Problem 3 - Calculus Early Transcendentals

Question

Describe the set of antiderivatives of \(f(x)=1\)

Step-by-Step Solution

Answer

Answer: The set of antiderivatives for the function \(f(x) = 1\) is \(\{x + C: C \in \mathbb{R}\}\), where C is any real number.

Step 1: Identify the type of function we're working with

We're working with a constant function of \(f(x) = 1\). To find the antiderivatives (also referred to as indefinite integrals), we will integrate it.

Step 2: Integrate the constant function

To find the integral of a constant function, we simply multiply the constant by x and add an arbitrary constant, denoted by C. This is because the derivative of a constant function is 0, and it's valid to add such constant terms to an antiderivative since the derivative of a constant is always 0. In this case, we have: \(\int{1 \, dx} = x + C\)

Step 3: Describe the set of antiderivatives

Since the antiderivative we found, \(x + C\), depends on the arbitrary constant C, we can describe the set of antiderivatives as: \(\{x + C: C \in \mathbb{R}\}\) This means that the antiderivatives of the function \(f(x) = 1\) are all functions of the form \(x + C\), where C is any real number.