1 Point
Suppose $f(t)$ is the temperature (in degrees Celsius) of a mysterious substance that you are studying in a lab after $t$ hours of observation. You measure the value of $f(t)$ in hourly intervals and record them in the following table:
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline$t$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline$f(t)$ & 4 & 6 & 14 & 11 & 7 & 16 & 18 & 14 & 17 & 5 \\
\hline
\end{tabular}
Determine the exact value of the definite integral
\[
\int_{3}^{9} f^{\prime}(t) d t
\]
Please be aware that the function in the integral is $f^{\prime}$, which is the derivative of $f$. You may assume that $f$ is differentiable and $f^{\prime}$ is continuous for all values of $x$.

Step 1 ：The Fundamental Theorem of Calculus states that if a function \(f\) is continuous on the interval \([a, b]\) and \(F\) is an antiderivative of \(f\) on the interval \([a, b]\), then \(\int_{a}^{b} f(t) dt = F(b) - F(a)\)

Step 2 ：In this case, we are asked to find the value of the definite integral \(\int_{3}^{9} f'(t) dt\)

Step 3 ：Since \(f'(t)\) is the derivative of \(f(t)\), we can say that \(f(t)\) is an antiderivative of \(f'(t)\). Therefore, we can apply the Fundamental Theorem of Calculus to find the value of the integral: \(\int_{3}^{9} f'(t) dt = f(9) - f(3)\)

Step 4 ：From the table, we know that \(f(3) = 14\) and \(f(9) = 17\). Substituting these values into the equation, we get \(\int_{3}^{9} f'(t) dt = 17 - 14 = 3\)

Step 5 ：So, the exact value of the definite integral is \(\boxed{3}\)