Hey, everyone. When working through problems with different exponential expressions and exponential functions, you may come across one that doesn't have a base of something like 2 or 1 half or ten or any other number but actually has a base of e, this lowercase e. And the first time you see that, you might be wondering why there is another letter in that function when we already have x right there and how we are going to work with this function with 2 different variables. But you don't have to worry about any of that because here I'm going to show you how e is literally just a number, and we can treat it just like we would any other exponential function and evaluate and graph it using all of the tools that we already know. So let's go ahead and get started. Now, like I said, e is not a variable at all but simply a number. And another similar number that you might be a bit more familiar with is pi. We know that pi is 3.1415 and so on, this long decimal that we don't write out. We just write pi. E is really similar. It's this long decimal, 2.71828 and so on, but we simply write it as e. Now because it's a number, we can treat it just like we would any other exponential function and do things like evaluate it for different values of x. So let's take a look at our function here,
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The Number e: Study with Video Lessons, Practice Problems & Examples
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Exponential functions with base e (approximately 2.71828) can be evaluated and graphed similarly to other bases. For example, to evaluate f(x) = ex, 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
Video transcript
Graph the given function.
g(x)=ex+3−1
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 ex 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.
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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) = ex, 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.
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How do you graph an exponential function with base e?
To graph an exponential function with base e, such as f(x) = ex, 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 ex 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.
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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.
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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 ex. Using e as the base ensures that the derivative of ex is simply ex, 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.
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