Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.
ƒ(𝓍) = 1/(𝓍² + 1) on [―1, 1]

Verified step by step guidance

Step 1: Recall the formula for the average value of a function ƒ(𝓍) on an interval [a, b]. The average value is given by:

\frac{1}{(b - a)} \int f_{x} (x) d x

. Here, a = -1 and b = 1.

Step 2: Set up the integral for the average value. Substitute ƒ(𝓍) = 1/(𝓍² + 1) into the formula:

\frac{1}{2} \int x_{-1} 1 \frac{1}{x^{2} + 1} d x

. Note that the factor

\frac{1}{2}

comes from

\frac{1}{(b - a)}

, where b - a = 2.

Step 3: Evaluate the integral. The integral of

\frac{1}{x^{2} + 1}

is a standard result in calculus. It is

arctan (x)

. Apply this result to the integral:

\frac{1}{2} [arctan (x)] x_{-1} 1

Step 4: Compute the definite integral by substituting the limits of integration. Substitute x = 1 and x = -1 into

arctan (x)

\frac{1}{2} [arctan (1) - arctan (- 1)]

. Recall that

arctan (1)

and

arctan (- 1)

are standard values.

Step 5: Draw the graph of ƒ(𝓍) = 1/(𝓍² + 1) on the interval [―1, 1]. The function is symmetric about the y-axis and decreases as |𝓍| increases. Indicate the average value as a horizontal line on the graph, representing the computed average value.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Average Value of a Function

The average value of a continuous function over a closed interval [a, b] is calculated using the formula (1/(b-a)) * ∫[a to b] f(x) dx. This represents the mean value of the function's outputs over the specified interval, providing insight into the function's overall behavior.

Definite Integral

A definite integral computes the accumulation of a function's values over a specific interval [a, b]. It is represented as ∫[a to b] f(x) dx and can be interpreted as the area under the curve of the function between the limits a and b, which is essential for finding the average value.

Graphing Functions

Graphing a function involves plotting its output values against input values on a coordinate plane. This visual representation helps in understanding the function's behavior, identifying key features such as intercepts, maxima, minima, and the average value, which can be marked on the graph for clarity.

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