Problem 7 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to evaluate the expression. Round your result to three decimal places.\(e^{4}\)

Step-by-Step Solution

Answer

54.598

Step 1: Access the 'e' function on the calculator

On most scientific calculators, you can access the value of \(e\) through a specific function which is usually found by pressing the '2nd' or 'shift' key followed by a button labeled 'e' or 'exp'.

Step 2: Raise 'e' to the power of 4

After accessing the 'e' function, you will input the exponent 4, often by pressing a key marked '^' or 'EXP' and then entering 4.

Step 3: Calculate and round the result

After pressing the '=' button or the 'enter' button, the calculator will display the result. Round this number to three decimal places. If the digit in the fourth decimal place is 5 or above, round up the third decimal place by adding 1. If it's less than 5, leave the third decimal place as is.

Key Concepts

Using Scientific Calculators

Scientific calculators are invaluable tools for students and professionals alike, especially when it comes to tackling mathematical problems involving exponential expressions. When it’s time to calculate something like the exponential function e to a power, a scientific calculator comes to the rescue by providing a straightforward way to perform the computation.

For example, when evaluating an expression like e^{4}, you can locate the 'e' or 'exp' button, which is often behind a '2nd' or 'shift' function on the calculator. This special button allows you to access the mathematical constant e easily. Next, you would input your exponent, in this case, 4, often by pressing a button like '^' or entering the sequence indicated for exponentiation on your specific calculator model. Finally, hit the '=' or 'enter' key to see the result. It's essential to get comfortable with this process and your specific calculator's layout to make solving exponential expressions an effortless part of your routine.

Exponential Function e

The exponential function e is a fundamental part of mathematics, often referred to as Euler's number. It’s approximately equal to 2.71828 but, like π (pi), it's an irrational number that goes on indefinitely without repeating.

The function e^{x} has unique properties, such as being its own derivative, which makes it extremely important in calculus. It often appears in natural growth and decay problems, finance to calculate compound interest, and even in complex numbers. Getting familiar with e and its properties enhances your understanding of continuous growth processes and can greatly benefit you in advanced math and applied sciences. When you see e to the power of any number, remember that you're dealing with exponential growth or decay, which scales rapidly.

Rounding Numbers

Rounding numbers is a necessary skill to master in mathematics, science, and even in daily life when precise values are not required or not available. It simplifies numbers while still keeping them close to the original value.

To round a number to a specific number of decimal places, like three decimal places, look at the fourth decimal place. If it’s 5 or greater, increase the third decimal place by one. If it’s less than 5, leave the number as it is. This process is straightforward but crucial to ensure that your calculations remain as accurate as needed for the context but are also manageable for interpretation and further calculations. For instance, rounding e^{4} to three decimal places helps you focus on the significant figures without the distraction of an excessively long decimal tail.