1. Mathematical Models How mathematics is used to simulate some aspect of the world

2. Mathematical Models Mathematical models can be Formulas Functions Data Graphs Geometric figures

3. Mathematical Models: Handshake For example, remember than handshake problem: If n people shake hands with each other, how many handshakes occur? This is a geometry model.

4. Mathematical Models: Handshake Another model for the handshake is H(n) = n = # of people n-1 = # of people each person shakes hands with Division by 2, because each handshake is otherwise counted twice This is a quadratic function as a model.

5. Mathematical Models: Handshake A more advanced way of thinking abut the number of handshakes is to use combinatorics. We will learn about permutations and combinations, which involves some repeated multiplication in a process called factorials.

6. Mathematical Models The process of building a mathematical model is to think about the question that is asked, perhaps produce some data to simulate the phenomena, and then decide which function, formula, or geometric figure best matches the real world situation.

7. Mathematical Models The process of building a model is accomplished in several steps. I = identify the problem B = build a simple model S = simulate the situation with the model E= evaluate how well the model matches the real situation then….. upgrade the model to make it more complex but better simulate and match the real situation.

8. Mathematical Models We will look at some of these situations with our mathematical functions: 1. Musical composition (and function transformations) 2. Ferris wheels ( and trig functions) 3. Heating costs 4. Parking garage charges

9. Mathematical Models 5. Unit and marginal costs of factory production and publishing 6. Predator-prey populations 7. Growth of one species (i.e., deer) 8. Income tax obligations 9. Telephone costs 10. Medication levels (for chronic illness)