Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


a. F(x) = x³ - 4x + 100 and G(x) = x³ - 4x - 100 are antiderivatives of the same function.

Verified step by step guidance

1. Recall that two functions are antiderivatives of the same function if their derivatives are equal, which means they differ by only a constant.
2. Find the derivative of \( F(x) = x^3 - 4x + 100 \). Using the power rule and constant rule, \( F'(x) = 3x^2 - 4 \).
3. Find the derivative of \( G(x) = x^3 - 4x - 100 \). Similarly, \( G'(x) = 3x^2 - 4 \).
4. Since \( F'(x) = G'(x) = 3x^2 - 4 \), both \( F(x) \) and \( G(x) \) are antiderivatives of the same function \( f(x) = 3x^2 - 4 \).
5. The difference between \( F(x) \) and \( G(x) \) is a constant (\( 200 \)), which confirms they are 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 reverse process of differentiation and is expressed as an indefinite integral, including an arbitrary constant C since differentiation eliminates constants.

Constant of Integration

When finding antiderivatives, any constant term disappears upon differentiation. Therefore, all antiderivatives of a function differ by a constant, called the constant of integration, which accounts for all possible vertical shifts of the antiderivative graph.

Verifying Antiderivatives by Differentiation

To determine if two functions are antiderivatives of the same function, differentiate both and compare their derivatives. If the derivatives are identical, the functions differ only by a constant and are antiderivatives of the same function.