Exercises

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

   `t^2e^t` `text[(] t^3+t text[)]e^(-3t)` ` text[(] t-3 text[)]e^(2t)`

   `x^3e^x` ` text[(] x^2-x text[)]e^(2x)` ` text[(] x+2 text[)]e^(-x)`

   `2^x text[(] x^2+1 text[)]` `7t^3-5t^2+13t-sqrt(2)` `(3/5)^t`

2. Calculate the second derivative of each of the following functions.
   

   `t^2e^t` `text[(] t^3+t text[)]e^(-3t)` ` text[(] t-3 text[)]e^(2t)`

   `x^3e^x` ` text[(] x^2-x text[)]e^(2x)` ` text[(] x+2 text[)]e^(-x)`

   `2^x text[(] x^2+1 text[)]` `7t^3-5t^2+13t-sqrt(2)` `(3/5)^t`

3. Calculate each of the following derivatives.

   `d/(dx)xe^x`

   `d^2/(dx^2)xe^x`

   `d/(dy)text[(]y+1text[)]e^(-2y)`

4. Given `y=text[(]1+t^2text[) (]t^3-3t^2+1text[)]`, calculate `dytext[/]dt` two ways:
   1. using the Product Rule.
   2. multiplying the two factors before differentiating.
   
   Note that the two answers you get should be equivalent.
5. 1. Calculate `dytext[/]dt`, where `y=text[(]1+2t text[)]e^(-3t)`.
   2. Check your formula for the derivative by calculating `Delta ytext[/]Delta t` at `t = 1` with `Delta t=0.01`.
6. When you cough, your windpipe contracts. The velocity `v` at which air comes out depends on the radius `r` of your windpipe. If `R` is the normal (rest) radius of your windpipe, then (for each possible radius `r`), `v=alphatext[(]R-r text[)]r^2`, where `alpha` is a constant.
   
   
   1. What value of `r` maximizes the velocity?
   2. What is that maximum velocity?
   
   
   Both answers may involve either or both of the constants `alpha` and `R`.
7. Your local pizza delivery uses square boxes that are made from rectangular pieces of corrugated cardboard, each 47 by 90 cm. The box is made by cutting out six small squares, three from each of the 90-cm edges, one square at each corner and one in the middle of the edge. The result is then folded into a box in the obvious fashion. (Even kids do it.)
   1. Find the largest volume for a box designed in this fashion.
   2. What is the length of the side of one of the four squares cut out when making the the box of maximum volume in (a)?