Approximate the function f(t)=(1)/(8)(2 t-1)^3+(1)/(4) by a quadratic polynomial on the interval

Approximate the function f(t)=(1)/(8)(2 t-1)^3+(1)/(4) by a...

Key Concepts

Least Squares Approximation

This method involves finding the approximating function that minimizes the L2 norm of the error – that is, it minimizes the integral of the square of the difference between the original function and the approximation. It is a fundamental technique in numerical analysis and statistics for obtaining the best fit under a given metric.

L2 Norm

The L2 norm is a measure of the magnitude of functions in an integral sense, defined as the square root of the integral of the square of the function over a specified interval. When used in approximation, minimizing the L2 norm of the error ensures that the average error (in a mean-square sense) is minimized.

Orthogonal Projection in Hilbert Spaces

When approximating a function by another from a subspace (such as quadratic polynomials), the best approximation in the L2 norm is obtained by projecting the function onto that subspace. This process establishes that the error is orthogonal to every function in the chosen subspace, ensuring the optimal approximation in the mean-square sense.

Polynomial Approximation

This involves representing a function with a polynomial of a particular degree. It is a key concept in numerical analysis and is used to simplify complex functions into easier-to-analyze forms, especially when combined with least squares methods to determine the best coefficients.

Error Analysis

After obtaining an approximation, error analysis studies where the approximation deviates most from the original function. This often involves comparing the graphs of the function and its approximation to identify regions or points (typically at the interval’s boundaries or where the function behavior changes rapidly) where the absolute error is highest.

Transcript

00:01 For this problem, we are asked to compute the taylor polynomial for the function f of x equals x the power of 11 over 2, and use the error bound to find the maximum possible size of the error.

00:11 We're then asked to verify the result with the calculator.

00:14 So to begin, we are going to have to take a bunch of derivatives here.

00:18 The first derivative of x the power of 11 over 2 is going to be 11 over 2 times x to the power of 9 over 2.

00:24 The second derivative is going to be 9.

00:29 I'll write it as 11 over 2 times 9 over 2 times x the power of 7 over 2.

00:36 The third derivative is going to be 11 over 2 times 9 over 2 times 7 over 2 times x the power of 5 over 2.

00:49 The fourth derivative is going to be 11 times 9 times 7 times 5 divided by 2 to the power of 4 times x the power of 3 over 2.

01:03 And we'll also want the fifth derivative, which is going to be 11 times 9, times 7, times 5 times 3, divided by 2 to the power of 5 times x the power of 1⁄2.

01:23 Now, the relevance here is that to determine the error bound, we are going to want to have the maximum value of the fifth derivative between 1 and 1 .2, which is clearly going to be the fifth derivative evaluated at 1 .2, because x power of 1 half is an increasing function.

01:44 So i'll note ahead of time here that we are going to have that k equals the fifth derivative there, evaluated at 1 .2, where also we were asked for the taylor polynomial.

01:56 So the general form of the taylor polynomial is going to be, since we're evaluating at 1, that's going to be starting off with just, just 1.

02:06 Then we'd have plus 11 over 2 times x minus 1...

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