AP Calc BC Exam Review- Non-Calc Day 1

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What is the Fundamental Theorem of Calculus?

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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 limit in calculus.

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

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What is the derivative of a function?

Back

The derivative of a function at a point is the slope of the tangent line to the graph of the function at that point, representing the rate of change of the function.

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Explain the concept of continuity in a function.

Back

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. It must be defined, and the limit must exist.

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What is an asymptote?

Back

An asymptote is a line that a graph approaches but never touches. Vertical, horizontal, and oblique asymptotes are common types.

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Define a critical point in calculus.

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

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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, 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 the difference between a definite and an indefinite integral?

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A definite integral computes the 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.

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Explain the concept of concavity.

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Concavity describes the direction of the curvature of a graph. A function is concave up if its second derivative is positive and concave down if its second derivative is negative.

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What is a Taylor series?

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A Taylor series is an infinite series of terms calculated from the values of a function's derivatives at a single point, used to approximate functions.