**Differentiation and Integration Using Power Series — Transcript & Summary | SozAI**
Source: https://sozai.app/transcript/differentiation-integration-power-series/

Learn how to differentiate and integrate functions using power series with step-by-step examples including sine and cosine functions.

## Key Takeaways

- Power series can be used effectively to approximate derivatives and integrals of functions.
- Differentiating the sine power series yields the cosine power series.
- Integration of complex functions like (sin x)/x can be approximated using power series expansions.
- Error bounds can be controlled by including terms until the desired precision is reached.
- Power series provide a useful tool for handling functions without elementary antiderivatives.

## What the video covers

- Introduction to differentiating and integrating functions using power series.
- Using power series to approximate the derivative of sine x and verifying it equals cosine x.
- Step-by-step differentiation of the sine power series and simplification of terms.
- Deriving the power series formula for cosine x from the derivative of sine x's series.
- Using power series to approximate the integral of (sin x)/x over the interval [0, 0.5].
- Simplifying the power series for sine x divided by x to prepare for integration.
- Finding terms of the power series until the error is less than 0.001 for accurate approximation.
- Calculating the antiderivative term-by-term and evaluating the definite integral.
- Final approximation of the integral as approximately 0.493 with error control.
- Summary and conclusion thanking viewers.

## Chapters

1. 00:00 Introduction to Power Series Differentiation and Integration
2. 00:19 Approximating Derivative of Sine x Using Power Series
3. 00:43 Differentiating Terms of the Sine Power Series
4. 01:06 Obtaining the Power Series for Cosine x
5. 02:28 Simplifying the Derivative Terms
6. 03:27 Simplifying the Derivative Summation
7. 03:55 Using Power Series to Approximate an Integral
8. 04:08 Power Series for Sine x Divided by x
9. 04:30 Simplifying the Power Series for Integration
10. 05:02 Calculating Terms for Desired Accuracy

Answers

## Questions about this video

How can power series be used to find the derivative of sine x?

By differentiating the power series expansion of sine x term-by-term, we obtain the power series expansion for cosine x, which is the derivative of sine x.

Why is power series useful for integrating functions like (sin x)/x?

Functions like (sin x)/x do not have elementary antiderivatives, so using power series expansions allows us to approximate the integral by integrating the series term-by-term.

How is the accuracy of the power series approximation controlled?

Accuracy is controlled by including terms in the series until the magnitude of the next term is less than a specified error threshold, such as 0.001, ensuring the approximation is within the desired precision.

## Full Transcript — Download SRT & Markdown

00:01

Speaker A

Welcome to a video on differentiating and integrating with power series. By now, we should all be fairly familiar with the most common power series as we see listed here.

00:12

Speaker A

And, we can use these power series to approximate the derivative and anti-derivatives of a variety of functions.

00:19

Speaker A

For example, we can use our series to approximate the derivative of f of x equals sine x.

00:24

Speaker A

Now, of course, we know the derivative of sine x is equal to cosine x.

00:28

Speaker A

Let's go ahead and approximate this derivative using a power series. There's a couple ways of doing this.

00:34

Speaker A

We can either differentiate this summation here or we can differentiate the right side by differentiating these individual terms.

00:43

Speaker A

Let's start by finding the derivatives of the first several terms of the expansion of the power series.

00:49

Speaker A

So, we'll find the derivative of x minus x cubed divided by three factorial plus x to the fifth divided by five factorial minus x to the seventh divided by seven factorial and so on.

01:06

Speaker A

And, these derivatives should give us the first several terms of the power series for cosine x.

01:13

Speaker A

So, this would be one minus three x squared divided by three factorial plus five x to the fourth divided by five factorial.

01:28

Speaker A

Minus seven x to the sixth divided by seven factorial and so on. Let's go ahead and simplify these terms.

01:39

Speaker A

The three and the three factorial would simplify to two factorial. So, we'd have minus x squared over two factorial.

01:48

Speaker A

Plus, the five and the five factorial would simplify. We'd have a denominator of four factorial.

01:55

Speaker A

And again, the seven and the seven factorial simplify like so. This should be the first several terms of the power series for cosine x.

02:07

Speaker A

Before we go and take a look at this, let's go ahead and find the derivative of this summation with respect to x.

