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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblema 4.9.3
Capitolo 4, Problema 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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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.
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Antiderivatives

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.
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Exponential Functions

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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Antiderivatives
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