calculus

Front

What is the definition of a derivative in calculus?

Back

The derivative of a function measures how the function's output value changes as its input value changes. It represents the slope of the tangent line to the graph of the function at a given point.

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

How do you find the critical points of a function?

Back

Critical points occur where the derivative is zero or undefined. To find them, set the derivative equal to zero and solve for x.

Front

What is an inflection point?

Back

An inflection point is a point on the graph of a function where the concavity changes, which can be found by analyzing the second derivative.

Front

What is the area under a curve?

Back

The area under a curve can be found using definite integrals, which calculate the accumulation of quantities represented by the function over a specified interval.

Front

What is the relationship between the first derivative and the increasing/decreasing behavior of a function?

Back

If the first derivative of a function is positive, the function is increasing; if it is negative, the function is decreasing.

Front

What is optimization in calculus?

Back

Optimization involves finding the maximum or minimum values of a function, often subject to certain constraints, typically using derivatives to identify critical points.

Front

What is the formula for the area of a rectangle?

Back

The area A of a rectangle is given by the formula A = length × width.

Front

How do you determine the maximum area of a rectangle inscribed in a semicircle?

Back

To find the maximum area, express the area as a function of one variable, use calculus to find the critical points, and evaluate the area at those points.

Front

What is the product of two variables in terms of their linear relationship?

Back

If y = mx + b, the product xy can be expressed as a quadratic function, which can be maximized using calculus.