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

c. F(x) = x² + 10 and G(x) = x² - 100 are antiderivatives of the same function.

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Recall that two functions are antiderivatives of the same function if their derivatives are equal, differing only by a constant.

Find the derivative of \( F(x) = x^2 + 10 \). Using the power rule, \( F'(x) = 2x \).

Find the derivative of \( G(x) = x^2 - 100 \). Similarly, \( G'(x) = 2x \).

Since \( F'(x) = G'(x) = 2x \), both \( F(x) \) and \( G(x) \) have the same derivative, meaning they are antiderivatives of the same function.

The difference between \( F(x) \) and \( G(x) \) is a constant (\( 10 - (-100) = 110 \)), which confirms they differ by a constant and thus are antiderivatives of the same function.

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Antiderivative Definition

An antiderivative of a function f(x) is another function F(x) whose derivative is f(x). In other words, if F'(x) = f(x), then F(x) is an antiderivative of f(x). Antiderivatives differ by a constant since the derivative of a constant is zero.

Derivative of Polynomial Functions

The derivative of a polynomial function like x² is found using the power rule: d/dx[x^n] = n*x^(n-1). For example, the derivative of x² is 2x. Constants vanish when differentiating, so terms like +10 or -100 become zero.

Checking if Two Functions are Antiderivatives of the Same Function

Two functions are antiderivatives of the same function if their derivatives are identical. Since constants disappear upon differentiation, functions differing only by a constant are antiderivatives of the same function. If their derivatives differ, they are not.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.c. F(x) = x² + 10 and G(x) = x² - 100 are antiderivatives of the same function.