The Number e: Videos & Practice Problems

Exponential functions with base e (approximately 2.71828) can be evaluated and graphed similarly to other exponential expressions. For example, to evaluate $e^2$, use a calculator with the second ln function. The significance of e arises from its connection to compounding interest and its applications in population growth and radioactive decay. Understanding e enhances comprehension of various mathematical concepts, allowing for effective problem-solving in multivariable polynomials and exponential expressions.

The Number e

When dealing with exponential functions, you may encounter the base e, which is approximately equal to 2.71828. This number is not a variable but a constant, similar to how π represents a specific value. Understanding that e is just a number allows you to evaluate and graph functions involving it, just like any other exponential function.

For example, consider the function f(x) = e^x. To evaluate this function for a specific value of x, such as 2, you would calculate f(2) = e^2. Using a scientific calculator, you can find this value by pressing the second button followed by ln (natural logarithm), then entering the exponent. Rounding to the nearest hundredths place, f(2) yields approximately 7.39.

Similarly, for x = -3, you would compute f(-3) = e^{-3}. By applying the rules of exponents, this can also be expressed as 1/e^3. Again, using the calculator, you would find that f(-3) is approximately 0.05 when rounded to the nearest hundredths place.

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 these two functions. 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 in various real-world applications, such as compound interest calculations. The formula for continuously compounded interest leads to the derivation of e as the limit of (1 + 1/n)^n as n approaches infinity. This concept extends beyond finance, appearing in fields like population growth modeling and radioactive decay.

In summary, e is a fundamental constant in mathematics that simplifies the study of exponential growth and decay. By treating it like any other number, you can effectively evaluate and graph functions involving e, enhancing your understanding of exponential behavior in various contexts.

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 is a mathematical constant approximately equal to 2.71828. It is crucial in mathematics due to its unique properties in exponential growth and decay processes. Originating from the concept of continuously compounding interest, e is the limit of (1 + 1/n)ⁿ as n approaches infinity. This constant appears in various mathematical contexts, including finance, population growth, and radioactive decay. Its importance lies in its ability to model natural phenomena that exhibit exponential behavior. Despite its complex decimal representation, e can be treated like any other numerical base in exponential functions, making it a versatile tool in mathematical analysis.

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

To evaluate exponential functions with base e, such as f(x) = e^x, using a calculator, you typically use the 'second ln' function. This function allows you to input the power of e directly. For example, to calculate e², you would press 'second ln' followed by '2' on your calculator. This operation will provide the value of e raised to the power of 2, which is approximately 7.39 when rounded to the nearest hundredth. Similarly, for negative exponents, such as e^-3, you would input 'second ln' followed by '-3', resulting in approximately 0.05.

How does the graph of f(x) = e^x compare to other exponential functions?

The graph of f(x) = e^x has a similar shape to other exponential functions, such as 2^x or 3^x. It is characterized by a rapid increase as x becomes positive and approaches infinity, and a gradual decrease as x becomes negative. The base e, approximately 2.718, places the graph of e^x between the graphs of 2^x and 3^x. This positioning is due to e being a number between 2 and 3. Like other exponential functions, transformations can be applied to graph more complex functions involving e, using techniques such as shifting, stretching, or reflecting.

Where does the number e come from and what are its applications?

The number e originates from the concept of continuously compounding interest, represented by the limit of (1 + 1/n)ⁿ as n approaches infinity. This mathematical constant is used in various applications beyond finance, including modeling population growth, radioactive decay, and other scenarios involving exponential growth or decay. In these contexts, e helps describe natural phenomena that change at rates proportional to their current value. Its applications extend to fields such as biology, physics, and economics, where understanding exponential processes is essential. Despite its complex nature, e is a fundamental tool in mathematical analysis and problem-solving.

How can e be used to predict population growth?

The number e is used in population growth models to describe exponential growth processes. In these models, the population size at any given time can be represented by the equation P(t) = P₀e^rt, where P₀ is the initial population size, r is the growth rate, and t is time. This equation shows how the population grows continuously at a rate proportional to its current size. The constant e allows for accurate predictions of future population sizes based on current data and growth rates. This application is crucial in fields such as ecology and demography, where understanding population dynamics is essential.