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 continuity, derivatives, and integrals.

Front

What is the definition of a derivative?

Back

The derivative of a function at a point is the limit of the average rate of change of the function as the interval approaches zero. It represents the slope of the tangent line to the curve at that point.

Front

Explain the concept of continuity at a point.

Back

A function f is continuous at a point c if: 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 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 any 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).

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 interval, often using derivatives to identify critical points.