Taylor polynomial remainder (part 2) | Series | AP Calculus BC | Khan Academy

Khan Academy
15 Sept 201115:08
EducationalLearning
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TLDRThe video delves into the concept of an error function in approximation theory, emphasizing its distinction from the expected value. It explores the relationship between the error function and the derivatives of the function being approximated. The key focus is on bounding the error at a point 'b' by leveraging the properties of the nth derivative of the error function. The video outlines a method to calculate an upper bound for the error function over a closed interval, using integration and the maximum value of the function's derivative. This provides a powerful tool for assessing the accuracy of approximations.

Takeaways
  • 📈 The concept of an error function is introduced, distinct from expected value, focusing on the difference between a function and its approximation.
  • 🔢 The error function's absolute value is of interest, as it represents the distance between the function and the polynomial approximation at a given point x.
  • 📌 The goal is to bound the error at some point b, greater than a, by finding a constant value that the error will not exceed.
  • 📚 The (n+1)th derivative of the error function is shown to be equal to the (n+1)th derivative of the original function, which is crucial for bounding the error.
  • 🌟 If the (n+1)th derivative of the original function is bounded over a specific interval, this can be used to bound the (n+1)th derivative of the error function.
  • 🧮 Integration is suggested as a method to bound the error itself at a value b, starting from the bounded (n+1)th derivative.
  • 📊 An example is provided where the (n+1)th derivative of f(x) is bounded by a constant m over the interval of interest.
  • 🌐 The importance of the interval in which the derivative is bounded is emphasized, as it must include the point b for the bounding to be relevant.
  • 🔄 Through a series of integrations, the goal is to work down from the (n+1)th derivative to the 0th derivative, which represents the error function itself.
  • 🎯 A general formula for bounding the error function is derived, involving the constant m, the points a and b, the power of (n+1), and n+1 factorial.
  • 💡 The ability to determine the maximum value of the function allows for the application of this bounding formula to real-world problems, providing a powerful mathematical tool.
Q & A

  • What is the error function discussed in the video?

    -The error function discussed in the video is the difference between the actual function and its approximation, represented by the absolute value of the difference at a given point x.

  • Why is the error function important in approximations?

    -The error function is important because it quantifies the accuracy of the approximation. By bounding the error, we can ensure that the approximation is within acceptable limits for practical applications.

  • How does the video attempt to bound the error function?

    -The video attempts to bound the error function by integrating the absolute value of the nth derivative of the error function, which is assumed to be continuous and bounded over a given interval.

  • What is the significance of the nth derivative of the error function?

    -The nth derivative of the error function is significant because it allows us to find an upper bound for the error function itself. By integrating the nth derivative, we can derive a formula to estimate the maximum possible error within a given interval.

  • How does the video relate the nth derivative of the function to the error function?

    -The video establishes that the nth derivative of the error function is equal to the nth derivative of the actual function by absolute value. This relationship is crucial for bounding the error since it allows us to apply knowledge about the function's derivatives to the error function.

  • What is the role of the constant 'm' in bounding the error function?

    -The constant 'm' represents the maximum value of the absolute value of the nth derivative of the function over the interval of interest. It is used to ensure that the bound on the error function is as tight as possible.

  • What is the significance of the interval [a, b] in the context of bounding the error?

    -The interval [a, b] is the range over which the function is being approximated. The error is bounded over this interval, which includes the point 'b' where the error is to be evaluated.

  • How does the video handle the absolute value when integrating the nth derivative?

    -The video notes that integrating the absolute value of a function can lead to a larger result than integrating the function itself, especially when the function changes sign over the interval. The absolute value of the integral is therefore used to ensure that the error bound accounts for both positive and negative deviations from the approximation.

  • What is the final formula derived for bounding the error function?

    -The final formula derived for bounding the error function is the error function at 'b' is less than or equal to 'm times (b - a) raised to the power of (n + 1) divided by (n + 1) factorial'. This formula provides an upper bound on the error at any point 'b' within the interval [a, b].

  • How can the derived formula be applied in practical scenarios?

    -The derived formula can be applied in practical scenarios where an approximation of a function is used, and it is necessary to ensure that the approximation is within acceptable limits. By knowing the maximum value of the nth derivative and the interval over which the approximation is valid, one can calculate the maximum possible error at any point within that interval.

  • What is the role of the constant 'c' in the bounding formula?

    -The constant 'c' is introduced as part of the indefinite integral and represents an arbitrary constant. In the bounding formula, 'c' is chosen to be as small as possible to provide the tightest upper bound on the error function, which in this case is '-negative Ma', ensuring the bound is as low as possible.

Outlines
00:00
📈 Introduction to Error Function and Approximation

This paragraph introduces the concept of the error function, which measures the difference between a function and its approximation. It emphasizes that the error function is concerned with the absolute value of the difference, as the approximation may overshoot or undershoot the actual function at different points. The speaker aims to bound this error at a certain point 'b' by leveraging knowledge of the function's derivatives and using integration techniques.

