Math 300 problem

[Kcashew]

I am unsure about where to start with this problem, and I was hoping I could get some one on one help.

Is there anyone willing to Zoom with me?

If so, please message me.

 

[Deleted member 4993]

I am unsure about where to start with this problem, and I was hoping I could get some one on one help.

Is there anyone willing to Zoom with me?

If so, please message me.

Can you calculate the following:

g(t) = e^(a*t) * cos(b*t)

\(\displaystyle \frac{dg(t)}{dt} \ = \ ? \)

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem.

 

[Kcashew]

I have included the calculation.

Where should I go from here?

 

[Steven G]

Are you really serious about this one? [math]\dfrac{d}{dt}e^u = \dfrac{du}{dt}*e^u[/math]
So [math]\dfrac{d}{dt}e^{(a+bi)t} = (\dfrac{d}{dt}(a+bi)t)(e^{(a+bi)t})=(a+bi)e^{(a+bi)t}[/math]=????

 

[Kcashew]

This helps me with part A of my problem, thank you.

How do I identify the real and imaginary parts for this equation?

Should I substitute a variable for (a+bi)?

P.S.
I'm not sure what you mean when you ask if I am "really serious about this one".

 

[HallsofIvy]

He is wondering if you have taken a Calculus class. If you have, then you should know that the derivative of \(\displaystyle e^x\) is the easiest possible derivative: \(\displaystyle \frac{de^x}{dx}= e^x\) and that together with the "chain rule" gives, as Jomo says, \(\displaystyle e^{(a+ bi)t}= (a+ bi)e^{(a+ bi)t}\).

 

[Deleted member 4993]

This helps me with part A of my problem, thank you.

How do I identify the real and imaginary parts for this equation?

Should I substitute a variable for (a+bi)?

P.S.
I'm not sure what you mean when you ask if I am "really serious about this one".

You say:

.....,.How do I identify the real and imaginary parts for this equation?

Can you identify the real and imaginary parts of e^{(at + ibt)}

 

[Steven G]

In 3 + 5i, 3 is the real part and 5i is the imaginary part.
In your problem, a and b are real numbers just like 3 and 5 are real numbers.
Now can you tell what the real and imaginary parts are in your equation?

 

[HallsofIvy]

\(\displaystyle e^{u+ vi}= e^ue^{vi}= e^u(cos(v)+ i sin(v))= e^u cos(v)+ e^u sin(v)i\)

The "real part" is \(\displaystyle e^u cos(v)\) and the "imaginary part" is \(\displaystyle e^u sin(v)i\).