02:21

Speaker A

So, the negative one in the power of n will not be affected by finding the derivative.

02:26

Speaker A

And, to find the derivative of this term with respect to x, we would have two n plus one times x to the two n plus one minus one or x to the two n, and we still have a denominator

02:39

Speaker A

of two n plus one factorial. But, remember that would be two n plus one times two n factorial.

02:49

Speaker A

And, so we can simplify a factor of two n plus one. So, the derivative would be the summation from n equals zero to infinity.

02:59

Speaker A

Negative one to the n times x to the two n divided by two n factorial.

03:07

Speaker A

Now, let's go ahead and go back to our formulas and verify that both of these would be the power series for cosine x.

03:19

Speaker A

Well, here's cosine x. And, this does match the derivative that we found using the terms of the power series for sine x.

03:32

Speaker A

And, the formula for the summation is the derivative of the power series for sine x as we see here.

03:42

Speaker A

Now, let's take a look at how we can use power series to integrate. We want to integrate sine x divided by x with respect to x on the interval from zero to 0.5. Now, because of this integral, we don't have

03:55

Speaker A

a nice way to integrate this. We can use power series to approximate this definite integral.

04:00

Speaker A

This would be equal to the integral from zero to 0.5 of the power series for sine x divided by x.

04:08

Speaker A

Well, here's the power series for sine x. We want to divide this by x, which is the same as multiplying by one over x.

04:21

Speaker A

So, we'll have a factor of x here. Let's go ahead and simplify this and then we can integrate.

04:30

Speaker A

Well, here we have two n plus one factors of x. And, here we have one factor of x.

04:36

Speaker A

So, we can simplify this to just x to the power of two n. Now what we're going to do is find the first several terms of this power series, but when we start to evaluate this we'll keep finding terms until we find the term

05:02

Speaker A

that is less than 0.001. So, when n is equal to zero we're going to have one.

05:10

Speaker A

When n is equal to one we're going to have minus this will be x squared divided by three factorial.

05:21

Speaker A

When n is two we'll have plus x to the fourth divided by five factorial, minus and so on.

05:32

Speaker A

I think we have enough terms now. Let's go ahead and find the anti-derivative and then evaluate it at the upper and lower limits of integration.

05:39

Speaker A

So, the anti-derivative of one with respect to x would be x. Here, we have minus x cubed divided by three times three factorial.

05:51

Speaker A

Plus, this will be x to the fifth divided by five times five factorial and so on.

05:59

Speaker A

So, go ahead and evaluate this at 0.5 and then also at zero. So, when x is 0.5 we'll have 0.5 minus 0.5 cubed, divided by three times three factorial, plus 0.5 to the fifth divided by five times five factorial and so on.

06:25

Speaker A

And then, when x is equal to zero these will all be zero. And, since we want to approximate this with an error that's less than 0.001, as soon as we find the term that's less than this we have found the approximation that we're looking for.

06:41

Speaker A

So, this is 0.5 minus, let's see what the value of this fraction would be.

06:52

Speaker A

0.5 cubed divided by three and three factorial would be six. So, you have approximately 0.0069.

07:15

Speaker A

And, if this fraction here is less than 0.001, this value here will be our approximation.

07:23

Speaker A

Let's go ahead and check it. Divided by five times five factorial. And, this number here is less than 0.001 and therefore our approximation will be this difference here.

08:07

Speaker A

So, our approximation is approximately 0.493. Okay, that's going to do it for this video. Thank you for watching.

Topics: power series differentiation integration sine function cosine function Taylor series Maclaurin series approximation calculus Mathispower4u

Study this video

- [Flashcards from this video](https://sozai.app/tools/ai-flashcard-generator/?from=transcript&id=69799)
- [Quiz on this video](https://sozai.app/tools/quiz-generator/?from=transcript&id=69799)

Free, made by AI from this transcript. No signup.


---
This is the markdown twin of https://sozai.app/transcript/differentiation-integration-power-series/ — the same content, without the markup.
Published by SozAI (https://sozai.app). Reuse and quotation are allowed with attribution and a link back.
Machine-readable index: https://sozai.app/llms.txt · data API: https://sozai.app/api/