AP Calculus Unit 3 Review

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What is the Fundamental Theorem of Calculus?

Back

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

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Define a derivative.

Back

A derivative represents the rate of change of a function with respect to a variable. It is defined as the limit of the average rate of change of the function as the interval approaches zero.

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What is the definition of a limit?

Back

A limit is a value that a function approaches as the input approaches some value. It is fundamental in defining continuity, derivatives, and integrals.

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Explain the concept of continuity at a point.

Back

A function f(x) is continuous at a point x = c if: 1) f(c) is defined, 2) lim (x -> c) f(x) exists, and 3) lim (x -> c) f(x) = f(c).

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What is the difference between a definite and an indefinite integral?

Back

A definite integral computes the net area under a curve between two points and results in a number, while an indefinite integral represents a family of functions and includes a constant of integration (C).

Front

State the Power Rule for differentiation.

Back

The Power Rule states that if f(x) = x^n, then f'(x) = n*x^(n-1), where n is a real number.

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What is the Chain Rule in calculus?

Back

The Chain Rule is a formula for computing the derivative of the composition of two or more functions. If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

Front

Define the Mean Value Theorem.

Back

The Mean Value Theorem states that if a function f is continuous on [a, b] and differentiable on (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 an inflection point?

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An inflection point is a point on the curve of a function where the concavity changes, which can be determined by analyzing the second derivative.

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Explain the concept of optimization in calculus.

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Optimization involves finding the maximum or minimum values of a function within a given domain, often using derivatives to identify critical points.