In order to solve the problem, let's break down the task into manageable steps based on the provided information:

You have the function: $g(x) = \int_{-8}^{x} f(t) \, dt$ where you need to compute the following values:

1. $g(-2)$
2. $g'(-2)$
3. $g''(-2)$
4. $g(5)$
5. $g'(5)$

The graph of $f(t)$, which is the integrand, is also provided.

Let's approach this systematically:

Step 1: Understanding the Integral

The function $g(x)$ is the definite integral of $f(t)$ from $-8$ to $x$. This means that $g(x)$ accumulates the area under the curve of $f(t)$ between $-8$ and $x$.

Step 2: Computing $g(x)$

• $g(-2)$ is simply the area under $f(t)$ from $-8$ to $-2$.
• $g(5)$ is the area under $f(t)$ from $-8$ to $5$.

Step 3: Computing Derivatives

• $g'(x)$ is the derivative of the integral with respect to $x$. By the Fundamental Theorem of Calculus, $g'(x) = f(x)$.

  • $g'(-2)$ will be the value of the function $f(x)$ at $x = -2$.
  • $g'(5)$ will be the value of the function $f(x)$ at $x = 5$.

• $g''(x)$ is the derivative of $g'(x)$, which will be the derivative of $f(x)$, i.e., $g''(x) = f'(x)$.

Step 4: Evaluating the Values

Now, let’s evaluate the given values based on the graph:

1. $g(-2)$: The area under the curve of $f(t)$ from $-8$ to $-2$. From the graph, we can estimate this area.

2. $g'(-2)$: This is simply $f(-2)$, which is the value of the function at $x = -2$.

3. $g''(-2)$: This is the derivative of $f(t)$ at $x = -2$, which we can estimate based on the graph.

4. $g(5)$: The area under the curve of $f(t)$ from $-8$ to $5$.

5. $g'(5)$: This is $f(5)$, which we can read directly from the graph.

I'll now calculate these based on the graph you provided. Let me process the images. PIL import Image

Load and display the images to analyze the graph

img1_path = "/mnt/data/1739391955072kcrplsi2.jpg" img2_path = "/mnt/data/1739391955072zvpyj8wa.jpg"

img1 = Image.open(img1_path) img2 = Image.open(img2_path)

img1.show(), img2.show()I have displayed the images of the graph for analysis, but currently, I cannot interpret the visual data directly. Would you be able to describe the graph, or point out key features such as the values of the function $f(t)$ at certain points? Specifically, information about the shape of the graph around the relevant points will help in calculating the areas under the curve and the values of $f(x)$.

Let me know if you need help with this!