If `p(t)=2+(t)/(2)+t^(2)-(t^(3))/(3)`, then `p(-1)` is

Options

A. `(15)/(6)`
B. `(1)/(6)`
C. `(17)/(6)`
D. `(13)/(6)`

Answer

To find \( p(-1) \) for the function \( p(t) = 2 + \frac{t}{2} + t^2 - \frac{t^3}{3} \), we will substitute \( t = -1 \) into the equation and simplify step by step.

### Step-by-Step Solution:

1. **Substitute \( t = -1 \) into the function**:
\[ p(-1) = 2 + \frac{-1}{2} + (-1)^2 - \frac{(-1)^3}{3} \]

2. **Calculate each term**:
- The first term is \( 2 \).
- The second term is \( \frac{-1}{2} \).
- The third term is \( (-1)^2 = 1 \).
- The fourth term is \( -\frac{-1}{3} = \frac{1}{3} \).

So, we can rewrite the equation:
\[ p(-1) = 2 - \frac{1}{2} + 1 + \frac{1}{3} \]

3. **Combine the constant terms**:
- Combine \( 2 + 1 = 3 \).
- Now we have:
\[ p(-1) = 3 - \frac{1}{2} + \frac{1}{3} \]

4. **Find a common denominator**:
- The common denominator for \( 2 \) and \( 3 \) is \( 6 \).
- Convert each term:
- \( 3 = \frac{18}{6} \)
- \( -\frac{1}{2} = -\frac{3}{6} \)
- \( \frac{1}{3} = \frac{2}{6} \)

5. **Combine the fractions**:
\[ p(-1) = \frac{18}{6} - \frac{3}{6} + \frac{2}{6} = \frac{18 - 3 + 2}{6} = \frac{17}{6} \]

### Final Answer:
\[ p(-1) = \frac{17}{6} \]