Question

Suppose \(A\) is an area function of \(f .\) What is the relationship between \(f\) and \(A ?\)

Short answer

Answer: The relationship between the area function A and the function f is that the derivative of A(x) with respect to x is equal to f(x). The slope of the area function at any point x is equal to the value of the function f(x).

Step-by-step solution

Understand the area function A

The area function A(x) is defined as the accumulated area under the curve of the function f(t), from a fixed point to x. Mathematically, we can represent A(x) as: A(x) = ∫(f(t), dt, a, x)

Differentiating the Area Function A

We will use the Fundamental Theorem of Calculus to differentiate A(x) with respect to x: d(A(x))/dx = f(x) Here, d(A(x))/dx is the derivative of A(x) with respect to x.

Relationship between A and f

As we have already found the derivative, our final answer is: d(A(x))/dx = f(x) This means that the relationship between the area function A and the function f is that the derivative of A(x) with respect to x is equal to f(x). In other words, the slope of the area function at any point x is equal to the value of the function f(x).

Key concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) serves as a bridge between the concepts of differentiation and integration, two core operations in calculus. It consists of two related parts. The first part ensures that if we have a continuous function over an interval, then we can find its antiderivative, and the process of integration can reverse differentiation.

Specifically, if we have a continuous function, say, f(x), and we integrate it from a to b, and call this antiderivative function A(x), then the FTC tells us that the definite integral from a to b of f(x) dx is equal to A(b) - A(a). It confirms that integrating a function over an interval gives us the net area under the curve of that function.

The second part of the FTC is particularly relevant in relating the area function A to f. It tells us that the derivative of the antiderivative A(x) is the original function f(x), which is essential in understanding the relationship between a derivative and the area under a curve. This forms the cornerstone for solving problems related to the rate of change and cumulative quantities in various fields such as physics, engineering, and economics.

Definite Integral

The definite integral of a function provides a way to calculate things like areas, volumes, and other quantities that accumulate. Geometrically, for a function f(x) on the interval [a, b], the definite integral from a to b is the signed area between the function and the x-axis.

The 'signed' part means that areas below the x-axis subtract from the total, whereas areas above add to it. As observed in the Fundamental Theorem of Calculus, integration and differentiation are inverse processes, and the definite integral is a critical piece in underlining this relationship.

In the problem given, the function A(x) represents the accumulated area from a starting point a to an arbitrary point x under the graph of the function f(t). The notation like A(x) = ∫(f(t), dt, a, x) signifies that A(x) is computed using the definite integral of f from a to x. This concept helps students grasp how functions accumulate values, which is especially useful for understanding physical phenomena such as distance traveled given a velocity function.

Derivative

Derivatives are a fundamental tool in calculus, measuring how a function changes as its input changes — in simpler terms, it's the slope of the function at any given point.

The derivative of a function at a point x is defined as the limit of the average rate of change of the function over an interval as the interval gets vanishingly small. Mathematically, the derivative of A(x) with respect to x, denoted as d(A(x))/dx, is the instantaneous rate of change of A with respect to x.

In the context of the area function relationship, taking the derivative of A(x) tells us the value of the function f(x) at a point. This is what we see in the solution provided: d(A(x))/dx = f(x), confirming the powerful concept that differentiating a function gives us the slope at any point, or the rate of change, and relates it directly to the original function—the one we integrated to get the area under its curve.