05:03
🧮 Derivatives and Bounding the Error Function

The speaker discusses the relationship between the nth derivative of the error function and the function itself. By knowing the maximum value 'm' of the nth derivative over a given interval, the speaker outlines a method to bound the error function. This involves integrating the absolute value of the nth derivative and applying the fundamental theorem of calculus to find an upper bound for the error function in terms of the nth derivative and the interval [a, b].

10:04
📊 Minimizing the Upper Bound of the Error

In this section, the speaker refines the process of bounding the error function by minimizing the upper bound. The speaker explains that by choosing an appropriate constant 'c' based on the value of the nth derivative at 'a', the upper bound can be minimized. The process involves repeated integration and application of the properties of derivatives to eventually bound the error function in terms of 'm', 'x', 'a', and 'n'. The speaker also hints at the potential of applying this method to real-world examples.

Mindmap
Keywords
💡Error Function
The Error Function, denoted by E, is a mathematical concept used to describe the difference between a function and its approximation. In the context of the video, it is crucial for understanding the accuracy of the approximation and is defined as the absolute value of the difference between the function f(x) and its polynomial approximation at a given point x. The video focuses on bounding this error to ensure the approximation is within acceptable limits.
💡Absolute Value
The absolute value of a number is its non-negative value, regardless of its sign. In the video, absolute value is used to define the error function, as it captures the magnitude of the error without considering its direction (whether the polynomial is above or below the actual function). This concept is essential for bounding the error and ensuring the approximation's accuracy.
💡Derivative
A derivative represents the rate of change or the slope of a function at a specific point. In the video, the n plus 1th derivative of the error function is equated to the n plus 1th derivative of the original function, which is used to find a bound for the error. Understanding derivatives is critical for analyzing the behavior of the error function and its approximation.
💡Bounding
Bounding involves limiting a value or function within a certain range or by a constant value. In the video, the goal is to find a constant that bounds the error function at some point b, ensuring that the error remains within acceptable limits. Bounding is crucial for validating the accuracy and reliability of the approximation.
💡Integration
Integration is the process of finding the antiderivative or the area under the curve of a function. In the video, integration is used to move from the n plus 1th derivative back to the original error function, allowing the creation of a bound for the error. This mathematical technique is essential for connecting the rate of change (derivative) to the accumulated change (error function).
💡Interval
An interval is a set of numbers on the number line between two values, including the endpoints. In the video, the interval between a and b is the range over which the error function is being bounded. The concept of an interval is important for specifying the domain in which the approximation is valid.
💡Constant Value
A constant value is a number that does not change. In the context of the video, finding a constant value that bounds the error function is the objective, as it provides a maximum limit to the error, ensuring the approximation's reliability within the given interval.
💡Polynomial Approximation
Polynomial Approximation is the process of approximating a function using a polynomial. In the video, the focus is on the difference between the actual function and its polynomial approximation, which is the error function. Understanding polynomial approximation is key to analyzing the accuracy and limitations of the approximation.
💡Max Value
The max value, or maximum value, of a function is the highest value that the function reaches over a specified domain or interval. In the video, the max value is used to bound the n plus 1th derivative of the function, which in turn helps to bound the error function. Identifying the max value is crucial for establishing the tightest possible bounds on the error.
💡nth Derivative
The nth derivative of a function is the derivative taken n times. It represents the rate of change of the function's rate of change, and so on, n times. In the video, the nth derivative of the error function is used to establish bounds on the error, with the goal of minimizing this derivative to improve the approximation's accuracy.
💡Anti-derivative
The anti-derivative, also known as the integral, is the reverse operation of differentiation. It is used to find the original function from its derivative. In the video, the anti-derivative is used to move from the n plus 1th derivative back to the nth derivative and so on, until reaching the 0th derivative, which is the error function itself.
Highlights

Exploration of the error function begins, distinct from the expected value.

The error function represents the difference between the actual function and its approximation.

The absolute value of the error is of interest, not just the difference.

The goal is to bound the error at some point b, assuming b is greater than a.

The (n+1)th derivative of the error function is equal to the (n+1)th derivative of the function.

If the (n+1)th derivative of the function is bounded over an interval, it can help bound the error function.

The absolute value of the (n+1)th derivative is denoted as m for the interval between a and b.

The (n+1)th derivative of the error function's absolute value is also bounded by m.

Integration of both sides of the inequality may help bound the actual error function.

The integral of the absolute value is generally greater than the absolute value of the integral.

The nth derivative of the error function at a is equal to 0, which helps in bounding the error.

The error function's nth derivative is bounded by M times x minus a.

Further integration leads to bounding the error function by M times x minus a squared over 2.

The process can be continued until the bound on the error function of x is achieved.

The final bound on the error function is expressed as a formula involving m, x, a, n, and factorial.

Knowing the maximum value of the function allows bounding the error function over the interval between a and b.

The error function at b can be bounded by a specific formula involving m, b, a, and n plus 1 factorial.

Transcripts

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