Problems

1. Use either the factoring technique or the expansion technique to explain how the Power Rule follows for arbitrary \(n\).
2. Use difference quotients to derive the Sum Rule.
3. It follows from your calculation of derivatives of polynomials that

   \(\displaystyle\frac{d}{dx} (mx+b)=m\).

   That is, the instantaneous rate of change of a linear function (at every \(x\)) is the same thing as its average rate of change, i.e., its slope. Explain why this is true.

4. Suppose \(f(t)\) is any function that has a derivative, and \(C\) is any constant. Show that the function \(g(t)= f(t)+C\) has the same derivative as \(f(t)\).
5. 1. Use the Graph tool to graph the curve \(y=1/t^2\).
   2. Zoom in on a representative selection of points, and estimate the derivative of \(1/t^2\) at those points.
   3. Use these estimations to formulate a rule for the derivative of \(1/t^2\).
   4. Check your rule by making graphical estimations of the derivative at two additional points.
6. Suppose the radius \(r\) of an inflating spherical balloon is given by the formula

   \(r(t)=2+t/2\) centimeters.

   1. What is the rate of change of the radius?
   2. Use your graphing tool to obtain graphical estimates of the rate of change of the surface area at \(t=2\), \(3\), and \(4\) seconds. Formulate a general rule for the rate of change of the surface area, and check it at two additional points.
   3. Use your graphing tool to obtain graphical estimates of the rate of change of the volume at \(t=2\), \(3\), and \(4\) seconds. Formulate a general rule for the rate of change of the volume, and check it at two additional points.
   4. Is this a realistic model for an inflating balloon? Why or why not?