Finding Taylor and Maclaurin Series
In Exercises 25–34, ...

Question

Finding Taylor and Maclaurin Series

In Exercises 25–34, find the Taylor series generated by f at x = a.

f(x) = cos(2x + π/2),a = π/4

Accepted Answer

Find the Taylor series generated by the function F X equals sin 2 x minus pi centered at A equals pi divided by 2. Now, to solve this, we need the formula for the Taylor series. This is given by the following f of x equals the sum. S N equals 0 to infinity. Of F to the n derivative of a. Divided by n factorial multiplied by X minus a. Race to the inn. We can expand this to give us FA plus FA. X minus A Plus F. In Eh Divided by in factorial. X minus A raised to the N. We'll put some dots there to imply that that continues. Now we just need to find our derivatives. So we have F of X. Equals sun. 2 x minus pi. Now we can find our first derivative. Our first derivative then. Will be To cosine 2 x minus pi. Which is just a chain rule. Then our 2nd derivative. Will be -4 sin to x minus pi. And we'll do one more, which is our 3rd derivative. To get -8. Cosine 2 x minus pi. Now, from here, we'll plug in our pi divided by 2. We get F of pi divided by 2. When we plug that in, we get 0. F pi divided by 2, we plug this in, gives us 2. F Pi divided by 2 equals 0, and the 3rd derivative equals. -8. Now, there is a pattern developing, and it looks like all of our even derivatives are going to go to zero, based on what we're seeing. So now let's set up our Our Taylor series. So, our first term then, what we notice F of X goes to 0, so we can ignore that one and go directly to Our next term F A. X minus a And the 3rd derivative is after that. Defied that By 3 factorial multiplied by x minus a raised to the 3rd. Now we can plug these in. F A. We know it is 2. So you have 2 multiplied by X minus pi divided by 2. And then our 3rd derivative is -8, divided by 3 factorial, multiplied by X minus pi, divided by 2, raised to the 3rd. And so on. Now we just seem to identify the pattern. This actually follows a pattern of the known sign Taylor series. But we just have 2 X minus pi divided by 2, replacing X. So We then know the sign series. Is given by the sum as n equals 0 to infinity. Of -1 to the n x 2N. Plus one Divided by 2 n plus 1 factorial. This is also multiplied. By x to the 2 n + 1. But we can rewrite our series to follow. Our new series. I will actually say X is you in this case, just for a bit more clarity. Now, because we have that 2 there, we're going to replace that X there with 2. So let's replace our entire series. We have the sum, and equals 0 to infinity. Of -1 ton, 2 2N + 1. Divided by 2 n plus 1 factorial multiplied. By x minus pi divided by 2. Race to the 2 n + 1. This is our tailor series. That is the answer to our problem. I hope to help you solve it. Thank you for watching. Goodbye.

Finding Taylor and Maclaurin SeriesIn Exercises 25–34, find the Taylor series generated by f at x = a.f(x) = cos(2x + π/2),a = π/4

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Finding Taylor and Maclaurin Series
In Exercises 25–34, find the Taylor series generated by f at x = a.
f(x) = cos(2x + π/2),a = π/4