Worked example: coefficient in Maclaurin polynomial | Series | AP Calculus BC | Khan Academy

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17 Dec 201603:05
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TLDRThe video script discusses the Maclaurin series for a function g, focusing on finding the coefficient of the x squared term. It explains that the nth derivative of g at x equals zero is given by the square root of n plus seven over n cubed, for n greater than or equal to one. The coefficient for x squared is derived by evaluating the second derivative at zero, resulting in 3/16.

Takeaways
  • 📚 The nth derivative of a function g at x equals zero is given by a formula involving the square root of n plus seven over n to the third power, for n greater than or equal to one.
  • 🔍 The focus is on finding the coefficient for the term containing x squared in the Maclaurin series of g.
  • 📝 The Maclaurin series for g is an approximation that can be expanded to include various terms, but only the coefficient of x squared is needed here.
  • 🧩 The coefficient for x squared in the Maclaurin series is derived from the second derivative of g evaluated at zero.
  • 📐 The second derivative of g evaluated at zero is calculated using the given formula with n set to two, resulting in the square root of nine over two to the third power.
  • 🔢 Simplifying the expression for the second derivative gives a value of 3/8, which is then divided by two to find the coefficient for x squared.
  • 🎓 The final coefficient for the x squared term in the Maclaurin series is found to be 3/16.
  • 📉 The process involves understanding how to derive coefficients in a Maclaurin series by evaluating derivatives at zero.
  • 📚 The script emphasizes the importance of focusing on specific terms in a series, rather than expanding the entire series.
  • 📘 The example provided demonstrates a method for finding coefficients in a series, which can be applied to other functions and series as well.
Q & A

  • What is the nth derivative of g at x equals zero?

    -The nth derivative of g at x equals zero is given by the formula \( \frac{\sqrt{n + 7}}{n^3} \) for \( n \geq 1 \).

  • What is the Maclaurin series for a function g(x)?

    -The Maclaurin series for a function g(x) is an infinite series that approximates the function as a sum of terms involving powers of x, starting from x=0. It is given by \( g(0) + g'(0)x + \frac{g''(0)}{2!}x^2 + \frac{g'''(0)}{3!}x^3 + \cdots \).

  • What is the coefficient for the term containing x squared in the Maclaurin series of g?

    -The coefficient for the term containing x squared in the Maclaurin series of g is \( \frac{g''(0)}{2!} \).

  • How do you find the coefficient for the x squared term in the Maclaurin series of g?

    -To find the coefficient for the x squared term, you need to evaluate the second derivative of g at x equals zero and divide it by 2! (which is 2).

  • What is the second derivative of g evaluated at x equals zero?

    -The second derivative of g evaluated at x equals zero is \( \frac{\sqrt{2 + 7}}{2^3} = \frac{3}{8} \).

  • What is the final coefficient for the x squared term in the Maclaurin series of g?

    -The final coefficient for the x squared term in the Maclaurin series of g is \( \frac{3}{16} \).

  • Why is the coefficient for the x squared term calculated as 3/16?

    -The coefficient is calculated as 3/16 because the second derivative evaluated at zero is 3/8, and dividing this by 2! (which is 2) gives 3/16.

  • Is the Maclaurin series always a polynomial?

    -No, the Maclaurin series is an infinite series. However, if you truncate it after a certain number of terms, it can be represented as a polynomial.

  • What is the purpose of the Maclaurin series in approximating functions?

    -The Maclaurin series is used to approximate functions around a point, typically x=0, by expressing the function as a sum of terms involving powers of x. This can simplify calculations and provide insights into the behavior of the function near the point of expansion.

  • Can the Maclaurin series be used to approximate any function?

    -The Maclaurin series can be used to approximate many functions, but not all. It is most effective for functions that are smooth and well-behaved near x=0. Some functions may not have a convergent Maclaurin series or may require a different point of expansion.

Outlines
00:00
📚 Maclaurin Series Coefficient Calculation

This paragraph introduces the problem of finding the coefficient for the x squared term in the Maclaurin series of a given function g. The function's nth derivative at x equals zero is provided as a formula involving the square root of n plus seven, divided by n cubed. The instructor guides through the thought process of constructing the Maclaurin series and emphasizes the need to focus only on the second derivative to find the coefficient for x squared. The solution involves substituting n with 2 in the given formula for the nth derivative, calculating the result, and then using it to find the coefficient in the series, which is determined to be 3/16.

