**Calculus Applications Project: Analyzing Age vs. Number of Jobs held**

For this project, you will collect data relating age and # of jobs held. You will try several mathematical models for curve fitting, and take the first and second derivatives of these models. You will provide a graph for each model. Then you will select the model that you think is the best fit.

*Collect data*

You will interview at least 12 or more people ranging in age from 18 to 65. Ask each of them how many jobs they have held so far. Display the data in a chart where x represents age and y represents the number of jobs held.

*Use that data and enter it in the website:*

Use the age for the x input and number of jobs for the y input.

Enter ages for the x data and the number of jobs for the y data. Remember, you are entering pairs. For example, if your data were (30,2), (20,4), (53,1) the you would list for the x

*30 20 53 * notice there is a space for each point.

And for y

*2 4 1* notice that you have to list them in order to match the way the x values are listed.

In the next part you select which regressions you want to do; select the first three.

You will want only the first three items checked.

When you hit the orange Calculate button on the right you will see something similar to this:

The only information you need from this are the three equations generated from linear, quadratic and cubic regressions.

Following this you will see the graph of these generated curves:

III. To turn in: the report

- Create a table to show your data with x in order of age one column and the corresponding y in the other:
- List the linear regression equation you got from the website. Explain what the slope means in terms of age and number of jobs. Explain what the y intercept means in terms of the number of jobs when the age is zero.
- Then take the derivative. Set it equal to zero and find the critical points; if it does not exist then state that. Take the second derivative.
- For the quadratic and cubic equations list the equations, take their first and second derivatives.
- For the quadratic only find the critical points for both derivatives. For the cubic find the critical points only for the second derivative.
- Include a graph of all three original regression equations. For the quadratic, look at the critical approximate location of the critical point from the first derivative. Write about what you think the graph is doing there (just a sentence or two). Now find the approximate location of the critical point for the second derivative and state what you think the graph is doing there. Finally, do the same thing for the second derivative of the cubic graph.
- State which of the three equations you have, linear, quadratic or cubic, best fits the original data and why you think this.
- Look under the Project button in Canvas to submit the project.