Question

In the following exercises, compute the Taylor series of each function around \(x=1\). $$ f(x)=e^{-x} $$

Short answer

The Taylor series is \[e^{-1} (1 - (x-1) + \frac{1}{2}(x-1)^2 - \frac{1}{6}(x-1)^3 + \cdots)\].

Step-by-step solution

Understand the Taylor Series Formula

The Taylor series of a function \(f(x)\) around a point \(a\) is given by the formula: \[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots \] In this exercise, \(a = 1\). We will compute the derivatives of \(f(x) = e^{-x}\) and evaluate them at \(x = 1\) to build the series.

Compute the First Derivative

The first derivative of \(f(x) = e^{-x}\) is \(f'(x) = -e^{-x}\). Evaluate it at \(x = 1\): \(f'(1) = -e^{-1}\).

Compute the Second Derivative

The second derivative is \(f''(x) = e^{-x}\). Evaluate it at \(x = 1\): \(f''(1) = e^{-1}\).

Compute the Third Derivative

The third derivative is \(f'''(x) = -e^{-x}\). Evaluate it at \(x = 1\): \(f'''(1) = -e^{-1}\).

Evaluate the Function at x=1

Evaluate the original function at \(x = 1\): \(f(1) = e^{-1}\).

Construct the Taylor Series

Now substitute the obtained values into the Taylor series formula: \[ f(x) = e^{-1} + (-e^{-1})(x-1) + \frac{e^{-1}}{2}(x-1)^2 + \frac{(-e^{-1})}{6}(x-1)^3 + \cdots \] Simplify: \[ e^{-1} \left( 1 - (x-1) + \frac{1}{2}(x-1)^2 - \frac{1}{6}(x-1)^3 + \cdots \right) \] This represents the expansion of \(f(x) = e^{-x}\) around \(x = 1\).

Key concepts

Derivatives

Derivatives are a foundational concept in calculus, representing the rate at which a function changes. For any given function, its derivative at a certain point gives us the slope of the tangent line at that point. Calculating derivatives is essential when developing a Taylor series, because it tells us how the function behaves near a specific value.In our exercise, the function of interest is the exponential function \( f(x) = e^{-x} \). To find the Taylor series, derivatives of this function are computed.

• First Derivative: The rate of change of \( e^{-x} \) is \( f'(x) = -e^{-x} \). This reflects how the function decreases as \( x \) increases.
• Second Derivative: Differentiating again gives \( f''(x) = e^{-x} \), indicating a change in the rate at which the function decreases.
• Third Derivative: Continuously differentiating yields \( f'''(x) = -e^{-x} \), showing a periodic change in acceleration.

Evaluating these derivatives at \( x = 1 \) helps construct the Taylor series, providing insights into the function's behavior right around this specific value.

Series Expansion

Series expansion is a powerful mathematical technique that allows us to express complex functions as infinite sums of simpler terms. This is incredibly useful, because it can simplify calculations and provide approximations for functioning behavior over a range.A Taylor series is a specific type of series expansion centered around a point, commonly denoted as \( a \). For our function \( f(x) = e^{-x} \), we expand it around the point \( x = 1 \). This means that we express the function as an infinite sum involving powers of \( (x-1) \). The formula for this expansion is:\[ f(x) = f(1) + f'(1)(x-1) + \frac{f''(1)}{2!}(x-1)^2 + \frac{f'''(1)}{3!}(x-1)^3 + \dots \]Each term introduces a higher power of \( (x-1) \). The coefficients are obtained from the derivatives of the function at the point \( x = 1 \). Ultimately, this series provides a local representation of the function, which is very accurate for values of \( x \) near 1.

Exponential Function

The exponential function, commonly denoted as \( e^x \), is one of the most important functions in mathematics. It has unique properties that make it a central object of study, particularly within calculus and series theory.For this exercise, we are considering its variant, \( e^{-x} \), known for describing processes like decay or damping in natural systems. Here are some of its key characteristics:

• Continuous Growth or Decay: Depending on whether the exponent is positive or negative, the function exhibits exponential growth or decay respectively.
• Self-derivative: The unique property of the exponential function is that it is its own derivative. When differentiated, \( e^x \) becomes \( e^x \) again, and \( e^{-x} \) differentiates to \( -e^{-x} \).
• Ubiquitous in Mathematics: Exponential functions appear in various fields, from solving differential equations to describing statistical distributions.

Understanding how \( e^{-x} \) behaves is crucial when finding its Taylor series, as these mathematical patterns dictate how the series terms are formed and the function's proximity fidelity.