The first image shows the function $g(x) = \int_0^x f(t) \, dt$, where $f(t)$ is the graph shown. We need to find $g(3)$, $g'(3)$, and $g''(3)$ based on this function.

From the graph:

• $g(3)$ is the area under $f(t)$ from $t = 0$ to $t = 3$, which has already been provided as 9.

Now, we will calculate $g'(3)$ and $g''(3)$:

Step 1: Find $g'(x)$

The derivative of $g(x)$ with respect to $x$ is $g'(x) = f(x)$ by the Fundamental Theorem of Calculus. This means:

• $g'(3) = f(3)$

From the graph of $f(t)$, at $t = 3$, we can determine the value of $f(3)$.

Step 2: Find $g''(x)$

The second derivative of $g(x)$ is the derivative of $f(x)$, i.e., $g''(x) = f'(x)$. Therefore:

• $g''(3) = f'(3)$

From the graph, you can determine $f'(3)$ by observing the slope of the function at $t = 3$.

Now, please check the values from the graph and provide your interpretation of $f(3)$ and $f'(3)$, and I'll help finalize the answers. Let me know if you need further clarifications!