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The Maclaurin series is a template that allows you to express many other functions as power series. It is the source of formulas for expressing both sin x and cos x as infinite series.

Without further ado, here it is:

The notation f⁽ⁿ⁾ means 'the nth derivative of f.' This becomes clearer in the expanded version of the Maclaurin series:

The Maclaurin series allows you to express functions as power series by following these steps:

1. Find the first few derivatives of the function until you recognize a pattern.

2. Substitute 0 for x into each of these derivatives.

3. Plug these values, term by term, into the formula for the Maclaurin series.

4. If possible, express the series in sigma notation.

For example, suppose that you want to find the Maclaurin series for e^x.

1. Find the first few derivatives of e^x until you recognize a pattern:

2. Substitute 0 for x into each of these derivatives.

3. Plug these values, term by term, into the formula for the Maclaurin series:

4. If possible, express the series in sigma notation:

To check this formula, use it to estimate e⁰ and e¹ by substituting 0 and 1, respectively, into the first six terms:

This exercise nails e⁰ exactly, and approximates e¹ to two decimal places. The Maclaurin series for e^x allows you to calculate this function for any value of x to any number of decimal places.

However, the Maclaurin series for e^x works best when x is close to 0. As x moves away from 0, you need to calculate more terms to get the same level of precision.

But now, you can begin to see why the Maclaurin series tends to provide better approximations for values close to 0: The number 0 is 'hardwired' into the formula as f(0), f‘(0), f'(0), and so forth.

The figure illustrates this point. The first graph shows sin x approximated by using the first two terms of the Maclaurin series — that is, as the third-degree polynomial

The second graph shows an approximation of sin x with four terms.

As you can see, each successive approximation improves upon the previous one. Furthermore, each equation tends to provide its best approximation when x is close to 0.

PowerDirector is one of the best loved video editing tools on Windows. Its ease of use and powerful suite of features make it suitable for both ordinary users and professionals. It offers you complete control over what aspects of the video content can be changed. To Mac user's disappointment, there is no Cyberlink PowerDirector for Mac OS X available yet. If you just made the move from Windows to Mac, you would definitely be missing the convenience and flexibility of PowerDirector. But that's not the end of the world, you can always find some Mac alternatives to PowerDirector for Mac, and Filmora Video Editor is one of the best alternatives to PowerDirector for Mac (macOS 10.13 High Sierra included).

The Best Alternative to PowerDirector for Mac - Filmora Video Editor

Power Of Two Download For Macbook Air

The Maclaurin series is a template that allows you to express many other functions as power series. It is the source of formulas for expressing both sin x and cos x as infinite series.

Without further ado, here it is:

The notation f⁽ⁿ⁾ means 'the nth derivative of f.' This becomes clearer in the expanded version of the Maclaurin series:

The Maclaurin series allows you to express functions as power series by following these steps:

1. Find the first few derivatives of the function until you recognize a pattern.

2. Substitute 0 for x into each of these derivatives.

3. Plug these values, term by term, into the formula for the Maclaurin series.

4. If possible, express the series in sigma notation.

For example, suppose that you want to find the Maclaurin series for e^x.

1. Find the first few derivatives of e^x until you recognize a pattern:

2. Substitute 0 for x into each of these derivatives.

3. Plug these values, term by term, into the formula for the Maclaurin series:

4. If possible, express the series in sigma notation:

To check this formula, use it to estimate e⁰ and e¹ by substituting 0 and 1, respectively, into the first six terms:

This exercise nails e⁰ exactly, and approximates e¹ to two decimal places. The Maclaurin series for e^x allows you to calculate this function for any value of x to any number of decimal places.

However, the Maclaurin series for e^x works best when x is close to 0. As x moves away from 0, you need to calculate more terms to get the same level of precision.

But now, you can begin to see why the Maclaurin series tends to provide better approximations for values close to 0: The number 0 is 'hardwired' into the formula as f(0), f‘(0), f'(0), and so forth.

The figure illustrates this point. The first graph shows sin x approximated by using the first two terms of the Maclaurin series — that is, as the third-degree polynomial

The second graph shows an approximation of sin x with four terms.

As you can see, each successive approximation improves upon the previous one. Furthermore, each equation tends to provide its best approximation when x is close to 0.

PowerDirector is one of the best loved video editing tools on Windows. Its ease of use and powerful suite of features make it suitable for both ordinary users and professionals. It offers you complete control over what aspects of the video content can be changed. To Mac user's disappointment, there is no Cyberlink PowerDirector for Mac OS X available yet. If you just made the move from Windows to Mac, you would definitely be missing the convenience and flexibility of PowerDirector. But that's not the end of the world, you can always find some Mac alternatives to PowerDirector for Mac, and Filmora Video Editor is one of the best alternatives to PowerDirector for Mac (macOS 10.13 High Sierra included).

The Best Alternative to PowerDirector for Mac - Filmora Video Editor

Power Of Two Download For Macbook Air

Filmora Video Editor for Mac is the best alternative to PowerDirector video editor for Mac (macOS High Sierra included) in the market and provides the best methods, tools and techniques to perform the video editing easily. Filmora Video Editor enables you to easily make movies , edit audio, apply effects and share your new movie on YouTube, Facebook, Vimeo, DVD, TV, iPhone, iPad, iPod, and more!

Filmora Video Editor for Mac is updated to support the latest macOS 10.13 (High Sierra).

Power Of Two Download For Macbook Pro

Key Features of Filmora Video Editor:

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• You can personalize your videos with features like green screening or adding titles.
• A huge library offers you a wide range of effects and themes to choose from.
• Stylize your videos with filters, overlays, face-offs and motion graphics.
• Export your video to any extension you need, depending on your purpose.

Why Choose This PowerDirector for Mac Alternative

Live Video Editing and Intuitive Interface

Filmora Video Editor has come with a user-friendly interface, which includes a most intuitive timeline located at the bottom of the video editing window. That timeline interface makes it extremely easy for user to identify the timeline to organize, sync and edit the video and audio. you are also provided the option of dragging and dropping videos into the video editing window and you can drag a soundtrack from iTunes to embed it in the videos. Different types of video formats are supported on Filmora Video Editor, so you can import any type of video formats for editing.

Visual Effects

Filmora Video Editor offers a slew of video editing tools allowing you to crop, rotate, apply effects, merge, split and trim, etc. In addition, you can use different types of enhancement tools along with the visual effects and video editing tools for application on video and audio enhancement. The users can immediately preview the edited videos in a side-by-side window to check the appearance of the end result.

Publish Your Finished Work to the Web

The basic idea behind creating videos is to share the videos with other web users on various social networking websites, and Filmora Video Editor allows you to publish edited videos directly to popular social networking websites such as Facebook and YouTube. You can also create preset profiles on the video editor so that you can easily export their edited videos to various social networking websites. Another convenience of using Filmora Video Editor is that the edited videos can be transferred to other devices such as iPhone, iPad and iPod for playback.