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. $$ f(x)=\frac{1}{(x-3)^{2}} \quad[0,2] $$

Short answer

The graph of the function \( f(x)=\frac{1}{(x-3)^{2}} \) over the interval \([0,2]\) shows its behavior. The average value of the function over the interval can be calculated by integrating \(f(x)\) from 0 to 2 and then dividing by the length of the interval. The \(x\)-values for which the function equals its average value can be found by solving the equation \(f(x) = A\) where \(A\) is the average value obtained from integration.

Step-by-step solution

Graph the Function

Start by graphing the function \( f(x)=\frac{1}{(x-3)^{2}} \) over the interval \([0,2]\) using a graphing utility. The graph will help visualize the function's behavior over the given interval.

Find the Average Value of Function

To find the average value of \(f(x)\) over \([0,2]\), integrate \(f\) from 0 to 2 and then divide by the length of the interval. The average value \(A\) on \([0,2]\) is given by \(A=\frac{1}{2-0}\int_{0}^{2}\frac{1}{(x-3)^{2}}\,dx\). Use the integral \(\int \frac{1}{u^{2}}\,du=-\frac{1}{u}+C\) to compute the integral and calculate the average value.

Find x-values where Function Equals Average Value

To find the \(x\)-values at which the function equals its average value, set \(f(x)\) equal to the average value and solve for \(x\). So, solve the equation \(f(x) = A\), where \(A\) is the average value obtained in step 2. After that, check if the solution lies in the interval \([0,2]\).

Key concepts

Average Value of a Function

Understanding the average value of a function is akin to finding the 'middle ground' of its output over a certain interval. In essence, it offers a single representative value for that stretch of the function's domain. To capture the average, we use integration, which helps us sum up infinitely many values the function takes on. For a continuous function like \( f(x) \), defined on the interval \( [a, b] \), the average value \( A \) is computed using the formula:
\[ A = \frac{1}{b-a} \int_{a}^{b} f(x) dx \].
This formula essentially tells us to integrate the function between two points and then normalize the result by the interval's length, giving us the 'mean height' of the curve over \( [a, b] \).

To apply this to our example \( f(x) = \frac{1}{(x-3)^2} \) over \( [0, 2] \), the average value is the definite integral of \( f \) from 0 to 2, divided by the interval's length, which in this case is 2 units.

Calculating it provides a single value that, when graphed, would create a horizontal line cutting across the curve, symbolizing the average output of the function across our interval of interest.

Definite Integral

In the realm of calculus, the definite integral is a fundamental concept utilized to calculate the accumulated value, such as area under a curve, between two points on a graph. This is more than just finding an area in a geometric sense; it comprehensively accumulates the function's value across the span of the interval.

To find a definite integral, you would typically need to:

• Identify the interval over which you are integrating.
• Find the antiderivative (indefinite integral) of the function.
• Apply the Fundamental Theorem of Calculus, which tells us that the definite integral from \( a \) to \( b \) is the antiderivative evaluated at \( b \) minus the antiderivative evaluated at \( a \).

In the context of our example with \( f(x) = \frac{1}{(x-3)^2} \), to find the definite integral over \( [0, 2] \), you would compute the antiderivative \( -\frac{1}{x-3} \), then evaluate this expression at 2 and subtract its value when evaluated at 0. This process nets the total 'area' ─ though it can represent much more ─ under the function from \( x = 0 \) to \( x = 2 \).

Graphing Calculator Utility

The graphing calculator remains an indispensable tool for visualizing mathematical concepts and functions. Its utility is especially pronounced when dealing with complex functions or wanting a graphical representation to complement algebraic solutions.

Using a graphing calculator, you can quickly plot a function to observe its behavior within a specified interval. Features commonly found on these calculators include:

• Plotting graphs of functions.
• Finding intersections or solutions graphically.
• Calculating definite integrals over intervals.
• Zooming and tracing to closely examine specific portions of a graph.

For the function \( f(x) = \frac{1}{(x-3)^2} \), a graphing calculator not only helps you visualize the curve over the interval \( [0, 2] \) but also aids in identifying the particular points where the function reaches its average value, as previously computed. This visual aid is particularly helpful for confirming solutions you've found algebraically and understanding the function's overall behavior.