Analyzing a Damped Trigonometric Graph

[nycmathguy]

Section 4.6

Can you show me how to do 73 as a guide for me to do a few on my own?

 

[MathLover1]

Damped Trigonometric Graphs

A product of two functions can be graphed using properties of the individual
functions. The graph of y=f (x) cosbx or y =f (x)sin( bx) oscillates between
the graphs of y= f( x) and y =-f( x) . When this reduces the amplitude of the
wave, it is called damped oscillation. The factor f (x)is called the damping factor.
Consequently,

−|f(x)| ≤ f(x) sin (x) ≤ |f(x)|
or
−|f(x)| ≤ f(x) cos (x )≤ |f(x)|

Which means that the graph of:

f(x) = x* sin(x) lies between lines y = −f(x) and y = f(x) .

Examples:

73.

g(x)=e^(-x^2/2)*sin(x)

damping occurs between e^(-x^2/2) and - e^(-x^2/2)
determine the dumping factor: e^(-x^2/2)

so, -e^(-x^2/2) <= e^(-x^2/2)*sin(x) <= e^(-x^2/2)

 

[nycmathguy]

Damped Trigonometric Graphs

A product of two functions can be graphed using properties of the individual
functions. The graph of y=f (x) cosbx or y =f (x)sin( bx) oscillates between
the graphs of y= f( x) and y =-f( x) . When this reduces the amplitude of the
wave, it is called damped oscillation. The factor f (x)is called the damping factor.
Consequently,

−|f(x)| ≤ f(x) sin (x) ≤ |f(x)|
or
−|f(x)| ≤ f(x) cos (x )≤ |f(x)|

Which means that the graph of:

f(x) = x* sin(x) lies between lines y = −f(x) and y = f(x) .

Examples:

View attachment 1065


73.

g(x)=e^(-x^2/2)*sin(x)

View attachment 1066

damping occurs between e^(-x^2/2) and - e^(-x^2/2)
determine the dumping factor: e^(-x^2/2)

so, -e^(-x^2/2) <= e^(-x^2/2)*sin(x) <= e^(-x^2/2)

View attachment 1067

Can we say that this idea of a damped trigonometric function is slightly similar to the Squeeze Theorem in calculus? I work on 74 and 76 when time allows.

 

[MathLover1]

you could say that
here are some facts:

The squeeze theorem

The squeeze theorem allows us to find the limit of a function at a particular point, even when the function is undefined at that point. The way that we do it is by showing that our function can be squeezed between two other functions at the given point, and proving that the limits of these other functions are equal to one another.

If you think about it, if you can show that two functions have the same value at the same point, and you know that your original function has to run through the other two (be squeezed, or pinched, or sandwiched between them), then the original function can’t take on any possible value other than the value of the other two at that particular point.

We assume that our original function is h(x) , and that it’s squeezed between two other functions, f(x) and g(x) , so
f(x) ≤ h(x) ≤ g(x)
We also assume that the limits of our other two functions are equal as we approach the point we’re interested in, so
lim(x→c, f(x))=lim(x→c,g(x))=L

What are damping functions?

A damped sine wave is a sinusoidal function whose amplitude approaches zero as time increases. Damped sine waves are commonly seen in science and engineering, wherever a harmonic oscillator is losing energy faster than it is being supplied.

The best way to explain them is to show you some examples:

Look at the function f(x) = x*sin(10x). (The * is being used to indicate multiplication.)

Ignoring the first factor, x, for a minute, the graph of g(x) = sin(10x) looks like:

So, what does multiplying by the x do? Let's find out by graphing the whole thing!

f(x) = x*sin(10x)

It really changed! Look back up at our first graph. Do you see what happened?

The graph of g(x) = sin(10x) is getting squished (or damped) between the graphs of y = x and y = -x !!

Then check it out! Let's graph f(x) = x*sin(10x), y = x and y = -x all on the same graph:

We see that our sine graph is, indeed, bounded between them!

In the function

this x is called the damping factor.

 

[nycmathguy]

you could say that
here are some facts:

The squeeze theorem

The squeeze theorem allows us to find the limit of a function at a particular point, even when the function is undefined at that point. The way that we do it is by showing that our function can be squeezed between two other functions at the given point, and proving that the limits of these other functions are equal to one another.

If you think about it, if you can show that two functions have the same value at the same point, and you know that your original function has to run through the other two (be squeezed, or pinched, or sandwiched between them), then the original function can’t take on any possible value other than the value of the other two at that particular point.

