We are given that $F(t)$ is the antiderivative (integral) of $f(t)$, with $F(0) = 5$, and the graph of $f(t)$ is shown. We are tasked with finding $F(1)$ and $F(9)$.

Key idea:

Since $F(t)$ is the antiderivative of $f(t)$, the value of $F(t)$ at any point is: $F(t) = F(0) + \int_0^t f(x) \, dx$ This means that the change in $F(t)$ from 0 to any other value of $t$ is given by the area under the curve of $f(t)$.

Let's compute $F(1)$:

1. The graph of $f(t)$ shows a straight line between $t = 0$ and $t = 1$, dipping down to $f(t) = -2$. This forms a triangle below the x-axis with a base of 1 and a height of 2.
2. The area of this triangle is: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times 2 = 1$ Since the area is below the x-axis, it counts as negative: $-1$.

Thus: $F(1) = F(0) + \int_0^1 f(t) \, dt = 5 + (-1) = 4$

Now let's compute $F(9)$:

To find $F(9)$, we need to calculate the area under the curve from $t = 0$ to $t = 9$.

1. From $t = 1$ to $t = 2$, the graph shows a triangle above the x-axis with a base of 1 and height of 2. The area is: $\frac{1}{2} \times 1 \times 2 = 1$
2. From $t = 2$ to $t = 6$, the graph is flat at $f(t) = 0$, so the area is 0.
3. From $t = 6$ to $t = 9$, the graph stays at $f(t) = 1$, which forms a rectangle with a base of 3 and height of 1. The area is: $3 \times 1 = 3$

Adding up these areas: $\int_0^9 f(t) \, dt = (-1) + 1 + 0 + 3 = 3$

Thus: $F(9) = F(0) + \int_0^9 f(t) \, dt = 5 + 3 = 8$

Final answers:

$F(1) = 4 \quad \text{and} \quad F(9) = 8$

Would you like a more detailed breakdown of these calculations or have any questions? Here are some related questions to explore further:

1. How would the calculation change if the area under the curve was more complex, such as involving curves instead of straight lines?
2. Can you describe how the Fundamental Theorem of Calculus applies in this situation?
3. How would you handle finding $F(t)$ if $f(t)$ had discontinuities?
4. What happens if the graph of $f(t)$ includes a curve that crosses the x-axis multiple times?
5. Can we estimate $F(15)$ based on the behavior of $f(t)$ after $t = 9$?

Tip: Always keep track of whether areas under the curve are above or below the x-axis, as it directly affects the sign of the integral!