Problem 38 - Precalculus with Limits

Question

Evaluating a Natural Exponential Function In Exercises \(35-38\) , evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. $$functions$$ $$f(x)=250 e^{0.05 x}$$ $$values$$ $$x=20$$

Step-by-Step Solution

Answer

To get the result, substitute \(x = 20\) into the function, simplify, and then round the result to three decimal places.

Step 1: Understand the Function

The given function is \(f(x) = 250e^{0.05x}\). In this function, \(e\) is the base of the natural logarithm, approximately equal to \(2.71828\), and \(x\) is the exponent. The number \(250\) is a coefficient.

Step 2: Substitute the value of x into the Function

The function should be evaluated at \(x = 20\). Therefore, substitute \(20\) for \(x\) in the function: \(f(20) = 250e^{0.05*20}\).

Step 3: Evaluate the Exponent

The exponent is \(0.05*20 = 1\). Therefore, the function is reduced to: \(f(20) = 250e^{1}\).

Step 4: Evaluate the Product

Finally, multiply \(250\) by \(e\) to get the final result.

Step 5: Round the Result

After calculating the product, round the result to three decimal places as per the exercise's instruction.

Key Concepts

Understanding Exponential Functions

Exponential functions are a type of mathematical expression where a constant base is raised to a variable exponent. These functions have the form:

• The base is usually a constant number, often greater than 1.
• The exponent is typically a variable, which allows the output of the function to change.

One notable example is the natural exponential function where the base is the mathematical constant, Euler's number, denoted as \(e\). Euler's number, \(e\), is approximately 2.71828 and is used extensively in calculus and complex analysis due to its unique mathematical properties.
In our function, \(f(x) = 250e^{0.05x}\), \(e\) is used as the base. This is what makes the function a natural exponential function. The constant \(250\) is a coefficient that stretches or compresses the graph of the exponential function, depending on its value. This function increases as \(x\) increases, which is typical behavior for exponential functions when the exponent is positive.

Exploring the Natural Logarithm

The natural logarithm, denoted as \(\ln\), is the logarithm to the base \(e\). Essentially, if \(e^y = x\), then \(\ln(x) = y\).

• It essentially helps to "undo" the exponential function, acting as its inverse.
• For example, if you have \(f(x) = e^x\), the natural logarithm can help you find the value of \(x\) given \(f(x)\).

This relationship plays an essential role in solving equations involving exponential growth or decay.
In our function \(f(x) = 250e^{0.05x}\), if we want to solve for \(x\) when \(f(x)\) is known, the natural logarithm can be used to re-arrange the equation and find the solution for \(x\). It allows us to handle the exponent and linearize the exponential growth.
Understanding both the natural exponential function and natural logarithm is crucial for evaluating and solving real-world problems involving exponential change.

Techniques for Evaluating Functions

Evaluating functions involves substituting a given value into an expression and performing the necessary calculations to find the result. Here’s how you can effectively evaluate functions like \(f(x) = 250e^{0.05x}\):

• First, substitute the specified value of \(x\) into the function. In our example, substitute \(x = 20\) into the equation, changing it to \(f(20) = 250e^{0.05 \times 20}\).
• Next, simplify the exponent by performing any arithmetic: \(0.05 \times 20 = 1\).
• Then evaluate the expression by replacing any computed values: \(f(20) = 250e^{1}\).
• Use a calculator to calculate the value of \(e\). The value of \(e^1\) is simply \(e\), which is approximately 2.71828.
• Multiply this result by the coefficient \(250\) to get the final output.
• Finally, round the outcome to the required decimal places, if necessary.

.These steps ensure you handle exponential functions efficiently and accurately, be it for homework or real-world applications.