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  2. AP Calculus
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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.

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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 ∫abf(x),dx\int_{a}^{b} f(x) , dx∫ab​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.

What is the formula for the average value of a function f(x) on the interval [a, b]?

Average Value=1b−a∫abf(x),dx\text{Average Value} = \frac{1}{b-a} \int_{a}^{b} f(x) , dxAverage Value=b−a1​∫ab​f(x),dx

How do you calculate the area under a curve?

∫abf(x),dx\int_{a}^{b} f(x) , dx∫ab​f(x),dx

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.