Exercises

1. Differentiate each of the following functions.

   `ftext[(]t text[)]=t^3` `ftext[(]t text[)]=e^(3t)` `ftext[(]t text[)]=3-e^t` `ftext[(]t text[)]=1-e^(3t)` `ftext[(]t text[)]=t-e^(3t)`

   `ftext[(]t text[)]=t^3-e^t` `ftext[(]t text[)]=t^3-3^t` `ftext[(]t text[)]=t^3-3t^2+1` `ftext[(]t text[)]=e^(3t)-3t^2+3t` `ftext[(]t text[)]=3+7t-2t^2+3t^3`

2. Which of the following functions are solutions of the differential equation `dy text[/] dt=4y`?

   `ftext[(]t text[)]=e^(4t)` `ftext[(]t text[)]=e^(2t)`

   `ftext[(]t text[)]=2t^2` `ftext[(]t text[)]=5e^(4t)`

   `ftext[(]t text[)]=e^(4t)+5` `ftext[(]t text[)]=2t^2+3`

3. Which of the following functions are solutions of the differential equation `dy text[/] dt=4t`?

   `ftext[(]t text[)]=2t^2` `ftext[(]t text[)]=2t^2+3`

   `ftext[(]t text[)]=3t^2` `ftext[(]t text[)]=5e^(4t)`

   `ftext[(]t text[)]=e^(4t)` `ftext[(]t text[)]=2+2t^2`

4. Which of the following functions is the solution of the initial value problem `dy text[/] dt=4t` with `y=4` at `t=1`?

   `ftext[(]t text[)]=2t^2` `ftext[(]t text[)]=2t^2+1`

   `ftext[(]t text[)]=2t^2+4` `ftext[(]t text[)]=4e^(4t)`

   `ftext[(]t text[)]=3+2t^2` `ftext[(]t text[)]=2+2t^2`

5. Which of the following functions is the solution of the initial value problem `dy text[/] dt=4t` with `y=4` at `t=0`?

   `ftext[(]t text[)]=2t^2` `ftext[(]t text[)]=2t^2+1`

   `ftext[(]t text[)]=2t^2+4` `ftext[(]t text[)]=4e^(4t)`

   `ftext[(]t text[)]=3+2t^2` `ftext[(]t text[)]=2+2t^2`

6. Which of the following functions is the solution of the initial value problem `dy text[/] dt=4y` with `y=3` at `t=0`?

   `ftext[(]t text[)]=e^(4t)` `ftext[(]t text[)]=2e^(4t)`

   `ftext[(]t text[)]=3e^(4t)` `ftext[(]t text[)]=4e^(4t)`

   `ftext[(]t text[)]=3+2t^2` `ftext[(]t text[)]=3t^2`

7. Which of the following functions is the solution of the initial value problem `dy text[/] dt=4y` with `y=4` at `t=0`?

   `ftext[(]t text[)]=e^(4t)` `ftext[(]t text[)]=2e^(4t)`

   `ftext[(]t text[)]=3e^(4t)` `ftext[(]t text[)]=4e^(4t)`

   `ftext[(]t text[)]=3+2t^2` `ftext[(]t text[)]=3t^2`

8. Find three solutions of each of the following differential equations.

   `(dy)/(dt)=7y` `(dy)/(dt)=7t`

   `(dy)/(dt)=1.07y` `(dy)/(dt)=1.07t`

9. Find the solution of each of the following initial value problems.

   `(dy)/(dt)=7y` with `y=2` at `t=0` `(dy)/(dt)=7t` with `y=2` at `t=0` `(dy)/(dt)=1.07y` with `y=2.3` at `t=0` `(dy)/(dt)=1.07t` with `y=2.3` at `t=0`

10. Table E1 shows the number of disintegrations per minute counted by a Geiger counter that is testing a sample of radioactive barium-137. The disintegration rate of a radioactive substance is proportional to the amount of the substance not yet disintegrated.

    1. Find a formula for a function that approximately fits the data.
    2. Find the half-life of barium-137, that is, the time it takes for a given sample to decay to half the original amount.

    Table E1   Disintegrations per minute
    for radioactive barium-137

    Time (minutes)	Counts per minute
    0	10,034
    1	8105
    2	5832
    3	4553
    4	3339
    5	2648
    6	2035

    
    This problem is adapted from the Core-Plus Mathematics Project.