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\title{Higher-order derivatives}
\date{September 28, 2022}

\begin{document}
\ \vspace{-.3in}

\maketitle

\vspace{-.2in}

\prob{
  On the first worksheet we dealt with a mathematical model for the vertical position of a model rocket.
  The altitude (in meters) $t$ seconds after launch is given by
  \[
  h(t) = \begin{cases} 40t^2 & \text{ if } t \leq 2 \\ 160+160(t-2)-4(t-2)^2 & \text{ if } t > 2 \end{cases} 
  \]
  This is a piecewise function because the rocket engine stops 2 seconds into the flight, after which the rocket moves only under the influences of gravity and friction.

  \begin{enumerate}[a)]

  \item Calculate the derivative $h'(t)$ and interpret what you find.
    Does the derivative exist at $t=2$?
  
    \vfill

  \item Calculate the second derivative $h''(t)$ and interpret what you find.
    Does the second derivative exist at $t=2$?
    
    \vfill
    
    
  \item Calculate the third derivative $h^{(3)}(t)$ and interpret what you find.

    \vfill
    
  \end{enumerate}
}

\newpage

\prob{
  Real mathematical models for the absorption and metabolism of a drug are too complex for us right now, but we can work with a simplified example.
  Suppose the concentration of a intravenously inject drug (in mg/l) $t$ minutes after injection is given by
  \[
  C(t) = \frac{10t}{t^2+1}
  \]
  
  \begin{enumerate}[a)]
    
  \item Calculate the derivative $C'(t)$ and interpret what you find. What are the units of $C'(t)$?
    For which values of $t$ is $C'(t)$ positive? For which values is it negative?

    \vfill
    
  \item Use Desmos to plot the graph $y=C(t)$. Compare the features of the graph with your answers for part a.
    
    \vspace{.5in}
    
  \item Calculate the second derivative $C''(t)$. What are the units of $C''(t)$?

    \vfill 
    
  \end{enumerate}
}

\chal{
  Continue working with $C(t)$ from the last problem.
  Simplify $C''(t)$ and find the values of $t$ for which $C''(t)$ is positive and the values for which it is negative.
  Compare with the graph of $y=C(t)$.
  }

\end{document}
