Problem 85 - Calculus Single Variable

Question

In each of Exercises \(85-88\), a function \(f\) is given. Let \(A_{f}(c)\) denote the average value of \(f\) over the interval \([c-1 / 4, c+1 / 4]\) Plot \(y=f(x)\) and \(y=A_{f}(x)\) for \(-1 \leq x \leq 1 .\) The resulting plot will illustrate the gain in smoothness that results from averaging. $$ f(x)=|x| $$

Step-by-Step Solution

Answer

Calculate the average of \( f(x) = |x| \) on intervals around \( c \), then plot both \( f(x) \) and \( A_f(x) \) to observe smoothing.

Step 1: Understanding the Function

We are given the function \( f(x) = |x| \), which represents the absolute value of \( x \). This function outputs the non-negative value of \( x \).

Step 2: Define the Interval for Averaging

We need to find the average value of \( f \) over the interval \([c-\frac{1}{4}, c+\frac{1}{4}]\). This interval has a length of \( \frac{1}{2} \).

Step 3: Set up the Average Value Formula

The average value of a function \( f \) over an interval \([a, b]\) is given by \[ A_f(c) = \frac{1}{b-a} \int_a^b f(x) \, dx \]. Here \( a = c - \frac{1}{4} \) and \( b = c + \frac{1}{4} \). Thus, \[ A_f(c) = \frac{1}{\frac{1}{2}} \int_{c-\frac{1}{4}}^{c+\frac{1}{4}} |x| \, dx \].

Step 4: Compute the Definite Integral

For \( |x| \), the integral will differ depending on whether \( x \) is positive or negative. We evaluate separately for \( c - \frac{1}{4} < 0 < c + \frac{1}{4} \) (if \( c \approx 0 \)), using \( |x| = -x \) when \( x < 0 \) and \( |x| = x \) when \( x > 0 \). Thus, for small \( c \), the integral becomes: \[ \int_{c-\frac{1}{4}}^{c} -x \, dx + \int_{c}^{c+\frac{1}{4}} x \, dx \].

Step 5: Evaluate the Integrals

Calculate separately: \[ \int_{c-\frac{1}{4}}^{c} -x \, dx = \left[ -\frac{x^2}{2} \right]_{c-\frac{1}{4}}^{c} \] and \[ \int_{c}^{c+\frac{1}{4}} x \, dx = \left[ \frac{x^2}{2} \right]_{c}^{c+\frac{1}{4}} \]. Evaluate these integrals and add them to get the total. Simplify these to get the expression for \( A_f(c) \).

Step 6: Plot the Graphs

After computing \( A_f(c) \), plot \( y = |x| \) (which is a V-shaped graph with its vertex at the origin) and \( y = A_f(x) \), which is smoothed around the vertex due to averaging. Ensure the plot is shown for \(-1 \leq x \leq 1\).

Key Concepts

Average Value of a Function

The average value of a function over a certain interval helps to give us a simplified look at complicated functions. Essentially, we're finding a single value that best represents the function over that interval, much like finding an average score for a test.
Imagine stretching a consistent value across an interval to equate to the area under the curve. This is what the average does—it provides a level version of the function's behavior.

To calculate the average value of a function, we use the formula for a function \( f \) over an interval \([a, b]\):

• \( A_f(c) = \frac{1}{b-a} \int_a^b f(x) \, dx \)

This formula is crucial as it gives a weighted measure of the area. In the case of the function \( |x| \), averaging over each smaller interval smooths out sharp turns, leading us right into our next topic.

Absolute Value Function

The function \( f(x) = |x| \) stands out because it's one of the most fundamental forms, showcasing symmetry about the y-axis. Whenever you see \(|x|\), think of it as making everything positive—transforming each negative input into its non-negative counterpart.
This creates a unique V-shaped graph with a vertex right at the origin (0,0). It's equally sharp on either side, highlighting why absolute values often signal a change in direction or magnitude.

The absolute value function is used in many applications:

• Handling deviations in data
• Maintaining only the magnitude in equations
• Analyzing signals in electrical engineering

These uses make learning the nature of \( |x| \) essential in both basic and advanced studies.

Definite Integral

The definite integral plays a pivotal role in calculus, as it's the tool we need to calculate the area under a curve between two points, \( a \) and \( b \).
In the simplest terms, it sums up infinitely many infinitesimally small areas under a curve, providing the total area:

• \( \int_a^b f(x) \, dx \)

In our exercise, dealing with \( |x| \) means altering our approach depending on the interval.
Because \(|x|\) is dependent on whether \(x\) is positive or negative, we split the integral at 0, if it lies within the limits, to properly account for the change in the function's behavior.
This underlines the flexible nature of definite integrals in handling complex functions.

Function Smoothing

Function smoothing is all about making jagged graphs look softer and more continuous. It involves averaging to reduce fluctuations and creating a more streamlined representation.
Think of it like evening out a bumpy road, making it easier to travel on and understand. In the context of our exercise, plotting \( y = A_f(x) \) against \( y = |x| \) shows this smoothing.

Smoothing can significantly aid in:

• Improving data clarity
• Enhancing visual analysis of trends
• Simplifying models for a better understanding

As you plot \( A_f(x) \), notice how the jaggedness at the absolute value's vertex gets softened into a rounded peak, illustrating how averages ease transitions in functions.