Problem 1 - Thomas Calculus

Question

Use substitution (as in Example 4) to find the Taylor series at \(x=0\) of the functions in Exercises \(1-10\) . $$e^{-5 x}$$

Step-by-Step Solution

Answer

The Taylor series for \( e^{-5x} \) at \( x = 0 \) is \( 1 - 5x + \frac{25x^2}{2} - \frac{125x^3}{6} + \ldots \).

Step 1: Identify the Function

We are asked to find the Taylor series for the function \( e^{-5x} \) at \( x = 0 \), which is also called the Maclaurin series.

Step 2: Recall the General Series Formula

The Maclaurin series for a function \( f(x) \) is given by \( f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \ldots \).

Step 3: Start With the General Taylor Series of \( e^x \)

The Taylor series of \( e^x \) at \( x = 0 \) is \( \sum_{n=0}^{\infty} \frac{x^n}{n!} \), which is \( 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots \).

Step 4: Apply Substitution in the Series

To find the series for \( e^{-5x} \), substitute \(-5x\) for \(x\) in the series for \( e^x \) to get: \( e^{-5x} = \sum_{n=0}^{\infty} \frac{(-5x)^n}{n!} \), which is \( 1 - 5x + \frac{(5x)^2}{2!} - \frac{(5x)^3}{3!} + \ldots \).

Step 5: Simplify the Series

Rewrite the series by simplifying each term: \( e^{-5x} = 1 - 5x + \frac{25x^2}{2} - \frac{125x^3}{6} + \ldots \).

Key Concepts

Maclaurin Series

The Maclaurin Series is a special case of the Taylor Series, focused on the point where the function is expanded around zero, that is, at \( x = 0 \). This series allows you to express complex functions as infinite sums of terms calculated from the derivatives of the function, evaluated at zero. This is useful when you want to approximate a function near this point.
Unlike the general Taylor Series, which can be expanded about any point \( a \), the Maclaurin Series simplifies calculations significantly by using \( a = 0 \).
When constructing the Maclaurin Series for any function \( f(x) \), you start with the value of the function at zero, \( f(0) \), then add terms involving the derivatives of the function at zero:

• First derivative term: \( f'(0)x \)
• Second derivative term: \( \frac{f''(0)}{2!}x^2 \)
• Third derivative term: \( \frac{f'''(0)}{3!}x^3 \)
• And so on...

Up to infinity, where each term's power and factorial in the denominator increase accordingly. This expansion helps in understanding the behavior of the function near zero and is widely used in calculus and various applications in physics and engineering.

Exponential Functions

Exponential Functions are functions where the variable appears as an exponent. The most common base for these functions is Euler's number, \( e \), which is approximately equal to 2.71828. A classic example is \( e^x \), representing continuous growth or decay, depending on whether the exponent is positive or negative.
When working with exponential functions in calculus, especially in Taylor or Maclaurin Series, they present unique properties. For example, the exponential function \( e^x \) is equal to its own derivative, making differentiation easy and leading to an interesting series representation.

• For positive exponents, exponential functions increase rapidly.
• For negative exponents, they decrease and approach zero.

These properties make them crucial in modeling natural growth processes, like population growth or radioactive decay, and in calculating compound interest in finance.

Power Series

A Power Series is an infinite series of the form \( \sum_{n=0}^{\infty} a_n (x - c)^n \), where \( a_n \) represents the coefficients and \( c \) is the center or point of expansion. In a Maclaurin series, \( c \) is zero, simplifying the series to \( \sum_{n=0}^{\infty} a_n x^n \).
Power Series is a powerful mathematical concept because it allows expressing functions as sums of infinitely many terms. These terms simplify complex functions into forms that are easier to handle analytically or computationally.

• Power series can converge within a certain radius, beyond which they don't approximate the function accurately.
• Power series can represent a wide variety of functions, including polynomial, exponential, and trigonometric functions.

Their convergence and ability to approximate functions over an interval make power series indispensable tools in both theoretical and applied mathematics.

Differentiation

Differentiation is a fundamental concept in calculus, concerned with finding the derivative of a function. The derivative measures how a function changes as its input changes, giving the rate of change or slope of the function at any given point.
Differentiation is the process used to obtain derivatives, and it's defined by the limit of the difference quotient:
\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]
In the context of Taylor or Maclaurin Series, differentiation focuses on evaluating the derivatives at a particular point, which enables constructing the terms of the series.

• By taking successive derivatives of a function, one can build higher-degree polynomial approximations that are increasingly accurate near the expansion point.
• The zeroeth derivative is simply the original function value.
• The first derivative provides a linear approximation (the tangent line).

These concepts are not only essential in theoretical calculus but also have practical applications in physics, engineering, economics, and beyond, where understanding rates of change is crucial.