Exercises

1. In this exercise, \(f(t)\) is the reciprocal function, \(f(t)=1/t\).

   1. Calculate the difference quotient \(\Delta f/\Delta t\) algebraically for an arbitrary \(\Delta t\).
   2. Simplify your algebraic expression for \(\Delta f/\Delta t\) until the \(\Delta t\) in the denominator cancels. What does the difference quotient approach as \(\Delta t \to 0\)?
   3. What is the derivative of the reciprocal function?

2. In this exercise, \(f(t)\) is the cubing function, \(f(t)=t^3\).

   1. Calculate the difference quotient \(\Delta f/ \Delta t\) algebraically for an arbitrary \(\Delta t\).
   2. Simplify your algebraic expression for \(\Delta f/ \Delta t\) until the \(\Delta t\) in the denominator cancels. What does the difference quotient approach as \(\Delta t \to 0\)?
   3. What is the derivative of the function \(f(t)=t^3\)?
3. If the position \(s\) of a falling body at time \(t\) is given by \(s=c t^2\), where \(c=4.90\), find the instantaneous speeds at the following times.

   a.  \(t=3\)

   b.  \(t=2.34\)

   c.  \(t=\sqrt{7}\)

4. 
   Suppose \(s=3t^2+5\). Use our Graph tool or the zooming feature on your calculator to determine the values of \(ds/dt\) at each of the following times.

   a.  \(t=0.5\)

   b.  \(t=4\)

   c.  \(t=7\)
5. If \(a\) and \(b\) are constants, find a formula for the derivative of \(y=at+b\).
6. If \(a\), \(b\), and \(c\) are constants, find a formula for the derivative of \(y=at^2+bt+c\).

In Exercises 7–12, use a graphical approach to approximate the derivative of the function at \(t=1\) and \(t=2\).