We assume that our original function is h(x) , and that it’s squeezed between two other functions, f(x) and g(x) , so
f(x) ≤ h(x) ≤ g(x)
We also assume that the limits of our other two functions are equal as we approach the point we’re interested in, so
lim(x→c, f(x))=lim(x→c,g(x))=L

What are damping functions?

A damped sine wave is a sinusoidal function whose amplitude approaches zero as time increases. Damped sine waves are commonly seen in science and engineering, wherever a harmonic oscillator is losing energy faster than it is being supplied.

The best way to explain them is to show you some examples:

Look at the function f(x) = x*sin(10x). (The * is being used to indicate multiplication.)

Ignoring the first factor, x, for a minute, the graph of g(x) = sin(10x) looks like:

So, what does multiplying by the x do? Let's find out by graphing the whole thing!

f(x) = x*sin(10x)

It really changed! Look back up at our first graph. Do you see what happened?

The graph of g(x) = sin(10x) is getting squished (or damped) between the graphs of y = x and y = -x !!

Then check it out! Let's graph f(x) = x*sin(10x), y = x and y = -x all on the same graph:

We see that our sine graph is, indeed, bounded between them!

In the function

View attachment 1069

this x is called the damping factor.

Excellent study notes. This is exactly why I joined the site and then invited you. I want to know the how and why things happen in a topic. Anyone can learn to be mechanical in terms of solving problems.

Most students are mechanical but have no idea why and/or how the process works out as it does. Honestly, ask a Calculus 1 student what a limit is. The student does not know.

Ask the same student to find the limit of (x + 5x) as x tends to 2, this is not a problem. The student will say the limit is 12. Yes, correct but why is it 12? The student does not know. Trust me, they don't know.

You say?

 

[MathLover1]

The student will say the limit is 12 because knows that must substitute 2 for x.
why is it 12: Everyone who gives you that answer, knows why
2+5*2=12 is not hard to conclude

 

[nycmathguy]

The student will say the limit is 12 because knows that must substitute 2 for x.
why is it 12: Everyone who gives you that answer, knows why
2+5*2=12 is not hard to conclude

You are right but 12 means something. The limit 12 means the height the function (x + 5x) will travel in terms of the line x = 0. No?

 

[MathLover1]

A limit tells us the value that a function approaches as that function's inputs get closer and closer to some number.

 

[nycmathguy]

A limit tells us the value that a function approaches as that function's inputs get closer and closer to some number.

The function gets close to some limit as x gets close to some value.

 

[MathLover1]

The limit of a function as x approaches a is equal to the value of the function at x=a or f(a)

 

[nycmathguy]

The limit of a function as x approaches a is equal to the value of the function at x=a or f(a)

Very good. We will be in Calculus 1 hopefully in April 2022. I will speed up Precalculus by posting the essentials of the course. No need to know every single topic. I just need to know enough of the material to give me a boost when going through the James Stewart Calculus textbook 5 or 6 months from now.

 

[nycmathguy]

Damped Trigonometric Graphs

A product of two functions can be graphed using properties of the individual
functions. The graph of y=f (x) cosbx or y =f (x)sin( bx) oscillates between
the graphs of y= f( x) and y =-f( x) . When this reduces the amplitude of the
wave, it is called damped oscillation. The factor f (x)is called the damping factor.
Consequently,

−|f(x)| ≤ f(x) sin (x) ≤ |f(x)|
or
−|f(x)| ≤ f(x) cos (x )≤ |f(x)|

Which means that the graph of:

f(x) = x* sin(x) lies between lines y = −f(x) and y = f(x) .

Examples:

View attachment 1065


73.

g(x)=e^(-x^2/2)*sin(x)

View attachment 1066

damping occurs between e^(-x^2/2) and - e^(-x^2/2)
determine the dumping factor: e^(-x^2/2)

so, -e^(-x^2/2) <= e^(-x^2/2)*sin(x) <= e^(-x^2/2)

View attachment 1067

Question 74

Question 76

 

[MathLover1]

74. what do you mean by " as x increases f(x) oscillates"?

you see from the graph as x -> infinity, curved line is flattening because

you see same in 75

 

[nycmathguy]

74. what do you mean by " as x increases f(x) oscillates"?

you see from the graph as x -> infinity, curved line is flattening because

you see same in 75

I will practice a few more.