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) = 𝓍ⁿ on [0,1] , for any positive integer n

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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} d x

. In this case, a = 0 and b = 1.

Step 2: Substitute ƒ(𝓍) = 𝓍ⁿ into the formula. The integral becomes:

\frac{1}{1} \int x^{n} d x

, where the limits of integration are from 0 to 1.

Step 3: Compute the integral of 𝓍ⁿ. Use the power rule for integration:

\int x^{n} d x = \frac{x^{n + 1}}{n + 1}

. Apply this rule to the integral.

Step 4: Evaluate the definite integral by substituting the limits of integration (0 and 1) into the result from Step 3. This gives:

\frac{1}{n + 1} [\frac{1^{n + 1}}{n + 1} - \frac{0^{n + 1}}{n + 1}]

Step 5: The average value of the function is the result of the evaluation in Step 4. To complete the problem, draw the graph of ƒ(𝓍) = 𝓍ⁿ on the interval [0, 1] and mark the average value as a horizontal line across the graph.

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주요 개념

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Average Value of a Function

The average value of a function over a given interval is calculated using the formula (1/(b-a)) * ∫[a to b] f(x) dx, where [a, b] is the interval. This concept helps in understanding how the function behaves on average across the specified range, providing insight into its overall trend rather than just its individual values.

Definite Integral

A definite integral represents the accumulation of quantities, such as area under a curve, over a specific interval. It is denoted as ∫[a to b] f(x) dx and is fundamental in calculating the average value of a function, as it quantifies the total output of the function across the interval [a, b].

Graphing Functions

Graphing a function involves plotting its values on a coordinate system, which visually represents its behavior. For the function f(x) = x^n, where n is a positive integer, the graph will show a curve that starts at (0,0) and rises to (1,1) as n increases, illustrating how the function's average value can be interpreted visually in relation to its shape.

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) = 𝓍ⁿ on [0,1] , for any positive integer n