Problem 16 - Precalculus : Building Concepts and Connections

Question

Evaluate each expression to four decimal places using a calculator. $$e^{-3.2}$$

Step-by-Step Solution

Answer

After correctly typing in the expression and pressing equals, the calculator provides the result. Rounded to four decimal places, the solution to \(e^{-3.2}\) is approximately 0.0408.

Step 1: Understand the expression

The given expression \(e^{-3.2}\) signifies the mathematical constant e raised to the power of -3.2. The symbol 'e' stands for the base of the natural logarithm, which is approximately equal to 2.71828. The exponent of -3.2 signifies taking the reciprocal of e raised to 3.2.

Step 2: Use a scientific calculator

To calculate this expression, a scientific calculator is needed. First, type in 'e' (most calculators have a dedicated button for this constant). Then type in the exponent -3.2. Make sure to include the negative sign.

Step 3: Calculate and round the result

After typing in the expression, press equals to find the result. The calculator will display the result, which needs to be rounded to four decimal places as per the requirement of the problem.

Key Concepts

Understanding the Natural Exponential Function

When dealing with the concept of the natural exponential function, it's pivotal to recognize 'e' as one of the most important constants in mathematics. The constant 'e' is approximately 2.71828, and it serves as the base for the natural logarithms. This number plays a crucial role in various areas of mathematics and science, such as compound interest, population growth, and calculus.

The natural exponential function is written as \(e^x\), where 'x' is any real number. This function describes growth or decay at a continuous and consistent rate. In the expression \(e^{-3.2}\), the negative exponent indicates a decay process or a decrease. It's the equivalent of 1 divided by \(e\) raised to the positive power of 3.2, reflecting how inversely proportional relationships are represented in exponential functions.

To fully grasp this concept, visualization can be very helpful. Plotting the curve of \(y = e^x\) on a graph shows an exponential growth as 'x' increases. As 'x' becomes negative, the curve approaches zero, never touching the horizontal axis, which embodies the never-ending decrease described by negative exponents in this function.

Using a Scientific Calculator Efficiently

Scientific calculators are a treasure trove of functionality for students tackling complex mathematical expressions. They go beyond the basic operations of addition, subtraction, multiplication, and division, providing the ability to compute advanced functions like exponentials, trigonometric functions, and logarithms.

When using a scientific calculator to evaluate \(e^{-3.2}\), it's essential to familiarize yourself with its layout. Typically, these calculators have a dedicated button for the constant 'e', which might be labeled as \(e^x\) or similar. To compute an expression such as \(e^{-3.2}\), you'll need to:

• Press the \(e\) button to input the base of the natural logarithm.
• Enter the exponent value, in this case, -3.2, ensuring that the negative sign is included.
• Press the '=' or 'Enter' button to calculate the result.

Practicing with your scientific calculator on a range of problems will make these processes second nature and improve your efficiency and accuracy in various mathematical scenarios.

Rounding Decimal Places for Precision

Rounding to a specific number of decimal places is a fundamental skill, not just in math but also in real-world applications where precision is necessary. When you round a number, you're adjusting it to a specified degree of accuracy, which in this case is four decimal places.

Here's a quick guide for rounding:

• Identify the digit at the designated decimal place. For four decimal places, this is the fourth digit after the decimal point.
• Look at the digit immediately to the right (the fifth digit in this case).
• If this digit is 5 or higher, increase the fourth digit by one. If it's less than 5, leave the fourth digit as it is.
• Eliminate all digits to the right of the fourth place.

For instance, if a calculator shows a result of 0.025317 and we want to round it to four decimal places, the result would be 0.0253 because the fifth digit is 1, which is less than 5 hence the fourth digit remains unchanged.