion

Top Calculus 1 / AB Educators

Grace He

Catherine Ross

Heather Zimmers

Caleb Elmore

Calculus 1 / AB Courses

Recommended Videos

Watch More Solved Questions in Chapter 3

Video Transcript

James Stewart

Calculus: Early Transcendentals

Grace He

Catherine Ross

Heather Zimmers

Caleb Elmore

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

Calculus: Early Transcendentals

Grace He

Catherine Ross

Heather Zimmers

Caleb Elmore

James StewartCalculus: Early Transcendentals

Grace He

Catherine Ross

Heather Zimmers

Caleb Elmore

James Stewart

Calculus: Early Transcendentals