The Number e: Videos & Practice Problems

Exponential functions with base e (approximately 2.71828) can be evaluated and graphed similarly to other bases. For example, to evaluate f(x) = e^x, use a calculator with the second ln function. The significance of e arises from its role in compounding interest and applications in population growth and radioactive decay. Understanding e enhances comprehension of various mathematical and real-world phenomena, making it a crucial constant in exponential functions.

The Number e

When exploring exponential functions, you may encounter the base e, which is approximately equal to 2.71828. Unlike variables, e is a constant, similar to the well-known number π (approximately 3.1415). This means you can evaluate and graph functions involving e just like any other exponential function.

For example, consider the function f(x) = e^x. To evaluate this function for specific values of x, such as x = 2 or x = -3, you can use a scientific calculator. For f(2), you would compute e^2 by using the second ln function on your calculator, which gives you approximately 7.39 when rounded to the nearest hundredth. Similarly, for f(-3), you would calculate e^{-3}, which can also be expressed as 1/e^3. This results in approximately 0.05 when rounded.

The graph of f(x) = e^x shares the same general shape as other exponential functions, such as 2^x and 3^x. Since e is between 2 and 3, the graph of e^x will lie between the graphs of 2^x and 3^x. For more complex functions involving e, you can apply transformations similar to those used for other bases.

Understanding the significance of e is crucial, especially since it arises from the concept of continuous compounding in finance. The formula for continuous compounding is given by:

\[A = Pe^{rt}\]

where A is the amount of money accumulated after time t, P is the principal amount, r is the annual interest rate, and t is the time in years. As the number of compounding periods approaches infinity, the limit of the expression leads to the value of e.

Beyond finance, e is also essential in fields such as biology for modeling population growth and in physics for processes like radioactive decay. Thus, while e may seem like just another number, it plays a vital role in various mathematical applications and real-world phenomena.

Graph the given function.
$g\left(x\right)=e^{x+3}-1$

• A.
• B.
• C.
• D.

Answer: (A)

Here’s what students ask on this topic:

What is the number e and why is it important in mathematics?

The number e, approximately 2.71828, is a mathematical constant that is the base of natural logarithms. It is important because it arises naturally in various contexts, such as in the calculation of compound interest, population growth models, and radioactive decay. The number e is unique in that the function e^x has the same rate of growth as its value, making it crucial in calculus and differential equations. Understanding e helps in comprehending exponential growth and decay processes in both mathematics and real-world applications.

How do you evaluate an exponential function with base e using a calculator?

To evaluate an exponential function with base e, such as f(x) = e^x, using a calculator, follow these steps: First, locate the 'e' function on your calculator, which is often accessed by pressing the 'second' or 'shift' key followed by the 'ln' key. Then, enter the exponent value. For example, to evaluate e², press 'second', 'ln', and then '2'. The calculator will display the result, which is approximately 7.39 when rounded to the nearest hundredth. This method allows you to compute values of e raised to any power efficiently.

How do you graph an exponential function with base e?

To graph an exponential function with base e, such as f(x) = e^x, follow these steps: First, create a table of values by choosing several x-values and calculating the corresponding y-values using the function. Plot these points on a coordinate plane. The graph of e^x will have a characteristic shape, starting close to the x-axis for negative x-values and increasing rapidly for positive x-values. The graph will pass through the point (0,1) since e⁰ = 1. The curve will be asymptotic to the x-axis as x approaches negative infinity and will increase without bound as x approaches positive infinity.

What are some real-world applications of the number e?

The number e has numerous real-world applications. It is used in calculating compound interest, where interest is compounded continuously. In biology, e is used in models of population growth, where populations grow exponentially. In physics, e appears in the equations describing radioactive decay and half-lives. Additionally, e is used in various fields of engineering, economics, and statistics, such as in the analysis of growth rates and in the calculation of probabilities in certain distributions. Its natural occurrence in these diverse areas highlights its significance in both theoretical and applied mathematics.

Why is the number e used as the base for natural logarithms?

The number e is used as the base for natural logarithms because it simplifies many mathematical expressions and calculations. The natural logarithm, denoted as ln(x), is the inverse function of the exponential function e^x. Using e as the base ensures that the derivative of e^x is simply e^x, and the derivative of ln(x) is 1/x. This property makes calculus operations involving exponential and logarithmic functions more straightforward. Additionally, e naturally arises in various growth and decay processes, making it a convenient and practical choice for the base of natural logarithms.