Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.                                                                          
                                                                                                                                                                                     (c) The functions p(𝓍) = sin 3𝓍 and q(𝓍) = 4 sin 3𝓍 are antiderivatives of the same function. 

Verified step by step guidance

1. Recall the definition of antiderivatives: Two functions are antiderivatives of the same function if their derivatives are equal.
2. Find the derivative of the first function \(p(\mathcal{x}) = \sin 3\mathcal{x}\) using the chain rule: \(p'(\mathcal{x}) = 3 \cos 3\mathcal{x}\).
3. Find the derivative of the second function \(q(\mathcal{x}) = 4 \sin 3\mathcal{x}\) similarly: \(q'(\mathcal{x}) = 4 \cdot 3 \cos 3\mathcal{x} = 12 \cos 3\mathcal{x}\).
4. Compare the derivatives \(p'(\mathcal{x}) = 3 \cos 3\mathcal{x}\) and \(q'(\mathcal{x}) = 12 \cos 3\mathcal{x}\). Since they are not equal, \(p\) and \(q\) are not antiderivatives of the same function.
5. Conclude that the statement is false because the derivatives differ by a constant factor, so \(p\) and \(q\) cannot both be antiderivatives of the same function.

Key Concepts

Antiderivative (Indefinite Integral)

An antiderivative of a function f(x) is another function F(x) whose derivative is f(x). It represents the family of all functions differing by a constant, since differentiation eliminates constants. For example, if F'(x) = f(x), then F(x) + C is the general antiderivative.

Linearity of Differentiation

Differentiation is a linear operation, meaning the derivative of a constant multiple of a function is the constant times the derivative of the function. For instance, if g(x) = k·f(x), then g'(x) = k·f'(x). This property helps compare functions to determine if they share the same derivative.

Checking if Two Functions are Antiderivatives of the Same Function

Two functions are antiderivatives of the same function if their derivatives are identical. If their derivatives differ, they cannot be antiderivatives of the same function. For example, p(x) = sin(3x) and q(x) = 4 sin(3x) have different derivatives, so they are not antiderivatives of the same function.