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To solve for \(F(2)\) and \(F(7)\) given that \(F(t)\) has the derivative \(f(t)\) shown in the graph and \(F(0) = 5\), we need to integrate \(f(t)\) over the appropriate intervals. This involves finding the area under the curve of \(f(t)\) from \(0\) to \(2\) and from \(0\) to \(7\).

First, let’s analyze the graph of \(f(t)\):

1. From \(t = 0\) to \(t = 1\), \(f(t)\) is a linear function that increases from 0 to 3.
2. From \(t = 1\) to \(t = 3\), \(f(t)\) is a linear function that decreases from 3 to 0.
3. From \(t = 3\) to \(t = 4\), \(f(t)\) is a linear function that decreases from 0 to -3.
4. From \(t = 4\) to \(t = 7\), \(f(t)\) is constant at -3.

Now, let's find the areas under the curve of \(f(t)\) from \(t = 0\) to \(t = 2\) and from \(t = 0\) to \(t = 7\).

### For \(F(2)\):
1. **From \(0\) to \(1\)**: 
   - The area is a triangle with base 1 and height 3:
   \[ \text{Area} = \frac{1}{2} \times 1 \times 3 = 1.5 \]

2. **From \(1\) to \(2\)**: 
   - The area is a trapezoid with bases 3 and 0, and height 1:
   \[ \text{Area} = \frac{1}{2} \times (3 + 0) \times 1 = 1.5 \]

Adding these areas:
\[ F(2) = F(0) + \text{Area from } 0 \text{ to } 2 = 5 + (1.5 + 1.5) = 5 + 3 = 8 \]

### For \(F(7)\):
1. **From \(0\) to \(1\)**:
   - The area is the same as above:
   \[ 1.5 \]

2. **From \(1\) to \(3\)**:
   - The area is a trapezoid with bases 3 and 0, and height 2:
   \[ \text{Area} = \frac{1}{2} \times (3 + 0) \times 2 = 3 \]

3. **From \(3\) to \(4\)**:
   - The area is a triangle with base 1 and height -3:
   \[ \text{Area} = \frac{1}{2} \times 1 \times (-3) = -1.5 \]

4. **From \(4\) to \(7\)**:
   - The area is a rectangle with base 3 and height -3:
   \[ \text{Area} = 3 \times (-3) = -9 \]

Adding these areas:
\[ F(7) = F(0) + \text{Total Area from } 0 \text{ to } 7 = 5 + (1.5 + 3 - 1.5 - 9) = 5 - 6 = -1 \]

Thus:
\[ F(2) = 8 \]
\[ F(7) = -1 \]

Would you like a detailed explanation of each step, or do you have any specific questions?

Here are some related questions you might find helpful:

1. How do you find the area under a curve using integration?
2. What are the properties of definite integrals?
3. How do you handle areas below the x-axis in definite integrals?
4. What is the Fundamental Theorem of Calculus?
5. How do you interpret the graph of a derivative function?
6. Can you explain how to integrate piecewise linear functions?
7. What is the relationship between a function and its derivative?
8. How do you find the definite integral of a function with respect to its limits?

**Tip:** When integrating a function graphically, remember to account for areas above and below the x-axis, as areas below contribute negatively to the total integral.

Learn how to calculate F(2) and F(7) using definite integrals and the Fundamental Theorem of Calculus. Understand the area under the curve of f(t) graphically, with step-by-step solutions. Suitable for university-level students.

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