M119 2021 Fall Midterm

Front

What is the definition of a derivative in calculus?

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 function's graph 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 Fundamental Theorem of Calculus?

Back

The Fundamental Theorem of Calculus links the concept of differentiation and integration, stating that if a function is continuous on [a, b], then the integral of its derivative over that interval equals the difference in the values of the function at the endpoints.

Front

Define a continuous function.

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 is a formula for computing the derivative of the composition of two or more functions. If y = f(g(x)), then the derivative is dy/dx = f'(g(x)) * g'(x).

Front

What is an integral?

Back

An integral is a mathematical object that represents the area under a curve defined by a function. It can be thought of as the accumulation of quantities.

Front

Explain the difference between definite and indefinite integrals.

Back

A definite integral computes the accumulation of a quantity over a specific interval [a, b] and results in a number, while an indefinite integral represents a family of functions and includes a constant of integration (C).

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 the context of a function's graph.

Back

Concavity refers to the direction in which a function curves. A function is concave up if its second derivative is positive, and concave down if its second derivative is negative.

Front

What is the purpose of the Mean Value Theorem?

Back

The Mean Value Theorem states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point c in the interval where the derivative equals the average rate of change over that interval.