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Magazine Chapter 30: Problem 23
Evaluate the given functions by using three terms of the appropriate Taylor series. $$e^{\pi}$$
Short Answer
Expert verified
Using the first three terms of the Taylor series, \( e^\pi \approx 9.0764 \).
Step by step solution
01
Understand the Taylor Series
The Taylor series for the exponential function centered at zero is given by \( e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots \). In this problem, you need to find the value of \( e^\pi \) using the first three terms of the Taylor series. This means you need to substitute \( x = \pi \) into the series and calculate the first three terms.
02
Substitute and Calculate the First Term
The first term in the Taylor series for \( e^x \) is simply 1. Thus, for \( e^\pi \), the first term is \( 1 \). This is because when \( x = \pi \), the term \( 1 \) remains unchanged.
03
Calculate the Second Term
The second term of the Taylor series is \( \frac{x}{1!} \). Substitute \( x = \pi \) into this term to get \( \frac{\pi}{1} = \pi \). Therefore, the second term of the expansion for \( e^\pi \) is \( \pi \).
04
Calculate the Third Term
The third term of the Taylor series is \( \frac{x^2}{2!} \). Substituting \( x = \pi \) gives \( \frac{\pi^2}{2} \). Calculate \( \pi^2 \approx 3.1416^2 \approx 9.8696 \). So, the third term is \( \frac{9.8696}{2} \approx 4.9348 \).
05
Sum the First Three Terms
Add the first three terms together to approximate \( e^\pi \). Thus, the sum is \( 1 + \pi + \frac{\pi^2}{2} \approx 1 + 3.1416 + 4.9348 \approx 9.0764 \). This is the approximation of \( e^\pi \) using the first three terms of its Taylor series.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponential Function
The exponential function is a crucial concept in mathematics, frequently represented as \( e^x \), where \( e \approx 2.71828 \) is the base of the natural logarithm and \( x \) is the exponent. This function arises naturally in various fields such as biology, physics, and finance due to its unique properties.
- Unique Properties: The exponential function is unique because it is its derivative, meaning that the rate of change of \( e^x \) is proportional to its current value. This property makes it an essential component in modeling exponential growth or decay processes, such as population growth or radioactive decay.
- Smooth and Continuous: It is defined for all real numbers and is known to be smooth and continuous, with derivatives of all orders.
- Applications: Beyond natural occurrences, it is applied in calculating compound interest and in solving differential equations.
Function Evaluation
Function evaluation is the process of finding the value of a function for specific input values. For the exercise at hand, we are evaluating the Taylor series expansion for the exponential function \( e^x \) at \( x = \pi \). Let's explore what this process involves.By substituting \( x = \pi \) into the Taylor series formula, we calculate each term separately to approximate \( e^\pi \). Each term requires careful substitution and simplification:
- First Term: This term is always \(1\) since the first term of the expansion \( e^x = 1 + x + \frac{x^2}{2!} + \ldots \) when \( x \) is any real number starts with \( 1 \).
- Second Term: Computed as \( \frac{x}{1!} \). With \( x = \pi \), this becomes \( \pi \).
- Third Term: Involves calculating \( \frac{x^2}{2!} \). For \( x = \pi \), you compute \( \pi^2 \) and divide by \( 2 \).
Mathematical Approximation
Mathematical approximation allows us to estimate complex functions through simpler expressions. In this context, we use the Taylor series to approximate \( e^\pi \) using only the first three terms.
- Why Approximate?: Most mathematical expressions don't have a simple, exact analytical form. Approximations provide practical ways of using math to solve real-world problems.
- Taylor Series: A Taylor series expands a function into an infinite sum of terms calculated from the function's derivatives at a single point. For \( e^x \) around zero, it is expressed as \( e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots \).
- Significance of Terms: Each additional term gives a more refined estimate, as evident when approximating \( e^\pi \) by using the series expansion up to the third term. Each successive term diminishes the approximation error.
