Unit 5 Check 1 - theorems

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What is Rolle's Theorem?

Back

Rolle's Theorem states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and F(a) = F(b), then there exists at least one c in (a, b) such that F'(c) = 0.

Front

What are the conditions for the Mean Value Theorem?

Back

The Mean Value Theorem requires that a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b).

Front

What does it mean for a function to have a critical value?

Back

A critical value of a function occurs where its derivative is zero or undefined, indicating potential local maxima, minima, or points of inflection.

Front

How do you find critical values of a function?

Back

To find critical values, take the derivative of the function, set it equal to zero, and solve for x. Also, check where the derivative does not exist.

Front

What is the significance of a zero in a function's interval?

Back

A zero in a function's interval indicates that the function crosses the x-axis, which can be found using the Intermediate Value Theorem if the function is continuous.

Tags

CCSS.HSA.APR.B.3

Front

What is the Intermediate Value Theorem?

Back

The Intermediate Value Theorem states that if a function is continuous on [a, b] and takes on values F(a) and F(b), then it takes on every value between F(a) and F(b) at least once.

Front

What is the formula for the Mean Value Theorem?

Back

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Back

Front

How do you determine if a function is continuous on an interval?

Back

A function is continuous on an interval if it is defined at every point in the interval and the limit as x approaches any point in the interval equals the function's value at that point.

Front

What is a sign change in a function?

Back

A sign change occurs when the function changes from positive to negative or vice versa, indicating the presence of a root (zero) in that interval.

Tags

CCSS.HSF-IF.C.7C