Problem 30 - Calculus: One and Several Variables

Question

Use Taylor polynomials to estimate the following within 0.01. $$e^{0.8}$$

Step-by-Step Solution

Answer

By using Taylor polynomials, the estimate for \(e^{0.8}\) to within an error of 0.01 is approximately 1 + 0.8 + 0.32 = 2.12.

Step 1: Identify the Taylor series for \(e^{x}\)

Express \(e^{x}\) as its infinite Taylor series, which is \(e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots\). Here, 'x' is the number to which 'e' is raised.

Step 2: Substitute 'x' with the given value

Substitute \(x = 0.8\) into the Taylor series for \(e^{x}\) to get \(e^{0.8} = 1 + 0.8 + \frac{0.8^{2}}{2!} + \frac{0.8^{3}}{3!} + \cdots\).

Step 3: Calculate enough terms to achieve desired accuracy

Given that the error should be less than 0.01, calculate the series term by term until the absolute difference between two successive terms is less than 0.01.\nThe first few terms of the series are approximately: 1 (the 0th term), 0.8 (the 1st term), 0.32 (for 2nd term), and 0.044 (for the 3rd term). The absolute difference between the 2nd and 3rd terms is less than 0.01. Hence, it is enough to consider the terms up to the 3rd term.

Key Concepts

Taylor Series

The Taylor series is a powerful mathematical tool that allows us to approximate functions using polynomials. This approximation is very useful, especially when evaluating functions that are difficult to compute directly. For any function that is infinitely differentiable at a certain point, known as the center, we can express it as an infinite sum of terms. Each term is derived from the function's derivatives at this center.
For example, the Taylor series for the exponential function, \(e^x\), centered at 0, is written as:

• \( e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \)

This series uses factorials in the denominators, which make each term progressively smaller. This characteristic aids in converging towards the actual function value as more terms are included in the series.

Exponential Function

The exponential function, \(e^x\), is one of the most fundamental mathematical functions encountered in calculus. It describes situations of growth or decay that are exponential in nature such as population growth or radioactive decay.
A unique property of \(e^x\) is that its rate of change at any point is equal to its value at that point. This property makes its Taylor series expansion particularly simple and elegant:

• The function itself and all its derivatives are \(e^x\)
• It can be represented as \(e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\)

This series is what allows us to approximate \(e^{0.8}\) within a given accuracy, exploiting the structure of the exponential function and the nature of its series expansion.

Estimation Error

In mathematical approximations, ensuring the accuracy of your result is crucial. Estimation error refers to the difference between the approximated value obtained by using a limited number of terms in a series and the actual value of the function.
For the above problem, the goal was to estimate \(e^{0.8}\) with an error of less than 0.01. With Taylor polynomials, each added term reduces the error of the approximation.

• The first term is merely 1, the value of the function at \(x=0\)
• Additional terms such as \(0.8\), \(\frac{0.8^2}{2!}\), and \(\frac{0.8^3}{3!}\) were added progressively until the change became negligible

Thus, stopping after adding appropriate terms ensures the estimated value is close enough to the actual value, meeting the desired error tolerance of 0.01.