Worked example: power series from cos(x) | Series | AP Calculus BC | Khan Academy

Khan Academy
10 Oct 201408:00
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TLDRThis video explores finding the Maclaurin series representation of the function f(x) = x^3 * cos(x^2). The speaker initially discusses the complexity of deriving this series through differentiation, emphasizing the tedious nature of calculating higher-order derivatives. Instead, they suggest leveraging the known Maclaurin series for cos(x) and adapting it for cos(x^2). By substituting and simplifying, the video demonstrates how to obtain the first five non-zero terms of the Maclaurin series for the given function, providing a more efficient and insightful approach to solving such problems.

Takeaways
  • ๐Ÿง‘โ€๐Ÿซ The goal is to find the Maclaurin series representation for f(x) = x^3 * cos(x^2) with the first five non-zero terms.
  • ๐Ÿ“ The Maclaurin series is a Taylor series centered at zero.
  • ๐Ÿ”„ Finding derivatives directly can be complex and tedious.
  • ๐Ÿงฎ Using the known Maclaurin series for cos(x) can simplify the process.
  • ๐Ÿ“Š The Maclaurin series for cos(x) is 1 - x^2/2! + x^4/4! - x^6/6! + x^8/8! - ...
  • ๐Ÿ” Rewriting f(x) as x^3 * g(x^2) where g(x) = cos(x) helps in the series expansion.
  • ๐Ÿ”ข Substitute x with x^2 in the cos(x) series to get 1 - x^4/2! + x^8/4! - x^12/6! + x^16/8! - ...
  • ๐Ÿงฉ Multiply the resulting series by x^3 to obtain the Maclaurin series for f(x).
  • ๐Ÿ” The resulting series is x^3 - x^7/2! + x^11/4! - x^15/6! + x^19/8!
  • โœ… This approach avoids the need for finding higher-order derivatives directly.
Q & A

  • What is the goal of the Maclaurin series approximation in the video?

    -The goal is to find the first five non-zero terms of the Maclaurin series representation of the function f(x) = x^3 * cos(x^2).

  • Why might finding the Maclaurin series through derivatives be challenging for this function?

    -Finding the Maclaurin series through derivatives is challenging because taking the derivatives of the function f(x) = x^3 * cos(x^2) involves the product rule and chain rule, which becomes increasingly complex and painful for higher-order derivatives.

  • What is a Maclaurin series?

    -A Maclaurin series is a Taylor series centered at zero. It represents a function as an infinite sum of terms calculated from the values of its derivatives at zero.

  • What hint does the speaker provide to make finding the series easier?

    -The speaker hints that knowing the Maclaurin series for cosine of x can be useful. Specifically, using the series for cos(x) to help find the series for f(x) = x^3 * cos(x^2).

  • What is the Maclaurin series for cos(x)?

    -The Maclaurin series for cos(x) is: 1 - x^2 / 2! + x^4 / 4! - x^6 / 6! + x^8 / 8! - ... and so on.

  • How does substituting x^2 for x in the Maclaurin series of cos(x) help?

    -Substituting x^2 for x in the Maclaurin series of cos(x) allows us to find the series for cos(x^2). This substitution transforms the series into a new polynomial which can then be multiplied by x^3 to get the series for f(x).

  • What is the resulting series after substituting x^2 for x in the Maclaurin series of cos(x)?

    -After substituting x^2 for x, the resulting series is: 1 - x^4 / 2! + x^8 / 4! - x^12 / 6! + x^16 / 8! - ... and so on.

  • How do you obtain the Maclaurin series for f(x) = x^3 * cos(x^2)?

    -To obtain the Maclaurin series for f(x) = x^3 * cos(x^2), you multiply the transformed series (from the previous step) by x^3. This gives: x^3 - x^7 / 2! + x^11 / 4! - x^15 / 6! + x^19 / 8! - ... and so on.

  • Why is it beneficial to rewrite the function in terms of known series?

    -Rewriting the function in terms of known series, such as the Maclaurin series for cos(x), simplifies the process and avoids the tedious and complex task of finding high-order derivatives directly.

  • What is the key insight for constructing the Maclaurin series in this example?

    -The key insight is that if you can express your function as a product of a polynomial and a function with a known Maclaurin series, you can substitute appropriately and multiply to find the series representation more easily.

Outlines
00:00
๐Ÿง  Understanding Maclaurin Series for a Complex Function

The speaker introduces the problem of finding the Maclaurin series representation of the function f(x) = x^3 * cos(x^2). They emphasize the goal of deriving the first five non-zero terms of the series and mention the challenges involved in calculating derivatives for the function. The video aims to provide a more efficient method by leveraging the known Maclaurin series of cos(x).

