Determine the fourth Taylor polynomial of f(x)=ln(1-x) at x=0, and use it to estimate ln(.9).

Name: Problem 10, Chapter 11: Calculus and Its Applications
Uploaded: 2021-06-26T01:20:14-08:00
Duration: 5 min 19 s
Channel: Kian Manafi
Description: Determine the fourth Taylor polynomial of f(x)=ln(1-x) at x=0, and use it to estimate ln(.9).

Determine the fourth Taylor polynomial of f(x)=ln(1-x) at...

Key Concepts

Taylor Polynomial

A Taylor polynomial is an approximation of a smooth function using the function’s derivatives at a specific point. It represents the function as a polynomial where the coefficients are determined by the successive derivatives evaluated at that point. This concept is fundamental in analysis as it allows us to locally approximate more complicated functions by simpler polynomial expressions.

Maclaurin Series

A Maclaurin series is a special case of the Taylor series that is centered at zero. It expresses a function as an infinite sum of its derivatives at zero, weighted by powers of the variable divided by factorial terms. This series is widely used in practice because many functions can be easily expanded around zero, enabling straightforward approximations for values near the origin.

Series Expansion of ln(1 - x)

The series expansion of ln(1 - x) is a classical example in calculus that demonstrates how logarithmic functions can be expressed as an alternating series. By differentiating ln(1 - x) repeatedly and evaluating at zero, one obtains a series where each term involves increasing powers of x. This series provides insights into the behavior of logarithmic functions in a neighborhood around the expansion point.

Polynomial Approximation for Estimation

Polynomial approximation for estimation involves using a truncated Taylor or Maclaurin series to estimate the value of a function near a given expansion point. This method relies on summing a finite number of terms from the series to provide an approximate value, which is particularly useful when the exact function is difficult to compute directly. It also highlights the balance between approximation accuracy and computational simplicity.

Transcript

00:01 Okay, so we're going to find the fourth tailor polynomial of this function, f of x.

00:05 The first thing we're going to want to do is find the first four derivatives of our function, because that's going to be needed when we're actually constructing our tailor polynomial.

00:16 So the first derivative is the derivative of natural log of 1 minus x, so that's 1 divided by 1 minus x, multiplied by negative 1, which is equal to negative 1, divided by 1 minus x.

00:30 The second derivative is equal to the derivative of negative 1 divided by 1 minus x.

00:36 And this is equal to negative negative 1 minus x to the negative 1 power not x minus 1.

00:44 And so we can just use power rule and chain rule to find this derivative.

00:48 Bring down the negative 1.

00:49 We multiply by the already negative 1 that we have.

00:53 So we're going to have just 1 minus x to the negative 2 and then multiplied by the derivative of what's in this parentheses, which is just negative 1.

01:01 So this is negative 1 minus x to the negative 2.

01:05 The third derivative, it's going to be negative 2 times negative 1, which is just going to be 2 times 1 minus x to the negative 3, and then multiplied by negative 1.

01:17 And then the fourth one, it's going to be 2 times negative 3 times negative 1 times negative 1 again.

01:25 So we're going to have negative 6 times 1 minus x to the negative 4.

01:30 Power.

01:33 So now let's figure out what all of these derivatives are equal to an x is equal to zero.

01:37 Since we're looking at the taylor polynomial about a is equal to zero, so the first derivative is just negative one divided by one minus zero, which is just going to be negative one.

01:52 The second derivative is negative one minus zero to the negative two, which is just going to be negative one divided by one squared, which is just negative one...

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