Exercises

1. In this exercise, `ftext[(]t text[)]` is the reciprocal function, `ftext[(]t text[)]=1text[/]t`.

   1. Calculate the difference quotient `Delta ftext[/]Delta t` algebraically for an arbitrary `Delta t`.
   2. Simplify your algebraic expression for `Delta ftext[/]Delta t` until the `Delta t` in the denominator cancels. What does the difference quotient approach as `Delta t->0`?
   3. What is the derivative of the reciprocal function?

2. In this exercise, `ftext[(]t text[)]` is the cubing function, `ftext[(]t text[)]=t^3`.

   1. Calculate the difference quotient `Delta ftext[/]Delta t` algebraically for an arbitrary `Delta t`.
   2. Simplify your algebraic expression for `Delta ftext[/]Delta t` until the `Delta t` in the denominator cancels. What does the difference quotient approach as `Delta t->0`?
   3. What is the derivative of the function `ftext[(]t text[)]=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`. Adapt your computer algebra tool or use the zooming feature on your calculator to determine the values of `dstext[/]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`.