AP Calculus Unit 5 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).

Front

Define a limit in calculus.

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

What is 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 as the interval approaches zero.

Front

Explain the concept of continuity at a point.

Back

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

Front

What is the Chain Rule in differentiation?

Back

The Chain Rule states that if a function y = f(g(x)) is composed of two functions, then the derivative is 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 [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?

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.

Front

What is the difference between definite and indefinite integrals?

Back

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

Front

Define a critical point in calculus.

Back

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