05:01
๐Ÿ”„ Transforming and Applying Known Series

The speaker demonstrates how to transform the function using the known Maclaurin series for cos(x). By substituting x with x^2 and multiplying by x^3, they construct the Maclaurin series for the given function. This approach avoids the complexity of directly computing high-order derivatives and yields the desired first five non-zero terms. The explanation concludes with a general strategy for using known series expansions to simplify the derivation process for similar functions.

Mindmap
Keywords
๐Ÿ’กMaclaurin Series
The Maclaurin series is a type of Taylor series expansion of a function, which is centered at zero. It is used to approximate functions around a specific point, in this case, zero. In the video, the Maclaurin series is the main focus for approximating the function f(x) = x^3 * cos(x^2). The script mentions finding the first five non-zero terms of the Maclaurin series for this function.
๐Ÿ’กTaylor Series
The Taylor series is a mathematical tool used to represent a function as an infinite sum 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 zero. The video script discusses the process of finding the Taylor series for the given function, which becomes complex due to the need for multiple derivatives.
๐Ÿ’กDerivatives
Derivatives in calculus represent the rate at which a function changes with respect to its variable. In the context of the script, finding the derivatives of the function f(x) is a necessary step to determine the coefficients for the Taylor/Maclaurin series. The process is described as 'painful' due to the complexity of the function and the high order of derivatives required.
๐Ÿ’กProduct Rule
The product rule is a fundamental calculus rule used to find the derivative of a product of two functions. In the script, the product rule is applied when finding the first derivative of f(x), which involves multiplying x^3 by cos(x^2) and its derivative.
๐Ÿ’กCosine Function
The cosine function is a trigonometric function that describes a wave with a period of 2ฯ€. In the video, the cosine function is squared, as in cos(x^2), which affects the periodicity and the behavior of the function. The Maclaurin series for the cosine function is well-known and is used as a basis for finding the series for the given function.
๐Ÿ’กFactorial
A factorial, denoted by '!', is the product of all positive integers up to a given number. In the context of the Maclaurin series for cosine, factorials are used in the denominators of the series terms, such as 2!, 4!, etc. The script uses factorials to illustrate the pattern in the series expansion.
๐Ÿ’กPolynomial
A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, and multiplication. The script discusses how the Maclaurin series for the given function results in a polynomial when the x in the cosine series is replaced with x^2.
๐Ÿ’กApproximation
In mathematics, an approximation is a value that is close to the actual value but is easier to use or understand. The video's theme involves approximating the function f(x) using its Maclaurin series, which is an approximation that can be made more accurate by including more terms.
๐Ÿ’กExponentiation
Exponentiation is the operation of raising a number to the power of another number. In the script, the function involves exponentiation, such as x^3 and x^2, and the process of finding the Maclaurin series involves dealing with powers of x and x^2.
๐Ÿ’กNon-zero Terms
Non-zero terms refer to the terms in a series that do not equal zero. The script specifically asks for the first five non-zero terms of the Maclaurin series, emphasizing the importance of including only significant terms in the approximation.
๐Ÿ’กRe-expression
Re-expression in the context of the video means rewriting the original function in a form that makes it easier to find its Maclaurin series. The script suggests rewriting the function as a product of a power of x and a function whose Maclaurin series is already known, which simplifies the process.
Highlights

Introduction to finding the Maclaurin series representation of a function involving x cubed times the cosine of x squared.

Explanation of the Maclaurin series as the Taylor series centered at zero.

Challenge of finding the Taylor series due to the complexity of deriving the function.

Derivation of the first derivative using the product rule and its expression.

The potential frustration of finding higher order derivatives for the Taylor series.

The strategy of evaluating derivatives at zero for coefficients in the series.

Hint to use the known Maclaurin series for cosine(x) to simplify the problem.

Reminder of the Maclaurin series for cosine(x) from a previous video.

The insight to express the function as x cubed times the square of g(x), where g(x) is cosine(x).

Method to substitute x with x squared in the known Maclaurin series to simplify calculations.

Illustration of how to distribute x cubed across the Maclaurin series of g(x) squared.

Derivation of the first five non-zero terms of the Maclaurin series for the given function.

Avoidance of brute force by re-expressing the function in terms of known series.

General strategy for finding the Maclaurin series of a function expressed as a power of x times another function.

The importance of recognizing patterns and leveraging known series to simplify complex problems.

Final expression of the first five non-zero terms of the Maclaurin series for the given function.

Conclusion emphasizing the efficiency of the method used compared to brute force.

Transcripts

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