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] .

Guida verificata passo dopo passo

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 b} f (x) d x

Step 2: For a linear function, f(x) = mx + c, substitute this into the formula for the average value. The integral becomes:

\int_{a b} (m x + c) d x

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.

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 (\frac{(}{a}) + c

, which is the function value at the midpoint of [a, b].

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.

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