Average value with a parameter Consider the function ƒ(𝓍) = a𝓍 (1―𝓍) on the interval [0, 1], where a is a positive real number.
(a) Find the average value of ƒ as a function of a .

Verified step by step guidance

1. Step 1: Recall the formula for the average value of a function ƒ(𝓍) on the interval [a, b], which is given by: $\frac{1}{(b-a)}\int f_{x}dx$. In this case, the interval is [0, 1] and the function is ƒ(𝓍) = a𝓍(1 - 𝓍).
2. Step 2: Substitute the interval [0, 1] and the function ƒ(𝓍) = a𝓍(1 - 𝓍) into the formula for average value: $\frac{1}{1}\int 0^1\left(ax\right(1-x\left)\right)dx$. This simplifies to: $\int 0^1ax(1-x)dx$.
3. Step 3: Expand the integrand a𝓍(1 - 𝓍) to simplify the integral. This becomes: $a(x-x^2)$. The integral now looks like: $a\int 0^1(x-x^2)dx$.
4. Step 4: Break the integral into two separate parts: $a(\int 0^1xdx-\int 0^1x^{2}dx)$. Compute each integral separately: $\int 0^1xdx$ and $\int 0^1x^{2}dx$. Use the power rule for integration: $\int x^{n}dx=\frac{x^{n+1}}{n+1}$.
5. Step 5: After computing the integrals, combine the results and multiply by the constant 'a' to find the average value of ƒ(𝓍) as a function of 'a'. The final expression will represent the average value of the function over the interval [0, 1].

Key Concepts

Average Value of a Function

The average value of a continuous function ƒ over an interval [a, b] is calculated using the formula (1/(b-a)) * ∫[a to b] ƒ(x) dx. This concept is essential for determining how the function behaves on the specified interval, providing a single representative value that summarizes the function's overall trend.

Definite Integral

A definite integral represents the accumulation of quantities, such as area under a curve, over a specific interval. In this context, it is used to compute the integral of the function ƒ(x) = a𝓍(1 - 𝓍) from 0 to 1, which is necessary for finding the average value of the function.

Parameter in Functions

A parameter is a variable that influences the behavior of a function but is not the primary variable of interest. In this case, 'a' is a parameter that affects the shape and scale of the function ƒ(x) = a𝓍(1 - 𝓍), and understanding its role is crucial for expressing the average value as a function of 'a'.