Unit 5 Calculus AB Multiple Choice

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 derivative. What does it represent?

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.

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 specific points and results in a number, while an indefinite integral represents a family of functions and includes a constant of integration (C).

Front

Explain the concept of limits in calculus.

Back

Limits describe the behavior of a function as it approaches a certain point from either side. They are fundamental in defining derivatives and integrals.

Front

What is the Chain Rule in differentiation?

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

What is 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.

Front

Define concavity and how it relates to the second derivative.

Back

Concavity refers to 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.

Front

What is a critical point?

Back

A critical point of a function occurs where its derivative is zero or undefined. Critical points are used to find local maxima and minima.

Front

Explain the Mean Value Theorem.

Back

The Mean Value Theorem states that if a function is continuous on a 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 the difference between local and absolute extrema?

Back

Local extrema are the highest or lowest points in a small neighborhood of a function, while absolute extrema are the highest or lowest points over the entire domain of the function.