AB Calculus Semester 1 review

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What is the limit definition of the first derivative?

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The limit definition of the first derivative of a function f at a point a is given by: f'(a) = lim (h -> 0) [(f(a+h) - f(a)) / h].

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What is a Left Riemann Sum?

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A Left Riemann Sum is an approximation of the area under a curve by dividing the interval into sub-intervals and using the left endpoint of each sub-interval to calculate the height of the rectangles.

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What is a Right Riemann Sum?

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A Right Riemann Sum is an approximation of the area under a curve by dividing the interval into sub-intervals and using the right endpoint of each sub-interval to calculate the height of the rectangles.

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What is the Mean Value Theorem?

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The Mean Value Theorem states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).

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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], then for any value L between f(a) and f(b), there exists at least one c in (a, b) such that f(c) = L.

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What is the Squeeze Theorem?

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The Squeeze Theorem states that if f(x) ≤ g(x) ≤ h(x) for all x in some interval around a, and if lim (x -> a) f(x) = lim (x -> a) h(x) = L, then lim (x -> a) g(x) = L.

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How do you calculate the area under a curve using Riemann Sums?

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To calculate the area under a curve using Riemann Sums, divide the interval into n sub-intervals, choose either left or right endpoints (or midpoints), calculate the height of the rectangles, and sum the areas of the rectangles.

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What is the formula for the Trapezoidal Rule?

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The Trapezoidal Rule for approximating the area under a curve is given by: Area ≈ (b - a) / 2n * [f(a) + 2Σf(xi) + f(b)], where n is the number of sub-intervals.

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What is the difference between definite and indefinite integrals?

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A definite integral computes the net area under a curve between two specific limits, while an indefinite integral represents a family of functions and includes a constant of integration.

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What is the Fundamental Theorem of Calculus?

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The Fundamental Theorem of Calculus links differentiation and integration, stating that if F is an antiderivative of f on an interval [a, b], then ∫[a to b] f(x) dx = F(b) - F(a).