Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problem 4.9.3

Chapter 4, Problem 4.9.3

Describe the set of antiderivatives of ƒ(x) = 1

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Recognize that the antiderivative of a function ƒ(x) is a function F(x) such that the derivative of F(x) equals ƒ(x). In this case, we are given ƒ(x) = 1.

Recall the basic rule of integration: the antiderivative of a constant c is c * x + C, where C is the constant of integration. Here, the constant c is 1.

Apply the rule to find the antiderivative of ƒ(x) = 1. The result is F(x) = x + C, where C represents an arbitrary constant.

Understand that the set of antiderivatives of ƒ(x) = 1 is represented by the family of functions F(x) = x + C, where C can be any real number.

Conclude that the set of antiderivatives is infinite, as it depends on the value of the constant C, which can vary freely.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Antiderivative

An antiderivative of a function f(x) is another function F(x) such that the derivative of F(x) equals f(x). In other words, if F'(x) = f(x), then F(x) is an antiderivative of f(x). Antiderivatives are essential in calculus as they are used to find the area under curves and solve differential equations.

Constant Functions

The function f(x) = 1 is a constant function, meaning it has the same value (1) for all x in its domain. The antiderivative of a constant function is a linear function, which can be expressed as F(x) = x + C, where C is a constant. This reflects the fact that the slope of the constant function is zero, leading to a linear increase in the antiderivative.

Family of Antiderivatives

The set of all antiderivatives of a function forms a family of functions that differ only by a constant. For the function f(x) = 1, the family of antiderivatives can be expressed as F(x) = x + C, where C represents any real number. This indicates that there are infinitely many antiderivatives, each corresponding to a different vertical shift of the linear function.

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