Problem 90 - A Graphical Approach to Precalculus with Limits

Question

Estimating Powers of \(e\) The series $$ e^{a} \approx 1+a+\frac{a^{2}}{2 !}+\frac{a^{3}}{3 !}+\dots+\frac{a^{n}}{n !} $$ where $$ n !=1 \cdot 2 \cdot 3 \cdot 4 \cdot \cdots \cdot n $$ can be used to estimate the value of \(e^{a}\) for any real number a. Use the first eight terms of this series to approximate each expression. Compare this estimate with the actual value. Give values to six decimal places. (a) \(e\) (b) \(e^{-1}\) (c) \(\sqrt{e}\)

Step-by-Step Solution

Answer

(a) 2.718253; (b) 0.367881; (c) 1.64871.

Step 1: Calculate for \( e \)

To approximate \( e \), set \( a = 1 \) and use the first eight terms of the series. Calculate:\[ e \approx 1 + 1 + \frac{1^{2}}{2!} + \frac{1^{3}}{3!} + \frac{1^{4}}{4!} + \frac{1^{5}}{5!} + \frac{1^{6}}{6!} + \frac{1^{7}}{7!} = 1 + 1 + 0.5 + 0.1667 + 0.0417 + 0.0083 + 0.0014 + 0.0002 = 2.718253.\]

Step 2: Compare with Actual Value for \( e \)

The actual value of \( e \) is approximately 2.718282. Comparing the two, the approximation 2.718253 is very close.

Step 3: Calculate for \( e^{-1} \)

Set \( a = -1 \) and use the first eight terms. Calculate:\[ e^{-1} \approx 1 - 1 + \frac{(-1)^{2}}{2!} - \frac{(-1)^{3}}{3!} + \frac{(-1)^{4}}{4!} - \frac{(-1)^{5}}{5!} + \frac{(-1)^{6}}{6!} - \frac{(-1)^{7}}{7!} = 1 - 1 + 0.5 - 0.1667 + 0.0417 - 0.0083 + 0.0014 - 0.0002 = 0.367881.\]

Step 4: Compare with Actual Value for \( e^{-1} \)

The actual value of \( e^{-1} \) is approximately 0.367879. The approximation 0.367881 is very close.

Step 5: Calculate for \( \sqrt{e} \)

Set \( a = 0.5 \) and use the first eight terms. Calculate:\[ \sqrt{e} \approx 1 + 0.5 + \frac{0.5^{2}}{2!} + \frac{0.5^{3}}{3!} + \frac{0.5^{4}}{4!} + \frac{0.5^{5}}{5!} + \frac{0.5^{6}}{6!} + \frac{0.5^{7}}{7!} = 1 + 0.5 + 0.125 + 0.0208 + 0.0026 + 0.0003 + 0.00002 + 0.000002 = 1.64871.\]

Step 6: Compare with Actual Value for \( \sqrt{e} \)

The actual value of \( \sqrt{e} \) is approximately 1.648721. The approximation 1.64871 is very close.

Key Concepts

Exponential Function

The exponential function, denoted as \( e^x \), is a fundamental mathematical function with essential properties. It is characterized by a constant base \( e \), known as Euler's number, which is approximately 2.71828. This function is incredibly significant in various fields, including mathematics, science, and engineering.

Some key characteristics of exponential functions include:

• Rapid Growth: The exponential function grows quickly; for positive \( x \), the value of \( e^x \) rises steeply.
• Inverse Function Relationship: The natural logarithm function, \( \ln(x) \), is the inverse of the exponential function.
• Continuity and Differentiability: \( e^x \) is continuous and differentiable across its entire domain.

These properties make the exponential function vital in modeling real-world phenomena like population growth, radioactive decay, and interest calculations in finance.

Approximation Methods

Approximating functions is a central theme in calculus, and the Taylor Series is one of the powerful methods used for this purpose. It allows us to approximate complex functions with polynomials, making calculations more manageable.

The Taylor Series for a function around a point \( a \) is given as:\[f(x) \approx f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n\]For the exponential function \( e^x \), substituting \( f(x) = e^x \) and \( a = 0 \) results in the series:\[e^x \approx 1 + x + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!}\]This series is exceptionally effective, as highlighted in exercises approximating \( e \), \( e^{-1} \), and \( \sqrt{e} \). The Taylor Series provides a robust alternative to directly calculating difficult function values, allowing us to work with simpler polynomials instead.

Calculus Concepts

Taylor Series and other approximation methods are deeply rooted in calculus concepts. Understanding calculus is crucial as it deals with changes and motion, which are applicable to countless areas such as physics, engineering, and economics.

Key calculus concepts involved in Taylor Series include:

• Derivatives: Indicate the rate at which a function changes at a given point. Taylor Series uses derivatives for polynomial approximations.
• Factorials: Represented as \( n! \), these are products of an integer and all positive integers below it. Factorials help weigh the terms in a Taylor Series to reflect their contribution appropriately.
• Convergence: Describes a condition where an infinite series approaches a finite limit. In the context of Taylor Series, it means our polynomial approximation gets closer to the function's actual value.

These concepts form the backbone of Taylor Series approximations and underscore their wide applicability and significance in solving real-world problems.