Problem 136 - Advanced Calculus

Question

Calculate \(\mathrm{e}^{4}\) within an error of \(10^{-3}\)

Step-by-Step Solution

Answer

The estimated value of \(e^{4}\) using the Taylor Series approximation to an error margin of \(10^{-3}\) is about \(30\).

Step 1: Recall Taylor series expansion formula for \(e^x\)

The Taylor series expansion of \(e^x\) around 0 is: \( e^x = \sum_{n=0}^{ \infty} \frac{x^n}{n!}\).

Step 2: Substitute \(x = 4\) into the equation

Substituting \(x = 4\) into the formula, we have \(e^4 = \sum_{n=0}^{ \infty} \frac{4^n}{n!}\).

Step 3: Start the series from \(n = 0\) and continue until the error margin is reached

Start adding the terms of the series from \(n=0\), and continue until the next term is less than \(10^{-3}\): At \(n = 0\): The term is \(\frac{4^0}{0!} = 1\) At \(n = 1\): The term is \(\frac{4^1}{1!} = 4\) and the series sum is \(1 + 4 = 5\) At \(n = 2\): The term is \(\frac{4^2}{2!} = 8\), and the series sum is \(5 + 8 = 13\) At \(n = 3\): The term is \(\frac{4^3}{3!} \approx 8.667\), and the series sum is \(13 + 8.667 = 21.667\) At \(n = 4\): The term is \(\frac{4^4}{4!} \approx 5.333\), and the series sum is \(21.667 + 5.333 = 27\) At \(n = 5\): The term is \(\frac{4^5}{5!} \approx 2.133\), and the series sum is \(27 + 2.133 = 29.133\) At \(n = 6\): The term is \(\frac{4^6}{6!} \approx 0.683\), and the series sum is \(29.133 + 0.683 = 29.816\) At \(n = 7\): The term is \(\frac{4^7}{7!} \approx 0.157\), and the series sum is \(29.816 + 0.157 = 29.973\) At \(n = 8\): The term is \(\frac{4^8}{8!} \approx 0.027\), and the series sum is \(29.973 + 0.027 = 30\) Adding the next term (\(n=9\)) into the sum would yield an addition of less than \(10^{-3}\), and therefore it should stop here as it's within our required error margin.

Step 4: Present the final result

The estimated value of \(e^{4}\) using the Taylor Series approximation to an error margin of \(10^{-3}\) is about \(30\).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. A notable example is the function \( e^x \), where \( e \) is an irrational and transcendental number approximately equal to 2.71828. This particular function is significant in various fields like natural sciences, engineering, and finance due to its unique properties. Just like how compounding interest or growth processes work continuously, the exponential function effectively models these real-world phenomena.
In calculus, \( e^x \) also holds a special place. Its derivative, \( rac{d}{dx} e^x = e^x \), is the same as itself, symbolizing a property of continuous growth without alteration in form. Therefore, understanding exponential functions is essential as they reflect constant relative growth and decay.

Error Margin Calculation

When approximating mathematical expressions, error margin calculation becomes crucial to ensure precision. It specifies an acceptable range within which the actual value is expected to lie. This is particularly important in applications where exact values are unattainable but precision remains critical.
The provided exercise aims for an approximation of \( e^4 \) and accepts an error margin of \( 10^{-3} \). This means the calculated value should differ by no more than 0.001 from the exact value. In practical terms, enhancing accuracy within this threshold allows computations to be reliable and useful, while minimizing unnecessary calculations.
Calculating the error involves stopping the series summation at a term whose next contribution would not exceed the defined threshold. Iteratively adding terms until the next term, when added, doesn't alter the result significantly is a common approach in ensuring the approximation fits the required precision.

Infinite Series

Infinite series are sequences of numbers added together indefinitely. A classic example is the Taylor series, which expresses functions as the sum of their derivatives at a single point raised to incremental powers. For the function \( e^x \), the Taylor series representation is:
\[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \]
This expansion allows us to express complex functions using simpler polynomial terms and sums. However, because it includes an infinite number of terms, direct computation isn't feasible. Instead, we compute partial sums that approximate the total.
These series are essential because they let us define and work with functions analytically that we'd otherwise only be able to assess graphically or numerically. The expansion of \( e^x \) into an infinite series provides a practical method for evaluating the function at any point with high precision.

Approximation Methods

Approximation methods, like the Taylor series, are mathematical techniques used to estimate function values when exact calculations are too complex. In our specific exercise, we used the Taylor series approximation to calculate \( e^4 \) to a defined error margin.
By iteratively adding more terms of the series, the approximation becomes closer to the actual function value. Initially, the term \( \frac{4^0}{0!} = 1 \) sets the baseline, and subsequent terms like \( \frac{4^1}{1!}, \frac{4^2}{2!} \), and others, add more refinement.
In practical scenarios, approximation allows more manageable calculations, making it possible to solve problems in areas like physics and finance where some degree of error is acceptable. The balance is to achieve adequate precision while reducing computational complexity — precisely what was applied in calculating \( e^4 \) within the error margin of \( 10^{-3} \).