Intermediate Value Theorem (IVT)

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What is the Intermediate Value Theorem (IVT)?

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The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b], and takes on different values at the endpoints, then it must take on every value between f(a) and f(b) at least once within that interval.

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What does it mean for a function to be continuous on an interval?

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A function is continuous on an interval if there are no breaks, jumps, or holes in the graph of the function over that interval.

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If g is continuous on [-1, 4] and g(-1) = -4, g(4) = 1, what can we conclude using IVT?

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By the IVT, there exists at least one c in the interval [-1, 4] such that g(c) = -3.

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What is the significance of the endpoints in the IVT?

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The values of the function at the endpoints of the interval are crucial because they determine the range of values that the function must achieve within that interval.

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If h(x) is continuous on [1, 6] and h(3) = 0, what does this imply?

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It implies that there is a solution to h(x) = 0 at x = 3.

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What is an example of a function that satisfies the conditions of the IVT?

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An example is f(x) = x^2 - 4, which is continuous on any interval. For the interval [0, 4], f(0) = -4 and f(4) = 12, so by IVT, there exists c in [0, 4] such that f(c) = 0.

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What is the conclusion of the IVT if f(1) = 1 and f(5) = -3?

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The IVT guarantees that there exists at least one c in the interval [1, 5] such that f(c) = -2.

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How can you determine if a function is continuous?

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A function is continuous if it is defined at every point in the interval, the limit exists at every point, and the limit equals the function's value at that point.

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What does the IVT guarantee about the values of a continuous function?

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The IVT guarantees that a continuous function will take on every value between its values at the endpoints of an interval.

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If a function is continuous on [a, b] and f(a) < k < f(b), what can we say about k?

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By the IVT, there exists at least one c in (a, b) such that f(c) = k.