Finding the Coefficients of a Taylor Polynomial: A Tricky Integration Question

[izelkay]

Homework Statement

Here's a screenshot of the problem: http://puu.sh/2Bta5

Homework Equations

The Attempt at a Solution

As can be seen by the screenshot, the answer's already given, but I'm not sure how to go about getting it. This one has me stumped since e-4x and sin(5x) are both Maclaurin series that I can easily find. But I'm not sure where to go from there, or if I'm even supposed to use both of the Maclaurin series.

 

[Mark44]

Homework Statement

Here's a screenshot of the problem: http://puu.sh/2Bta5

Homework Equations

The Attempt at a Solution

As can be seen by the screenshot, the answer's already given, but I'm not sure how to go about getting it. This one has me stumped since e-4x and sin(5x) are both Maclaurin series that I can easily find. But I'm not sure where to go from there, or if I'm even supposed to use both of the Maclaurin series.

It looks like you are supposed to write a few terms of the Maclaurin series for e^-4x in one area and a few terms for the series for sin(5x) in the other area.

 

[Dick]

Homework Statement

Here's a screenshot of the problem: http://puu.sh/2Bta5

Homework Equations

The Attempt at a Solution

As can be seen by the screenshot, the answer's already given, but I'm not sure how to go about getting it. This one has me stumped since e-4x and sin(5x) are both Maclaurin series that I can easily find. But I'm not sure where to go from there, or if I'm even supposed to use both of the Maclaurin series.

Using both the Maclaurin series is probably the easiest way to do it. Just multiply them together. Ignore products that will give you powers greater than 3.

 

[izelkay]

Using both the Maclaurin series is probably the easiest way to do it. Just multiply them together. Ignore products that will give you powers greater than 3.

Ok, that's what I thought I should do but wasn't sure if that was allowed or not.
Is this correct so far? :

http://puu.sh/2BySD

It looks like you are supposed to write a few terms of the Maclaurin series for e^-4x in one area and a few terms for the series for sin(5x) in the other area.

I'll try this too, thanks.

 

[Dick]

Ok, that's what I thought I should do but wasn't sure if that was allowed or not.
Is this correct so far? :

http://puu.sh/2BySD



I'll try this too, thanks.

No, not correct. You don't multiply series term by term. You multiply them like polynomials. Write out each series up to power x^3 and then foil them. Like polynomials. Drop powers bigger than 3.

 

[izelkay]

No, not correct. You don't multiply series term by term. You multiply them like polynomials. Write out each series up to power x^3 and then foil them. Like polynomials. Drop powers bigger than 3.

Ah, ok. I think I understand now.

 

[izelkay]

Never mind, still not getting it. Looking at these series for e^-4x and sin(5x), wouldn't that mean I need 4 terms (n=0 to n=3) for e^-4x to make that 3rd degree, and 2 terms (n=0 to n=1) for sin(5x) to make that 3rd degree?

So for e^-4x I have:
1 - 4x + 8x - (64/6)x³

For sin(5x):
5x + (125/6)x³

FOILing those out and ignoring powers greater than 3, I get:

5x + (125/6)x³ - 20x² + 40x³

Is there something else I'm doing wrong?

 

[Dick]

For sin(5x):
5x + (125/6)x³

For sin(5x):

5x - (125/6)x³

Not plus.

 

[izelkay]

Ok, never mind, found my error. That + (125/6)x³ should actually be a - (125/6)x³

 

[izelkay]

For sin(5x):

5x - (125/6)x³

Not plus.

Just caught it. Thanks for your help, much appreciated.