Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(c) The average value of a linear function on an interval [a, b] is the function value at the midpoint of [a, b] .

Verified step by step guidance

1. Step 1: Recall the formula for the average value of a function f(x) on an interval [a, b]. It is given by: $\frac{1}{(b-a)}{\int }_{a}f\left(x\right)dx$.
2. Step 2: For a linear function, f(x) = mx + c, substitute this into the formula for the average value. The integral becomes: ${\int }_a(mx+c)dx$.
3. Step 3: Compute the integral of the linear function. The integral of mx is $\frac{m}{2}{x}^2$, and the integral of c is cx. Evaluate these terms at the bounds a and b.
4. Step 4: Simplify the result of the integral and divide by (b - a) to find the average value. After simplification, the average value of the linear function will be: $m\left(\frac{(}{a}\right)+c$, which is the function value at the midpoint of [a, b].
5. Step 5: Conclude that the statement is true. The average value of a linear function on an interval [a, b] is indeed equal to the function value at the midpoint of [a, b], as shown by the calculation.

Key Concepts

Average Value of a Function

The average value of a function f(x) over an interval [a, b] is calculated using the formula (1/(b-a)) * ∫[a to b] f(x) dx. This concept is crucial for understanding how to find the mean value of a function across a specified range, which can differ from simply evaluating the function at a single point.

Linear Functions

A linear function is a polynomial function of degree one, typically expressed in the form f(x) = mx + b, where m is the slope and b is the y-intercept. The properties of linear functions, such as constant slope and direct proportionality, influence how their average value behaves over an interval.

Midpoint of an Interval

The midpoint of an interval [a, b] is calculated as (a + b)/2. For linear functions, evaluating the function at this midpoint can yield the average value, but this is not true for non-linear functions. Understanding this concept helps clarify why the statement may hold true specifically for linear functions.