Question

Use a graphing utility to graph the function over the interval. Find the average value of the function over the interval. Then find all \(x\) -values in the interval for which the function is equal to its average value. $$ f(x)=e^{x / 4} \quad[0,4] $$

Short answer

The average value of the function over the interval [0,4] is \(e - 1\). The only x-value in the interval at which the function equals its average value is at \(x = 4\).

Step-by-step solution

Graph the Function Using a Graphing Utility

Most electronic graphing utilities, like a TI-84 calculator or an online graphing calculator tool, can be used to plot this function within the interval [0,4]. Just input the function \(f(x) = e^{x / 4}\) and set the domain to [0,4].

Compute the Average Value

Compute the average value of the function over the interval by integrating the function on the interval and dividing by the length of the interval. The average value \(A\) is given by the formula \(A=\frac{1}{b-a}\int_{a}^{b}f(x)dx\). Applying this formula with \(f(x) = e^{x / 4}\), \(a=0\), and \(b=4\), the average value of the function over the interval is: \[A=\frac{1}{4}\int_{0}^{4}e^{x / 4}dx\] This integral can be solved using the method of substitution.

Calculate the Integral using Substitution

Let \(u = x / 4\). Then \(dx = 4du\). Substitute into the integral and solve to get: \[= \frac{1}{4} \int e^{u} \cdot 4du = \int e^{u} du = e^{u}\] Now substitute \(u = x / 4\) back in to get: \[= e^{x / 4}\] Evaluate this from 0 to 4 to get: \(A = e - 1\)

Find x-values where f(x) equals the average value

Finally, find all x-values in the interval [0,4] for which the function equals its average value. This involves solving the equation \(f(x) = A\): \[e^{x / 4} = e - 1\] Solving for x gives: \(x = 4 \ln(e - 1)\) Within the interval [0,4], the only solution is \(x = 4\), since \(\ln(e-1)\) approximately equals 1.

Key concepts

Graphing Functions

Understanding how to graph functions is a fundamental skill in mathematics that can give a visual insight into the behavior of functions. In our example, the function to graph and analyze is the exponential function, f(x) = e^{x / 4}, defined over the interval [0,4].

When graphing an exponential function, it's important to notice how the function grows. Since the exponent x/4 gets larger as x increases, the function rises more steeply as we move to the right along the x-axis. In this graph, you'll see a smooth, increasing curve that starts from f(0) = 1 (since any number to the power of zero equals one) and grows to f(4) = e.

With graphing utilities like a graphing calculator or software, you can visualize this process easily. Just make sure to input the function correctly and set the proper domain, which in this case, is between 0 and 4. Knowing how to graph functions helps not only in understanding their characteristics but also in solving related mathematical problems such as finding average values or specific points on the graph where the function reaches certain values.

Definite Integrals

The concept of a definite integral is crucial in calculating areas under curves and finding average values of functions over a specific interval. In our case, we're concerned with the average value of our exponential function over the interval [0,4].

The definite integral is represented by the symbol ∫, and it calculates the accumulation of quantities, which can be areas under a curve. For the average value, we use the formula A = (1/(b-a)) ∫_{a}^{b} f(x)dx, where [a,b] is the interval and f(x) is the function. The outcome is the total accumulated quantity divided by the length of the interval, giving us the average value.

By utilizing this powerful tool, you'll find not just areas but also understand the general behavior of functions over intervals, which is both interesting and extremely useful in various fields of study such as physics, engineering, and economics.

Exponential Functions

Exponential functions, such as our function f(x) = e^{x / 4}, are fascinating and appear in various real-world applications, from compound interest in finance to population growth models in biology.

An exponential function typically takes the form f(x) = a^{x}, where a is a positive constant, and x is the exponent. They are characterized by their rapid growth or decay, depending on whether the base is greater than one or between zero and one, respectively.

In terms of graphing, exponential functions have distinctive curved graphs that either consistently rise or fall, and they never become zero, approaching the x-axis asymptotically as x approaches negative infinity. Understanding the nature of exponential growth is essential for predicting long-term trends or behaviors in various scenarios.