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) = cos 𝓍 on [―π/2 , π/2]

Verified step by step guidance

1. 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: $\frac{1}{(b-a)}{\int }_a^bƒ\left(𝓍\right)d𝓍$
2. Step 2: Substitute the given interval [―π/2, π/2] into the formula. Here, a = ―π/2 and b = π/2. The formula becomes: $\frac{1}{(\pi /2--\pi /2)}{\int }_{-}^{\pi }\cos \left(𝓍\right)d𝓍$
3. Step 3: Simplify the denominator of the fraction. The length of the interval [―π/2, π/2] is π. The formula now becomes: $\frac{1}{\pi }{\int }_{-}^{\pi }\cos \left(𝓍\right)d𝓍$
4. Step 4: Compute the integral of cos(𝓍) over the interval [―π/2, π/2]. Recall that the integral of cos(𝓍) is sin(𝓍). Apply the Fundamental Theorem of Calculus: $\frac{1}{\pi }\left[\sin \right(𝓍\left)\right]|-\pi /2^\pi /2$
5. Step 5: Evaluate the definite integral by substituting the limits of integration (―π/2 and π/2) into sin(𝓍). Simplify the result to find the average value of the function. Finally, draw the graph of ƒ(𝓍) = cos(𝓍) on the interval [―π/2, π/2] and indicate the average value as a horizontal line on the graph.

Key Concepts

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 essential for calculating the total value of a function between two points, which is a key step in finding the average value.

Graphing Functions

Graphing a function involves plotting its values on a coordinate system, which visually represents its behavior over an interval. This is crucial for understanding the function's characteristics, such as peaks, troughs, and the average value, allowing for a more intuitive grasp of the function's overall performance.