Intermediate Value Theorem

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What is the Intermediate Value Theorem?

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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 (f(a) ≠ f(b)), 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?

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A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. A function is continuous on an interval if it is continuous at every point in that interval.

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What is a zero of a function?

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A zero of a function f(x) is a value x = c such that f(c) = 0. It represents the x-intercept of the function on a graph.

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How can you determine if a zero exists between two points using the Intermediate Value Theorem?

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To determine if a zero exists between two points a and b, check if f(a) and f(b) have opposite signs (f(a) * f(b) < 0). If they do, then by the Intermediate Value Theorem, there is at least one zero in the interval (a, b).

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What is a discontinuity in a function?

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A discontinuity occurs at a point in a function where it is not continuous. This can happen due to a jump, infinite behavior, or a removable discontinuity (like a hole in the graph).

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At x = 3, the function f(x) has a discontinuity because the denominator becomes zero, making f(3) undefined.

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This function models the cooling of an object over time, where 90 is the initial temperature, 70 is the ambient temperature, and the exponential term represents the rate of cooling.

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How do you find an interval where a function's value is below a certain threshold?

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To find an interval where a function's value is below a certain threshold, evaluate the function at various points and identify the interval where the function transitions from above to below the threshold.

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The natural logarithm introduces a non-linear component to the function, affecting its behavior and potentially creating zeros in the function.

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What does it mean if a function is increasing on an interval?

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A function is increasing on an interval if, for any two points x1 and x2 in that interval where x1 < x2, the function value at x2 is greater than the function value at x1 (f(x2) > f(x1)).