Problem 8 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to evaluate the expression. Round your result to three decimal places.\(e^{-5}\)

Step-by-Step Solution

Answer

The value of \(e^{-5}\) rounded to three decimal places is 0.007.

Step 1: Compute the Value Using a Calculator

Enter \(e^{-5}\) into the calculator. Ensure that the calculator is set to calculate to a sufficient number of decimal places. The exponent button on the calculator might be labeled as 'EXP', 'EE', or ^. Enter -5 as the exponent.

Step 2: Interpret the Result

The calculator should give a result approximately equal to 0.006737946999085467.

Step 3: Round to Three Decimal Places

Round the result to three decimal places. Since the fourth digit after the decimal point is less than 5, round down. The rounded number is 0.007.

Key Concepts

Calculations with Euler’s Number

Euler's number, denoted as \(e\), is a fundamental constant in mathematics with a value approximately equal to 2.718. It frequently appears in calculus, mathematical modeling, and exponential growth calculations.

In operations involving Euler's number, particularly in calculus and exponential functions, calculators become indispensable tools. Calculating expressions like \(e^{-5}\) requires understanding how to input the expression in the calculator correctly:

• To enter a negative exponent, select the exponentiation function. This is often denoted by 'EXP', 'EE', or the caret symbol (^).
• For this specific calculation, you would enter \(e\), use the exponent function, and then type -5.
• The calculator automatically computes the expression: \(e^{-5}\), which is a relatively small number, as exponential functions with negative exponents decay over time.

The computed decimal value must be interpreted and usually involves many digits. This is where rounding and precision settings come into play in many applications.

Rounding Decimals

Rounding involves modifying a number to a specific degree of precision. This precision is crucial when dealing with long decimal numbers. Precision is often required in financial calculations, scientific observations, and everyday computations.

In this context, when rounding \(e^{-5}\) to three decimal places, we take several clear steps:

• Locate the digit immediately after the desired decimal place, which in this case is the fourth digit after the decimal point.
• If this "deciding" digit is less than 5, as it is in our calculation (yielding the long expression 0.006737946...), we round down.
• This process results in the rounded value being 0.007.

Through these steps, rounding simplifies the representation of complex numerical results while maintaining a useful level of accuracy.

Calculator Use in Mathematics

Modern calculators are robust tools designed to help users in executing complex computations quickly and accurately.

Proper calculator use enhances mathematical ability and comprehension, particularly when dealing with functions and constants like \(e\). In practice:

• Understanding function keys is essential. This includes "EXP" or "EE" for exponents, and recognizing others like SIN, COS for trigonometric functions.
• Compatibility and settings checks are central. Ensure your calculator is in the correct mode (degrees or radians) for the problem's context and that it rounds large decimal outputs to the chosen precision automatically if necessary.
• The calculator displays must also be readable. Double-check entries, which aids in minimizing errors, especially with negative exponents.

Knowing these utility aspects makes calculators a central facet of mathematical problem-solving in both educational and professional scenarios.