Question

Fill in the blanks with either of the words the derivative or an antiderivative: If \(F^{\prime}(x)=f(x),\) then \(f\) is __________ - of \(F\), and \(F\) is __________ of \(f\)

Short answer

Question: Fill in the blanks: "If \(F'(x)=f(x),\) then \(f\) is the _____ of \(F\), and \(F\) is _____ of \(f\)." Answer: If \(F'(x)=f(x),\) then \(f\) is the derivative of \(F\), and \(F\) is an antiderivative of \(f\).

Step-by-step solution

Identify which term is the derivative and which is the antiderivative based on the given equation

Given \(F'(x)=f(x)\), we can infer that \(f(x)\) is in fact the derivative of some function \(F(x)\). This also implies that \(F(x)\) is an antiderivative of \(f(x)\).

Fill in the blanks with the appropriate terms

As we identified in step 1, we have \(f(x)\) as the derivative of \(F(x)\), and \(F(x)\) as an antiderivative of \(f(x)\). Therefore, the sentence should be filled in as follows: "If \(F'(x)=f(x),\) then \(f\) is the derivative of \(F\), and \(F\) is an antiderivative of \(f\)."

Key concepts

Derivative Function

A derivative function represents the rate at which a function changes at any given point. It's a fundamental concept in differential calculus, revealing how a function's value will change as its input changes. Let's consider a real-world example, such as a car's speedometer that shows the car's instantaneous speed, which is a derivative of the car's position with respect to time.

The derivative is usually denoted as \(f'(x)\), \(df/dx\), or \(Df(x)\), and calculated using various rules of differentiation, like the power rule, product rule, and chain rule. If we have a position function \(s(t)\), the derivative function \(v(t)=s'(t)\) gives us the velocity at time \(t\). Remember that the derivative can be understood geometrically as the slope of the tangent line to the graph of a function at a point.

Antiderivative Function

An antiderivative function, on the other hand, is essentially the reverse process of differentiation. It is part of integral calculus and is used to find a function whose derivative is the given function. This means that if \(F(x)\) is the antiderivative of \(f(x)\), then \(F'(x)=f(x)\). The antiderivative is also known as the indefinite integral and is denoted by the integral sign without limits, for example, \(\int f(x)\,dx\).

Antiderivatives are not unique—since taking the derivative of a constant yields zero, any antiderivative can have an arbitrary constant, \(C\), added to it. This family of functions is called the 'general antiderivative'. For instance, if \(f(x)=3x^2\), then its general antiderivative would be \(F(x)=x^3+C\). The process of finding antiderivatives is essential in solving many problems involving motion, area, and growth, among others.

Differential Calculus

Differential calculus is a branch of calculus concerned with the study of how functions change when their inputs change. The main object of study in differential calculus is the derivative. This area of calculus allows us to break things down into infinitely small pieces, and find how each piece changes in relation to the whole. It's a powerful tool for scientists and engineers for optimizing and finding rates of change.

Applying differential calculus can give us insights into a function's increasing or decreasing behavior, its concavity, and points of inflection. It aids in solving maxima and minima problems which are pivotal in economics, engineering, and science. Understanding how to differentiate various function types is crucial, as well as comprehending the meaning of higher-order derivatives such as acceleration, which is the second derivative of position with respect to time.

Integral Calculus

Integral calculus, in contrast, involves the accumulation of quantities, such as areas under curves and the accumulated change in position, which is total distance traveled. This branch deals with the concept of an integral, which is essentially the reverse of a derivative. Integrals can be divided into two categories: indefinite integrals (antiderivatives) and definite integrals.

The definite integral, denoted with limits of integration, is used to calculate the accumulation of a quantity, such as area or volume. It is represented as \(\int_a^b f(x)\,dx\), which gives us the accumulated quantity of \(f\) along the interval \[a, b\]. Integral calculus is not just about calculating numbers; it provides a way to understand and describe the world around us, for example, how materials accumulate over time or how the total energy of a system can be computed.