Mindmap
Keywords
💡Nth derivative
The 'Nth derivative' refers to the derivative of a function taken N times, where N is a non-negative integer. In the context of the video, the Nth derivative of a function g at x equals zero is given by a specific formula involving the square root of N plus seven over N to the third power. This concept is central to understanding how the function behaves at x=0, and it is used to determine the coefficients in the Maclaurin series.
💡Maclaurin series
A 'Maclaurin series' is a representation of a function as an infinite sum of terms, which are calculated from the derivatives of the function at a single point, typically zero. In the video, the instructor discusses the Maclaurin series for a function g, focusing on the coefficient of the term containing x squared. This series is a fundamental tool in approximating functions and is crucial for the problem at hand.
💡Coefficient
A 'coefficient' in a polynomial or series is a numerical factor that multiplies a variable raised to a certain power. In the script, the instructor is specifically interested in the coefficient of the term containing x squared in the Maclaurin series of g. This coefficient is essential for understanding the behavior of the function at small values of x.
💡Second derivative
The 'second derivative' of a function is the derivative of its first derivative. It provides information about the concavity and inflection points of the function. In the video, the second derivative of g at x equals zero is calculated using the given formula, which is key to finding the coefficient of the x squared term in the Maclaurin series.
💡Factorial
A 'factorial', denoted by n!, is the product of all positive integers up to n. In the context of the video, factorials are used in the denominators of terms in the Maclaurin series, indicating the number of ways to arrange n items. For example, the term with x squared in the series involves the second derivative divided by 2!, which simplifies to 1.
💡Function evaluation
In mathematics, 'function evaluation' refers to the process of substituting a value into a function to obtain a result. In the video, the instructor evaluates the Nth derivative of g at x equals zero, which is crucial for determining the coefficients in the Maclaurin series. This evaluation is a fundamental step in the process.
💡Polynomial
A 'polynomial' is an expression consisting of variables and coefficients, combined using only addition, subtraction, and multiplication. In the video, the instructor discusses the Maclaurin polynomial, which is a truncated version of the Maclaurin series, used to approximate functions. The focus is on finding the coefficient of the x squared term in this polynomial.
💡Infinite series
An 'infinite series' is a sequence of terms added together, where the number of terms is unbounded. The Maclaurin series is an example of an infinite series, where each term is derived from the derivatives of a function at a single point. The video discusses how the coefficients of these terms can be determined.
💡Principal root
The 'principal root' of a number is the non-negative square root, which is typically used when dealing with real numbers. In the video, the principal root is taken of the expression (n + 7) to determine the value of the Nth derivative at x equals zero, which is essential for calculating the coefficients in the Maclaurin series.
💡Approximation
In mathematics, 'approximation' refers to the process of finding a value that is close to the actual value of a quantity. In the video, the Maclaurin series is used to approximate the function g at values of x near zero. The coefficients of the terms in this series are crucial for the accuracy of the approximation.
Highlights

The nth derivative of g at x equals zero is given by a formula involving the square root of n plus seven over n to the third, for n greater than or equal to one.

The Maclaurin series for g is discussed, focusing on the terms that include x squared.

The general form of the Maclaurin series is presented, emphasizing the coefficients of each term.

The coefficient for the term containing x squared is the focus of the discussion.

The second derivative of g evaluated at x equals zero is calculated using the provided formula.

The calculation involves substituting n with two to find the second derivative.

The square root of two plus seven is taken, resulting in nine, and then the principal root is calculated.

The result of the principal root calculation is three over two to the third power.

The coefficient for the x squared term is simplified to 3/8.

The final coefficient for the x squared term in the Maclaurin series is found to be 3/16.

The process does not require finding the entire Maclaurin polynomial, only the coefficient for the x squared term.

The method involves evaluating the nth derivative at x equals zero and then applying it to the specific case of n equals two.

The importance of understanding the general formula for the nth derivative is emphasized.

The calculation process is detailed step by step, ensuring clarity in the derivation of the coefficient.

The final answer is presented clearly, highlighting the coefficient for the x squared term in the Maclaurin series.

The significance of the coefficient in the context of the Maclaurin series is discussed.

Transcripts

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