AP Calculus Unit 3 Review

Front

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 at which a function is changing at any given point, 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 derivatives and integrals.

Front

Explain the concept of continuity at a point.

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.

Front

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

Back

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

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 is continuous on the 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).

Front

What is an inflection point?

Back

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.

Front

Explain the concept of optimization in calculus.

Back

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