Maclaurin series of sin(x) | Series | AP Calculus BC | Khan Academy

Khan Academy
17 May 201106:33
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TLDRThis video delves into the mathematical concept of approximating the sine function using the Maclaurin series, a special case of the Taylor series centered at x=0. The presenter illustrates the process by computing the derivatives of the sine function and evaluating them at x=0, revealing a cyclical pattern. The video highlights the contrasting nature of sine and cosine, where cosine is represented by even powers of x, and sine by odd powers, switching signs. This exploration of sine and cosine through their polynomial representations showcases the intricate relationship between these fundamental trigonometric functions.

Takeaways
  • 📚 The Maclaurin series, a special case of the Taylor series, is used to approximate functions around x=0.
  • 🌀 The video demonstrates the process of approximating the sine of x using the Maclaurin series.
  • 🔄 Derivatives of sine of x show a cyclical pattern: cos(x), -sin(x), -cos(x), sin(x), and so on.
  • 👁️ Evaluating derivatives of sine at x=0 reveals a sequence of coefficients for the Maclaurin series: 0, 1, 0, -1, 0, ...
  • 📈 The Maclaurin series for sine of x is a polynomial representation involving odd powers of x and alternating signs.
  • 🔢 The series for cosine of x involves even powers of x, while sine of x involves odd powers.
  • 🎢 Adding more terms to the series improves the approximation of the sine and cosine functions.
  • 🌟 The video hints at the complementary nature of sine and cosine, filling each other's gaps in their polynomial representations.
  • 💡 The upcoming video will explore the Maclaurin series for e^x and its relation to sine and cosine when involving imaginary numbers.
  • 🌐 The mathematical concepts presented are foundational for understanding more advanced topics in calculus and analysis.
Q & A

  • What is a Maclaurin series?

    -A Maclaurin series is a type of Taylor series centered around x equals 0, used for approximating functions with polynomials.

  • How does the Maclaurin series approximation for cosine of x relate to its pattern?

    -The Maclaurin series approximation for cosine of x shows a pattern where the derivatives cycle through 0, 1, 0, -1, and then repeat, with the signs alternating after each cycle.

  • What is the first derivative of sine of x?

    -The first derivative of sine of x is cosine of x.

  • What happens to the second derivative of sine of x?

    -The second derivative of sine of x is the derivative of cosine of x, which is negative sine of x.

  • How does the third derivative of sine of x relate to the first derivative?

    -The third derivative of sine of x is the derivative of negative sine of x, which results in negative cosine of x.

  • What pattern do the derivatives of sine of x follow after being evaluated at x equals 0?

    -After evaluating the derivatives at x equals 0, the pattern follows 0, 1, 0, -1, and then repeats, cycling through the same sequence.

  • What is the polynomial representation of sine of x using the Maclaurin series?

    -The polynomial representation of sine of x using the Maclaurin series is x - (x^3)/3! + (x^5)/5! - (x^7)/7! + ..., where the sine terms alternate in sign and only odd powers of x are included.

  • How does the polynomial representation of sine of x differ from that of cosine of x?

    -The polynomial representation of sine of x includes only odd powers of x with alternating signs, while cosine of x includes even powers of x with signs that alternate after each cycle.

  • What is the significance of the factorial in the Maclaurin series polynomial representation?

    -The factorial in the Maclaurin series polynomial representation is used as a denominator to normalize the coefficients of each term, ensuring the polynomial approximates the function accurately.

  • What is the relationship between sine and cosine functions in terms of their Maclaurin series?

    -The Maclaurin series for sine and cosine show a complementary relationship, with cosine involving even powers of x and sine involving odd powers, both with alternating signs.

  • How does the Maclaurin series for e to the x differ from those of sine and cosine?

    -The Maclaurin series for e to the x includes a combination of even and odd powers of x, with a positive coefficient for each term, and it does not alternate signs like sine and cosine do.

  • What is the role of imaginary numbers in the combination of functions in the context of Maclaurin series?

    -When dealing with e to the x and involving imaginary numbers, the combination of real and imaginary parts can lead to more complex and fascinating patterns that are not present in the real-valued functions alone.

Outlines
00:00
📚 Approximating Sine with Maclaurin Series

This paragraph delves into the process of approximating the sine function using the Maclaurin series, which is a special case of the Taylor series centered around x=0. The discussion begins with a recap of the previous video on the Maclaurin series of the cosine function and transitions into exploring a similar approach for sine. The paragraph outlines the steps for finding the derivatives of sine and evaluating them at x=0, highlighting the cyclical pattern that emerges. The derivatives of sine follow a sequence of cosine, negative sine, negative cosine, and positive sine, and so on. The focus then shifts to the construction of the polynomial representation of sine using the Maclaurin series, emphasizing the pattern of alternating signs and the fact that sine's series involves odd powers of x. The comparison between the Maclaurin series of cosine (even powers of x) and sine (odd powers of x) underscores the complementary nature of these two fundamental trigonometric functions.

