Exercises

1. Calculate the derivative of each of the following functions.

   `ftext[(]t text[)]=e^(4t)`	`utext[(]t text[)]=t+1`
   `gtext[(]t text[)]=e^(-3t)`	`vtext[(]t text[)]=t^3+t^2+t+1`
   `htext[(]t text[)]=3^t-t^3`	`wtext[(]t text[)]=t^2+t+1/t`

2. For each of the following functions, use your calculations in the corresponding part of Exercise 1 to identify the range of values of `t` for which the graph is increasing, the range of values of `t` for which the graph is decreasing, and all local maxima and local minima.

   `ftext[(]t text[)]=e^(4t)`	`utext[(]t text[)]=t+1`
   `gtext[(]t text[)]=e^(-3t)`	`vtext[(]t text[)]=t^3+t^2+t+1`
   `htext[(]t text[)]=3^t-t^3`	`wtext[(]t text[)]=t^2+t+1/t`

3. Water is the only common liquid whose greatest density occurs at a temperature above its freezing point. (This phenomenon favors the survival of aquatic life by preventing ice from forming at the bottoms of lakes.) According to the CRC Handbook of Chemistry and Physics (now in its 87th edition), a mass of water that occupies one liter at 0°C occupies a volume of `1+aT+bT^2+cT^3` liters at `T`°C, where `0<=T<=30`, and where the coefficients are
         `a=-6.42*10^(-5)`,
         `b=8.51*10^(-6)`, and
         `c=-6.79*10^(-8).
   Find the temperature between 0 and 30°C at which the density of water is greatest.
   
4. Find the number `x` between 0 and 4 where `ftext[(] x text[)]=x^3-3x^2+9x+5` is the smallest.
5. Let `Ptext[(] x text[)]=x^4-9x^3+20x^2+1`.
   1. Find the maximum and minimum values of `P` on the interval `[0,4]`.
   2. Find the maximum and minimum values of `P` on the interval `[0,5]`.
   3. Find the maximum and minimum values of `P` on the interval `[0,6]`.
6. Suppose a can (a right circular cylinder) is to be made to hold 63 cubic inches of Blastola Cola. The material for the sides and bottom costs 0.12¢ per square inch, and the material for the easy-open top costs 0.5¢ per square inch.

   1. What height and radius will minimize the cost of the material?
   2. What is the minimum cost?
7. Find formulas for the radius and height of the least expensive can in Exercise 6 if the top costs `d`¢ per square inch more than the sides and bottom. (In Exercise 6, `d` was 0.38.)
8. Suppose the value of `d` in Exercise 7 is 0.7; i.e., the easy-open top costs 0.92¢ per square inch. Suppose also that design considerations restrict the radius to lie between 1.4 and 1.7 inches.
   1. What should the height and radius of the can be to minimize the cost of the material?
   2. What is the minimum cost?