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Magazine Chapter 9: Problem 11
Find a formula for an arbitrary Taylor polynomial of \(f\). $$ f(x)=e^{-x} $$
Short Answer
Expert verified
The formula for the Taylor polynomial of \( e^{-x} \) is \( P_n(x) = \sum_{k=0}^{n} \frac{(-1)^{k}x^{k}}{k!} \).
Step by step solution
01
Understand the Function and Taylor Series
The function we are given is \( f(x) = e^{-x} \). A Taylor series expansion for a function \( f(x) \) about a point \( a \) is given by: \[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^{2} + \frac{f^{(3)}(a)}{3!}(x-a)^{3} + \cdots \] where \( f^{(n)}(a) \) is the \( n \)-th derivative of \( f \) evaluated at \( a \).
02
Determine the Derivatives of the Function
To find the Taylor series, we need derivatives of \( f(x) \). Here are the first few derivatives:- \( f(x) = e^{-x} \), so \( f(0) = 1 \).- First derivative: \( f'(x) = -e^{-x} \), and \( f'(0) = -1 \).- Second derivative: \( f''(x) = e^{-x} \), and \( f''(0) = 1 \).- Third derivative: \( f^{(3)}(x) = -e^{-x} \), and \( f^{(3)}(0) = -1 \).- This cycle repeats for higher derivatives.
03
Plug the Derivatives into the Taylor Series Formula
We start constructing the Taylor series centered at \( a = 0 \): \[ f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f^{(3)}(0)}{3!}x^3 + \cdots \]Substituting the values:\[ e^{-x} = 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \cdots \]
04
General Formula for the n-th Taylor Polynomial
The pattern established suggests that the formula for the Taylor polynomial of order \( n \) is:\[ P_n(x) = \sum_{k=0}^{n} \frac{(-1)^{k}x^{k}}{k!} \] This captures the alternating signs and factorial denominators seen in the individual terms.
05
Verify the Formula for Specific Values of n
Ensure the formula works for specific cases.- For \( n = 0 \): \( P_0(x) = 1 \)- For \( n = 1 \): \( P_1(x) = 1 - x \)- For \( n = 2 \): \( P_2(x) = 1 - x + \frac{x^2}{2} \)These match each Taylor polynomial derived individually.
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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 powerful mathematical tool used to approximate complex functions. It expands a function into an infinite series of terms calculated using the function鈥檚 derivatives at a single point. The general formula for a Taylor Series of a function \( f(x) \) about a point \( a \) is:
This is especially useful when the function itself is difficult to work with directly.
- \( f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^{2} + \cdots \)
- Each term depends on the derivatives of the function \( f \).
- The quality of the approximation improves as we include more terms.
This is especially useful when the function itself is difficult to work with directly.
Derivatives
Derivatives are a fundamental concept in calculus that describe how a function changes at any given point. In the context of Taylor Series, derivatives allow us to see the behavior of a function's slope and curvature, which helps in creating the polynomial approximation.
Different patterns in the sign and magnitude of these derivatives will affect the form of the Taylor polynomial.
- The first derivative \( f'(x) \) gives the rate of change or the slope of \( f \).
- The second derivative \( f''(x) \) describes the change of the slope (concavity) of \( f \).
- Higher-order derivatives continue this pattern, analyzing changes in curvature.
Different patterns in the sign and magnitude of these derivatives will affect the form of the Taylor polynomial.
Maclaurin Series
A special case of the Taylor Series is the Maclaurin Series. This occurs when the series is expanded around \( a = 0 \).
Moreover, this method aligns with many standard functions and often appears in foundational exercises like finding the polynomial for \( e^{-x} \).
- A Maclaurin Series is simply a Taylor Series centered at zero.
- It simplifies the equation to \( f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^{2} + \cdots \)
Moreover, this method aligns with many standard functions and often appears in foundational exercises like finding the polynomial for \( e^{-x} \).
Exponential Function
The exponential function is one of the most interesting and widely used functions in mathematics, denoted by \( f(x) = e^x \). Its properties and behavior make it essential in many areas of maths and science, such as growth processes and complex number calculations. The function \( f(x) = e^{-x} \), which is just the inverse of \( e^x \), provides an excellent example illustration of Taylor Polynomials.
- \( e^x \) has the unique property that its derivative is itself, \( f'(x) = e^x \).
- This property of self-derivation greatly simplifies the calculation of Taylor Series.
