We are given that $F(t)$ is a function with its derivative $f(t)$ shown in the graph, and $F(0) = 2$. We need to find values for $F(2)$ and $F(7)$. To do this, we will apply the Fundamental Theorem of Calculus, which relates the integral of the derivative $f(t)$ to the change in the function $F(t)$.

The Fundamental Theorem of Calculus states: $F(b) = F(a) + \int_a^b f(t) \, dt$ This means that to find $F(2)$ and $F(7)$, we will need to integrate $f(t)$ (i.e., calculate the area under the curve) between the relevant points.

Step 1: Find $F(2)$

We are given $F(0) = 2$, so to find $F(2)$, we compute the area under $f(t)$ from $t = 0$ to $t = 2$.

From the graph:

• Between $t = 0$ and $t = 2$, the curve is below the x-axis and linear. This part of the curve forms a triangle with a base of 2 and a height of -2.
• The area of the triangle is: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2 \times (-2) = -2$

Using the Fundamental Theorem of Calculus: $F(2) = F(0) + \int_0^2 f(t) \, dt = 2 + (-2) = 0$

Step 2: Find $F(7)$

To find $F(7)$, we integrate $f(t)$ from $t = 2$ to $t = 7$.

Breaking it into sections:

• From $t = 2$ to $t = 3$, we have a small triangle with a base of 1 and height of 1. The area is: $\text{Area} = \frac{1}{2} \times 1 \times 1 = 0.5$
• From $t = 3$ to $t = 5$, the curve is a trapezoid with bases 1 and 2, and height of 2. The area is: $\text{Area} = \frac{1}{2} \times (1 + 2) \times 2 = 3$
• From $t = 5$ to $t = 7$, the curve is a rectangle with a base of 2 and height of 2. The area is: $\text{Area} = 2 \times 2 = 4$

Summing the areas from $t = 2$ to $t = 7$: $\int_2^7 f(t) \, dt = 0.5 + 3 + 4 = 7.5$

Now, using the Fundamental Theorem of Calculus: $F(7) = F(2) + \int_2^7 f(t) \, dt = 0 + 7.5 = 7.5$

Final Answers:

• $F(2) = 0$
• $F(7) = 7.5$

Would you like more details or further clarification on any step?

Here are some related questions you might find useful:

1. How do you calculate the integral geometrically for non-linear curves?
2. What happens if the derivative function crosses the x-axis multiple times?
3. How does the Fundamental Theorem of Calculus apply to more complex functions?
4. Can the integral still be calculated if the graph is not piecewise linear?
5. How would you interpret negative areas in real-world applications?

Tip: Always pay attention to whether the area is above or below the x-axis when using the Fundamental Theorem of Calculus!