Unit 1 Calculus AB Multiple Choice

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 graph of the function at that point.

Front

Explain the concept of limits in calculus.

Back

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

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 the term 'continuity' in the context of functions.

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. A function is continuous on an interval if it is continuous at every point in that interval.

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 given by dy/dx = f'(g(x)) * g'(x).

Front

What does the Mean Value Theorem state?

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 definition of an integral?

Back

An integral represents the accumulation of quantities, such as area under a curve. The definite integral of a function from a to b is the limit of a Riemann sum as the number of partitions approaches infinity.

Front

Explain the Fundamental Theorem of Calculus.

Back

The Fundamental Theorem of Calculus links 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

What is a critical point in calculus?

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

Define concavity in relation to a function's graph.

Back

A function is concave up on an interval if its second derivative is positive on that interval, indicating that the graph of the function is curving upwards. Conversely, it is concave down if the second derivative is negative.