05:00
🔢 Sin and Cos Patterns in Polynomial Representation

This paragraph further explores the patterns observed in the polynomial representation of sine and cosine functions, highlighting the alternating signs and the specific powers of x involved. It contrasts the series of cosine, which includes even powers of x with a positive sign, against the series of sine, which comprises odd powers of x that alternate in sign. This comparison reveals a fascinating interplay between the two functions, with sine and cosine seemingly filling gaps in each other's representations. The paragraph also hints at the upcoming exploration of the exponential function (e to the power of x) and its intriguing relationship with sine and cosine when imaginary numbers are introduced. This sets the stage for a deeper understanding of the mathematical concepts and their interconnectedness, promising a mind-expanding exploration in the subsequent video.

Mindmap
Keywords
💡Maclaurin series
The Maclaurin series is a type of Taylor series centered around 0, used to approximate functions as polynomials. In the video, it is used to approximate the functions of cosine and sine, demonstrating how these approximations become increasingly accurate as more terms are added. The Maclaurin series for cosine is shown to be an even power series, while that for sine is an odd power series, highlighting the complementary nature of these functions.
💡Taylor series
The Taylor series is a mathematical representation that allows functions to be expressed as infinite sums of terms calculated from the values of the function's derivatives at a single point. The Maclaurin series is a special case of the Taylor series where the point is 0. In the video, the Taylor series is mentioned as the foundation for the Maclaurin series, and it is used to approximate the sine and cosine functions around x=0.
💡Derivatives
Derivatives represent the rate of change of a function with respect to its variable. In the context of the video, derivatives are used to find the coefficients for the Maclaurin series of sine and cosine functions. The process involves taking successive derivatives and evaluating them at x=0 to determine the pattern of the series.
💡Polynomial representation
A polynomial representation is a mathematical expression that approximates a function using a sum of powers of the variable, each multiplied by a coefficient. In the video, the Maclaurin series for sine and cosine functions are examples of polynomial representations that provide close approximations of these functions, especially as more terms are included.
💡Sine of x
The sine of x, often written as sin(x), is a fundamental function in trigonometry that represents the ratio of the length of the side opposite to an angle in a right triangle to the length of the hypotenuse. In the video, the sine function is approximated using its Maclaurin series, revealing a pattern of alternating signs and odd powers of x.
💡Cosine of x
The cosine of x, often written as cos(x), is another fundamental trigonometric function that represents the ratio of the length of the adjacent side to an angle in a right triangle to the length of the hypotenuse. In the video, the cosine function is approximated using its Maclaurin series, which consists of even powers of x with alternating signs.
💡Cycles
In the context of the video, cycles refer to the repeating patterns observed when taking successive derivatives of the sine and cosine functions. The derivatives of sine and cosine cycle through a pattern of positive and negative values, eventually returning to their original function after a certain number of derivatives.
💡Approximation
Approximation in mathematics is the process of estimating a value or quantity that is close to the actual value or quantity but not exact. In the video, the Maclaurin series is used to approximate the functions sine and cosine, showing how the polynomial becomes a closer representation of the functions as more terms are added.
💡Factorial
The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. In the video, factorials are used as denominators in the Maclaurin series for sine and cosine functions, indicating how the coefficients are determined based on the power of x and the corresponding factorial.
💡Complementary nature
The complementary nature refers to the relationship between two things that are complete or make up a whole when combined. In the video, this term is used to describe the relationship between the sine and cosine functions, as their Maclaurin series show a pattern where one has even powers of x and the other has odd powers, complementing each other.
💡Imaginary numbers
Imaginary numbers are a class of numbers that, when squared, result in a negative number, and are used to extend the real number system. They are denoted by the symbol 'i', where i^2 = -1. In the video, the mention of imaginary numbers suggests a future discussion on how they interact with the exponential function e^x, leading to more complex and fascinating mathematical concepts.
Highlights

The Maclaurin series is used to approximate the cosine of x using a polynomial.

A similar pattern is sought to approximate the sine of x using the Maclaurin series.

The Maclaurin series is a special case of the Taylor series centered around x equals 0.

The first derivative of sine of x is cosine of x.

The second derivative of sine of x is negative sine of x.

The third derivative of sine of x is negative cosine of x.

The fourth derivative of sine of x returns to positive sine of x, showing a cyclical pattern.

The Maclaurin series for sine of x is derived by evaluating the function and its derivatives at x equals 0.

The sine of x at x equals 0 is 0, and its first derivative (cosine of x) at x equals 0 is 1.

The second derivative of sine of x at x equals 0 is 0, and the third derivative is negative 1.

The Maclaurin series for sine of x is a polynomial representation that alternates signs and includes only odd powers of x.

The cosine of x Maclaurin series includes only even powers of x with alternating signs and division by the corresponding factorial.

The sine of x series complements the cosine of x series, with sine filling the gaps of odd powers that cosine leaves out.

The Maclaurin series for sine of x can be expressed as a sum of odd powers of x divided by the factorial of the power, alternating in sign.

The next video will explore the Maclaurin series for e to the power of x, which will involve a combination of sine and cosine.

The inclusion of imaginary numbers in the e to the x series will lead to a mind-blowing combination of sine and cosine patterns.

Transcripts

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