Problem 29 - Thomas Calculus

Question

Find the Taylor series generated by \(f\) at \(x=a.\) \(f(x)=e^{x}, \quad a=2\)

Step-by-Step Solution

Answer

The Taylor series for \( e^x \) at \( x = 2 \) is \( \sum_{n=0}^{\infty} \frac{e^2}{n!}(x-2)^n \).

Step 1: Understand the Formula for Taylor Series

The Taylor series for a function \( f(x) \) about a point \( a \) is given by:\[T(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n\]where \( f^{(n)}(a) \) is the \( n \)-th derivative of \( f \) evaluated at \( x=a \).

Step 2: Calculate the Derivatives of the Function

For the function \( f(x) = e^x \), the derivatives are quite simple because the derivative of \( e^x \) is always \( e^x \). Therefore, the \( n \)-th derivative of \( f \) evaluated at any point \( x \) is \( e^x \).

Step 3: Evaluate Derivatives at \( a = 2 \)

Since \( f(x) = e^x \), we have \( f^{(n)}(x) = e^x \) for all \( n \). Thus, \( f^{(n)}(2) = e^2 \) for all \( n \).

Step 4: Write the Taylor Series

Substitute the derivatives and \( a = 2 \) into the Taylor series formula:\[T(x) = \sum_{n=0}^{\infty} \frac{e^2}{n!}(x-2)^n\]This represents the Taylor series for \( f(x) = e^x \) centered at \( x = 2 \), with each term being \( \frac{e^2}{n!}(x-2)^n \).

Key Concepts

Maclaurin Series

The Maclaurin series is a special case of the Taylor series, where the expansion occurs around zero. This means that in a Maclaurin series, the functions are expressed in terms of powers of x, rather than \(x-a\). You can think of it as a Taylor series with \(a = 0\). When \(a = 0\), the general Taylor series formula simplifies, making calculations less cumbersome without losing the essence.Some important points about Maclaurin series:

• Useful for simplifying functions so that they can be easily calculated or used in applications.
• Involves derivatives of all orders, just like Taylor series.
• Good for approximating values of functions near \(x = 0\).

The Maclaurin series is especially handy when dealing with functions like trigonometric functions and exponential functions. For example, the Maclaurin series for \(e^x\) is a widely known formula and appears in numerous mathematical situations.Understanding both Taylor and Maclaurin series is key in mathematics, especially in calculus and analytical contexts, providing tools for approximation and expansion.

Exponential Functions

Exponential functions are functions of the form \(f(x) = e^x\), where \(e\) is the base of the natural logarithm, approximately equal to 2.71828. These functions are prevalent in many scientific fields due to their properties of constant growth rates.Some key characteristics of exponential functions include:

• Rapid growth or decay, depending on the sign of the exponent.
• Their derivatives remain in the same form, which is unique compared to other functions.
• In many physical and natural sciences, they model processes such as population growth, radioactive decay, and interest calculations.

Because the derivative of \(e^x\) is \(e^x\) itself, exponential functions are particularly straightforward when working with calculus operations. This property significantly simplifies the derivation of series expansions like Taylor and Maclaurin, as seen in the exercise problem where derivatives are constant when evaluated.

Derivatives

Derivatives are fundamental tools in calculus, reflecting the rate at which a function is changing at any given point. For any function, calculating its derivatives gives us insight into its behavior and helps in analysis, prediction, and optimization tasks.When dealing with derivatives:

• They help in constructing Taylor series, where each derivative contributes to the accuracy of the approximation.
• For functions like \(e^x\), derivatives are particularly simple, always reverting back to \(e^x\) itself, making analysis uncomplicated.
• Derivatives of various orders (first, second, third, etc.) provide different levels of detail into the function's change and curvature.

In the context of the given exercise, derivatives are pivotal because they form the foundation upon which the entire series expansion sits. Calculating derivatives at a particular point, like \(x = 2\), facilitates the development of tailored series expansions for function approximation, as neatly demonstrated in constructing a Taylor series.