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Calculus - Average Function Value & Theorems

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Calculus - Average Function Value & Theorems

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Questions and Answers

What is the formula to compute the average value of a continuous function over the interval [a, b]?

$f_{avg} = (b - a) \cdot \int_a^b f(x) dx$

$f_{avg} = \int_a^b f(x) dx$

$f_{avg} = \frac{1}{b + a} \int_a^b f(x) dx$

$f_{avg} = \frac{1}{b - a} \int_a^b f(x) dx$ (correct)

In the Mean Value Theorem for Integrals, what condition must the function f(x) satisfy on the interval [a, b]?

f(x) must have a maximum value at a

f(x) must be continuous on the interval (correct)

f(x) must be differentiable on the entire interval

f(x) must be continuous only at a

What can be inferred about the value of c in the Mean Value Theorem for Integrals?

c exists in [a, b] such that f(c) equals the average value of the function (correct)

c does not exist within the interval [a, b]

c is the average of a and b

c is always greater than b

When determining the area between two curves y = f(x) and y = g(x) over an interval [a, b], which condition must be met?

f(x) and g(x) should be continuous on the interval [a, b] Signup and view all the answers

If f(t) = t^2 - 5t + 6cos(πt) on the interval [-1, 5/2], what is required to calculate its average value?

Finding the definite integral of f(t) from -1 to 5/2 Signup and view all the answers

What is the primary purpose of calculating the average value of a function?

To understand the general behavior of the function over an interval Signup and view all the answers

If f(x) = x^2 + 3x + 2, what is the first step to apply the Mean Value Theorem for Integrals on the interval [1, 4]?

Calculate the average value using $1/(4-1) \int_1^4 f(x) dx$ Signup and view all the answers

What does the integral $\int_a^b f(x) dx$ represent in the context of the Mean Value Theorem for Integrals?

The total change in the function value over the interval Signup and view all the answers

Study Notes

Average Function Value

The average value of a continuous function f(x) over the interval [a, b] is calculated as:
1/(b-a) * ∫_a^b f(x) dx

Mean Value Theorem for Integrals

If f(x) is continuous on [a, b], there exists a number c in [a, b] such that:
∫_a^b f(x) dx = f(c) * (b - a)
This implies there's a point c within the interval where the function's value equals its average value.
Simplified as: 1/(b-a) * ∫_a^b f(x) dx = f(c)

Area Between Curves

Calculate the area between two curves y = f(x) and y = g(x) within the interval [a, b]. Assume f(x) ≥ g(x).

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Description

This quiz covers key concepts related to the average value of continuous functions, including the Mean Value Theorem for Integrals and the calculation of areas between curves. Test your understanding of these essential calculus topics and practice applying theorems in real scenarios.