Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the Taylor series in \(x-a\) through the term \((x-a)^{3}\). \(e^{x}, a=1\)

Step-by-Step Solution

Answer

The Taylor series of \(e^x\) about \(x=1\) through \((x-1)^3\) is \(e \left[ 1 + (x - 1) + \frac{1}{2}(x - 1)^2 + \frac{1}{6}(x - 1)^3 \right]\).

Step 1: Identify the Function and Center

The function we need the Taylor series for is \(f(x) = e^x\) and the series is centered around \(a = 1\). First, recall the formula for the Taylor series centered at \(a\): \[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots\]

Step 2: Compute the Function Values at \(x = a\)

Calculate \(f(1), f'(1), f''(1),\) and \(f'''(1)\). Since \(f(x) = e^x\), each derivative of \(e^x\) is also \(e^x\). Therefore:- \(f(1) = e^1 = e\),- \(f'(1) = e^1 = e\),- \(f''(1) = e^1 = e\), and- \(f'''(1) = e^1 = e\).

Step 3: Write the Taylor Series Expression

Substitute the computed function values into the Taylor series formula:\[e^x \approx e + e(x - 1) + \frac{e}{2!}(x - 1)^2 + \frac{e}{3!}(x-1)^3\]

Step 4: Simplify the Taylor Series

Simplify the expression by calculating the factorials and factor out \(e\):- \( \, e(x - 1)\),- \( \frac{e}{2}(x - 1)^2 \),- \( \frac{e}{6}(x - 1)^3 \).Combine these:\[ e^x \approx e \left[ 1 + (x - 1) + \frac{1}{2}(x - 1)^2 + \frac{1}{6}(x - 1)^3 \right]\]

Key Concepts

Exponential Functions

Exponential functions are a fundamental part of mathematics and often appear in various forms across many fields, from science to economics. The most famous exponential function is the natural exponential function, denoted as \( e^x \). This function has a unique property: its derivative is the same as the function itself.

• The base \( e \) is an irrational number approximately equal to 2.71828. It is the only number where the growth rate of the function equals the value of the function.
• Exponential functions are used to model situations where something grows or decays at a constant relative rate, such as population growth or radioactive decay.

Understanding exponential functions is crucial for dealing with more complex problems in higher mathematics and it sets the foundation for working with series expansions like the Taylor series.

Derivatives Calculation

Calculating derivatives is a critical skill in calculus, which allows you to understand how functions change. When considering exponential functions like \( f(x) = e^x \), calculating the derivatives is particularly straightforward because the derivative of \( e^x \) is also \( e^x \).

• First Derivative: \( f'(x) = e^x \), which means the rate of change of the function is equal to its current value. This consistency makes applications of the exponential function much simpler in calculus.
• The ability to recognize this pattern is vital as it allows us to efficiently compute higher-order derivatives, which are needed for series expansion.

For the Taylor series centered at a point \( a = 1 \), this directly applies since all higher derivatives up to the third are simply \( e^1 = e \), simplifying the process considerably.

Mathematical Series Expansion

Mathematical series expansions allow us to approximate functions using polynomials, making them easier to work with in computation and analysis. The Taylor series is a powerful tool in this category that provides a polynomial approximation of a function centered at a certain point.

• The general formula for the Taylor series of a function \( f(x) \) centered at \( a \) is: \[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots \]
• This series expansion is particularly useful because each additional term provides a more accurate approximation of the function near \( a \).

The key here is substituting the derivatives calculated at the point \( a \), making use of the properties of exponential functions to form our polynomial expansion.

Higher Mathematics

Higher mathematics encompasses advanced concepts that extend beyond basic algebra and calculus. It includes the study of complex functions, series expansions like Taylor and Fourier series, and differential equations.

• The Taylor series is a cornerstone concept in higher mathematics, bridging the gap between simple polynomial approximations and complex function behavior analysis.
• By understanding how to expand functions into series, students gain insight into analyzing not just constants and variables, but intricate behaviors of functions across different points.

By mastering series expansion methods, you can solve more complex problems, giving you a robust toolkit for both theoretical exploration and practical application in fields like engineering, physics, and computational biology.