Let $g\left(x\right)={\int }_0^{x}f\left(t\right)dt$, where $f$ is the function whose graph is shown. Which of the following statements is true about $g\left(x\right)$?

• A. $g\left(x\right)$ is an antiderivative of $f\left(x\right)$.
• B. $g\left(x\right)$ is the derivative of $f\left(x\right)$.
• C. $g\left(x\right)$ is always equal to $f\left(x\right)$.
• D. $g\left(x\right)$ is always a constant function.

Answer: (A)

Verified step by step guidance

1. Step 1: Understand the problem. The function g(x) is defined as the integral of f(t) from 0 to x, which can be written mathematically as $g(x)\; =\; \int$₀^x f(t) dt. This means g(x) represents the accumulated area under the curve of f(t) from 0 to x.
2. Step 2: Recall the Fundamental Theorem of Calculus. The theorem states that if g(x) is defined as the integral of f(t) from a constant to x, then g'(x) = f(x). This implies that g(x) is an antiderivative of f(x).
3. Step 3: Analyze the given options. Based on the Fundamental Theorem of Calculus, the correct statement is that g(x) is an antiderivative of f(x). The other options can be ruled out because: (a) g(x) is not the derivative of f(x), (b) g(x) is not always equal to f(x), and (c) g(x) is not a constant function since it depends on x.
4. Step 4: Verify the reasoning. To confirm, note that the derivative of g(x) with respect to x is f(x), which aligns with the definition of an antiderivative. This relationship is fundamental in calculus and connects differentiation and integration.
5. Step 5: Conclude the solution. The correct answer is that g(x) is an antiderivative of f(x), as it satisfies the relationship g'(x) = f(x) derived from the Fundamental Theorem of Calculus.