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Chapter 9: Problem 20

Find the Maclaurin polynomial of degree \(n\) for the function. $$ f(x)=x^{2} e^{-x}, \quad n=4 $$

Short Answer

Expert verified
The Maclaurin polynomial of degree 4 for the function \(f(x) = x^{2}e^{-x}\) is \(P_{4}(x) = -x^{2} + \frac{1}{2}\cdot x^{3} - \frac{1}{6}\cdot x^{4}\).

Step by step solution

01

Define the function

The given function is \(f(x) = x^{2}e^{-x}\) and we need to find its Maclaurin polynomial of degree 4.
02

Find the \(n^{th}\) derivative for \(n=0,1,2,3,4\)

The consecutive derivatives at \(x=0\) can be derived as: \n - \(f^{(0)}(0) = 0\)\n - \(f^{(1)}(0) = 0\)\n - \(f^{(2)}(0) = -2\)\n - \(f^{(3)}(0) = 6\)\n - \(f^{(4)}(0) = -8\) This requires using the product and chain rules to differentiate the function, and then evaluating the derivatives at \(x = 0\).
03

Construct the Maclaurin polynomial

The Maclaurin polynomial of degree 4 can be constructed as follows based on Taylor series formula: \[P_{4}(x)=f^{(0)}(0)+f^{(1)}(0)\cdot x+ \frac{f^{(2)}(0)}{2!}\cdot x^{2}+\frac{f^{(3)}(0)}{3!}\cdot x^{3}+\frac{f^{(4)}(0)}{4!}\cdot x^{4}\] Substituting the values from step 2, we get: \[P_{4}(x) = -x^{2} + \frac{1}{2}\cdot x^{3} - \frac{1}{6}\cdot x^{4}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Taylor series
The Taylor series is a fundamental concept used to approximate functions with polynomials. It expresses a function as an infinite sum of terms calculated from the function's derivatives at a single point. When this point is zero, it's called a Maclaurin series, which is a specific case of the Taylor series.
  • Purpose: The Taylor series helps us approximate complex functions with simpler polynomial expressions. This is particularly useful in calculus and physics.
  • Formula: The general formula for a Taylor series of a function \( f(x) \) centered at \( a \) is: \[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots\]
  • Application to Maclaurin Polynomial: For a Maclaurin series, which is a Taylor series centered at 0, the formula simplifies to: \[ f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \ldots\]
In this exercise, we use the Maclaurin series (a type of Taylor series) to approximate \( f(x) = x^2 e^{-x} \) up to the fourth-degree polynomial.
Differentiation
Differentiation is the process of finding a derivative, which measures the rate at which a function's value changes as its input changes. For this Maclaurin series exercise, we need to differentiate the function multiple times.
  • Basic Derivative: If you have a function \( f(x) = x^n \), its derivative is given by \[ f'(x) = n x^{n-1}\]
  • Higher-order Derivatives: It's important to take derivatives multiple times to find the terms in a Taylor/Maclaurin series. For example, the second derivative \( f''(x) \) is the derivative of \( f'(x) \).
  • Product Rule: Used when differentiating products of functions, such as \( x^2 e^{-x} \). The product rule states: \[ (uv)' = u'v + uv'\] where \( u \) and \( v \) are functions of \( x \).
Differentiation allows us to compute the needed values for each term in the Maclaurin polynomial.
Chain rule
The chain rule is a crucial differentiation technique used when dealing with composite functions, such as \( e^{-x} \) in this problem.
  • Definition: The chain rule states that if a function \( y \) can be written as a composite function \( y = f(g(x)) \), then its derivative is: \[ \frac{dy}{dx} = f'(g(x)) \, g'(x)\]
  • Application: For \( f(x) = x^2 e^{-x} \), treating \( e^{-x} \) as the inner function and \( x^2 \) as the outer function, the chain rule helps us find the derivatives.
  • Step-by-Step: When applying the chain rule, differentiate the outer function, keeping the inner function unchanged, then multiply by the derivative of the inner function.
Using the chain rule, we can effectively break down and differentiate complex expressions.
Polynomial construction
Polynomial construction involves creating a polynomial that approximates a given function, such as in the Maclaurin series. Once derivatives are calculated for a function, each successive term of the polynomial is added based on these derivatives.
  • Process: After finding the necessary derivatives, they are plugged into the Taylor or Maclaurin formula.
  • Example: Based on the derivatives found: \[ f^{(0)}(0) = 0, \quad f^{(1)}(0) = 0, \quad f^{(2)}(0) = -2, \quad f^{(3)}(0) = 6, \quad f^{(4)}(0) = -8\]The Maclaurin polynomial is expressed as: \[ P_{4}(x) = -x^2 + \frac{1}{2}x^3 - \frac{1}{6}x^4\]
  • Result: This polynomial represents an approximation of the function \( f(x) = x^2 e^{-x} \) up to the fourth degree.
Polynomial construction simplifies complex functions into manageable approximations for analysis and problem-solving.

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Most popular questions from this chapter

Bessel Function The Bessel function of order 0 is \(J_{0}(x)=\sum_{k=0}^{\infty} \frac{(-1)^{k} x^{2 k}}{2^{2 k}(k !)^{2}}\) (a) Show that the series converges for all \(x\). (b) Show that the series is a solution of the differential equation \(x^{2} J_{0}{ }^{n}+x J_{0}{ }^{\prime}+x^{2} J_{0}=0\) (c) Use a graphing utility to graph the polynomial composed of the first four terms of \(J_{0}\). (d) Approximate \(\int_{0}^{1} J_{0} d x\) accurate to two decimal places.

The series represents a well-known function. Use a computer algebra system to graph the partial sum \(S_{10}\) and identify the function from the graph. $$ f(x)=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n}}{(2 n) !} $$

Prove that \(e\) is irrational. \([\) Hint: Assume that \(e=p / q\) is rational ( \(p\) and \(q\) are integers) and consider \(\left.e=1+1+\frac{1}{2 !}+\cdots+\frac{1}{n !}+\cdots\right]\)

Find the Maclaurin series for \(f(x)=\ln \frac{1+x}{1-x}\) and determine its radius of convergence. Use the first four terms of the series to approximate \(\ln 3 .\)

Probability A fair coin is tossed repeatedly. The probability that the first head occurs on the \(n\) th toss is given by \(P(n)=\left(\frac{1}{2}\right)^{n}\), where \(n \geq 1\) (a) Show that \(\sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^{n}=1\). (b) The expected number of tosses required until the first head occurs in the experiment is given by \(\sum_{n=1}^{\infty} n\left(\frac{1}{2}\right)^{n}\) Is this series geometric? (c) Use a computer algebra system to find the sum in part (b).
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