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  2. AP Calculus
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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

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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

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.

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.