Problem 14 - Intermediate Algebra

Question

Complete the table of values. Use a calculator when necessary and round to the nearest hundredth. $$ \begin{array}{|c|c|c|c|c|c|} \hline \boldsymbol{x} & -2 & -1 & 0 & 1 & 2 \\ \hline \boldsymbol{e}^{x} & & & & & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The completed table has \(e^x\) values: 0.14, 0.37, 1.00, 2.72, and 7.39.

Step 1: Understanding the Exponential Function

The given exercise involves evaluating the exponential function \(e^x\), where \(e\) is a mathematical constant approximately equal to 2.71828. For each given value of \(x\), we'll substitute it into the function to compute \(e^x\).

Step 2: Calculate \( \boldsymbol{e}^{-2} \)

Use a calculator to compute \(e^{-2}\). This is done by taking the reciprocal of \(e^2\). Calculate and round the result to the nearest hundredth: \[e^{-2} \approx 0.1353 \Rightarrow 0.14\]

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

Calculate \(e^{-1}\), which is the reciprocal of \(e\). Use a calculator and round to the nearest hundredth: \[e^{-1} \approx 0.3679 \Rightarrow 0.37\]

Step 4: Calculate \( \boldsymbol{e}^0 \)

Recall that any number raised to the power of 0 is 1. Thus, \[e^0 = 1\]

Step 5: Calculate \( \boldsymbol{e}^1 \)

Since raising \(e\) to the power of 1 simply results in \(e\) itself: \[e^1 = e \approx 2.7183 \Rightarrow 2.72\]

Step 6: Calculate \( \boldsymbol{e}^2 \)

Compute \(e^2\) using a calculator and round to the nearest hundredth: \[e^2 \approx 7.3891 \Rightarrow 7.39\]

Step 7: Complete the Table

Fill in the exponential values obtained: \[\begin{array}{|c|c|c|c|c|c|}\hline \boldsymbol{x} & -2 & -1 & 0 & 1 & 2 \\hline \boldsymbol{e}^{x} & 0.14 & 0.37 & 1.00 & 2.72 & 7.39 \\hline\end{array}\]

Key Concepts

Evaluating Exponential Expressions

Exponential expressions involve raising a base, like the mathematical constant \(e\), to a specific power. In this exercise, we evaluated \(e^x\) for various values of \(x\), including -2, -1, 0, 1, and 2. When evaluating exponential expressions, it’s useful to understand a few key rules:

• The base \(e\) is a transcendental number, approximately equal to 2.71828, widely used in calculus and natural logarithms.
• For negative exponents like \(e^{-2}\), compute by finding the reciprocal of \(e^2\). This involves dividing 1 by the result of \(e\) raised to the power of 2.
• When \(x = 0\), any base raised to the power of 0 is equal to 1. So \(e^0 = 1\).

These basic principles of exponential expressions help you evaluate and understand different power levels of constants like \(e\). When using a calculator, always ensure to enter the expression correctly to avoid errors.

Constants in Mathematics

In the realm of mathematics, constants are pivotal in simplifying and understanding complex equations. The constant \(e\) is one of the most important numbers after \(\pi\) and represents the base of the natural logarithm.

• It's approximately 2.71828, becoming more precise as decimal places increase. This constant often appears in compound interest calculations, differential equations, and growth patterns.
• Each time you encounter an \(e^x\) expression, you're dealing with the growth pattern or decay of the base \(e\).
• Understanding \(e\) is crucial for fields beyond mathematics, such as physics and engineering, where exponential growth or decay is a common concept.

These insights into constants in mathematics can help solidify understanding of their role and application across various problems.

Rounding Numbers

Rounding numbers is a key skill when working with mathematical calculations, especially when working with irrational numbers like \(e\). In this exercise, rounding was necessary for expressing numbers in a more manageable form. Here’s how you round numbers effectively:

• Identify which decimal place you need to round to; in this case, it was rounding to the nearest hundredth (two decimal places).
• Look at the digit immediately following your rounding place. If it is 5 or greater, round up; if it is less than 5, round down.
• When \(e^{-2} \approx 0.1353\), the digit in the thousandth place (3) does not require increasing the rounding digit, so it becomes 0.14.
• Accuracy in rounding ensures that calculations remain precise and results are understandable, especially when compiling datasets.

Practicing rounding enhances your ability to present numbers clearly, making them easier to work with in various contexts.