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Define the average value of a function.
The average y-value of a function over a given interval.
What does 'continuous on [a,b]' mean?
The function has no breaks, jumps, or undefined points on the closed interval from a to b.
Define definite integral.
The definite integral of a function f(x) from a to b represents the net signed area between the curve of f(x) and the x-axis from x=a to x=b.
What is the formula for the average value of a function f(x) on the interval [a, b]?
$\text{Average Value} = \frac{1}{b-a} \int_{a}^{b} f(x) , dx$
How do you calculate the area under a curve?
$\int_{a}^{b} f(x) , dx$
Explain the concept of the average value of a function.
It finds the height of a rectangle with the same width (b-a) as the interval [a, b] that has the same area as the area under the curve of the function on that interval.
What does the integral $\int_{a}^{b} f(x) , dx$ represent?
The area under the curve of f(x) from x=a to x=b.
Why is continuity important when finding the average value?
Continuity ensures that the definite integral exists and that the average value can be accurately calculated.