Problem 10 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to evaluate the expression. Round your result to three decimal places.\(e^{-2.7}\)

Step-by-Step Solution

Answer

Approximately 0.067

Step 1: Understanding the Expression

The given expression is \(e^{-2.7}\). The base 'e' is a constant known as Euler's Number, approximately equal to 2.71828. This is an exponential expression with base e and exponent -2.7.

Step 2: Solve using a Calculator

Use the 'exp' function on the calculator to evaluate the expression. Make sure your calculator is set to allow the use of Euler's number (e). Input the expression into your calculator to obtain the result.

Step 3: Round Off the Result

Round off the result to three decimal places. The number of decimal places is the number of digits following the decimal point. If needed, use your calculator's function for rounding to conform to these stipulated decimal places.

Key Concepts

Euler's Number

Euler's Number, denoted as \(e\), is a mathematical constant that is approximately equal to 2.71828. It is a crucial number in mathematics, particularly in calculus and complex analysis, because it forms the base of natural logarithms. One of the beauties of \(e\) is its role in describing exponential growth or decay, which appears frequently in fields such as biology, finance, and physics.
Euler's Number arises naturally in many calculations over continuous rates of growth or decay. For example, in compound interest calculations where interests compound continuously, \(e\) is often involved.

• Euler's Number is irrational, meaning it cannot be accurately expressed as a fraction.
• It is transcendental, which means it is not the root of any non-zero polynomial equation with rational coefficients.

Understanding \(e\) is fundamental for comprehending exponential functions and higher-level mathematics where continuous processes are modeled.

Exponential Functions

Exponential functions are mathematical functions in the form \(f(x) = a \cdot b^x\), where \(b\) is a positive real number, and \(a\) is a constant. When \(b = e\), the natural base, the function takes the form \(f(x) = e^x\). This particular function is fundamental in modeling natural processes that involve exponential growth or decay, such as radioactive decay, population growth, or cooling of hot objects.

• The base \(e\) gives rise to the concept of continuous growth; unlike simple interest, exponential growth compounds continuously.
• Exponential functions showcase unique properties, such as the rate of growth at any point being proportional to the total quantity.

In a mathematical expression such as \(e^{-2.7}\), the negative exponent indicates an inverse or decay process. Thus, inverting the growth into decay provides a real-world connection to decreasing populations or shrinking investments. An understanding of exponential functions equips you with powerful tools for modeling and predicting real-world phenomena.

Calculator Usage

Calculators are incredibly useful for evaluating complex mathematical expressions, such as exponential expressions involving Euler's Number. When working with expressions like \(e^{-2.7}\), follow these tips to use your calculator effectively:

• Ensure your calculator is set to `scientific` or `calculator` mode that supports \(e\) and exponential functions.
• Use the `exp` function key on your calculator, if available, to directly input and evaluate expressions such as \(e^{-2.7}\).
• If your calculator lacks an `exp` function, use pre-programmed functions or ensure \(e\) is correctly represented.
• Always double-check entries before pressing 'equal' to avoid errors.
• Don't forget to round your answer correctly, as required, after obtaining the result. For example, round the number to three decimal places for precision.

Calculator proficiency will not only save time but also improve accuracy in mathematical computations. Having these skills is essential, especially when dealing with sophisticated expressions in exams or complex analyses.