The problem asks which graph corresponds to $g(x)$ if $f(x)$ is the integral of $g(t)$ from $t = 2$ to $t = x$, given that the graph of $f(x)$ is linear with a positive slope.

Key Points:

1. The Fundamental Theorem of Calculus states: $f'(x) = g(x)$ So, $g(x)$ is the derivative of $f(x)$.

2. Observing the graph of $f(x)$, it is a straight line with a positive slope. A straight line has a constant slope, meaning $f'(x) = g(x)$ must be constant.

3. Hence, $g(x)$ must be a horizontal line because its value does not change with $x$.

Conclusion:

The correct choice is (A), where $g(x)$ is a horizontal line, representing a constant value.

Would you like further clarification on integrals, derivatives, or related concepts?

Here are 5 related questions to explore:

1. What is the geometric interpretation of the Fundamental Theorem of Calculus?
2. How does the slope of $f(x)$ influence the graph of $g(x)$?
3. Can the integral of a function ever result in a non-linear graph?
4. How do you identify the constant of integration in indefinite integrals?
5. How does the graph of a derivative look if the original function is quadratic?

Tip: Always analyze the relationship between a function and its derivative graphically to ensure consistency in calculus problems.