Assignment 6: Differential Equations with Two Important Applications

Week 6

In class on Monday, [day 15]:

·        Section 3.1: Differential Equations and Initial Values

o       In class on Monday, I gave an overview of Section 3.1.  This section is building on ideas that were first introduced in Chapter 2.  We are looking at two different families of functions: power functions (functions of the form ) and exponential functions (functions of the form  ).

o       Up to now, solving a mathematics problem has usually meant finding a number that makes a particular equation true.  However, solving a differential equation means finding a function (not just a number) that fits in the given problem situation.  Recall that back in Section 2.5.2, the authors give us a differential equation, and showed us one function that solves this equation (the function P₁) and another function that did not solve the given equation (the function P₂).  First they show why P₁ works, and then they show why P₂ does not work.

o       Each differential equation gives us a whole family of functions.  Additional information about the value of the function at a particular value of the independent variable – the initial value – makes it possible to pick out the one solution from this family of functions that applies in the particular situation.

o       Slope fields give us an additional tool for visualizing what is going on.  In the slope field we can “see” or visualize the family of functions.  As we become more comfortable in reading the slope fields, we can use this tool to help us verify the solution to a differential equation.

By class on Wednesday, [day 16]:

·        Your group work on the Project is due.  Each group is to turn in one copy of the CAS print-out for the first four parts of the project (through Checking Your Model Function). 

In class on Wednesday, [day 17]:

·        You will have time for questions about the Project, and your individual paper.  Your individual paper will address the questions in the last section, Using Your Model Function.

·        Section 3.2: An Initial Value Problem: A Cooling Body

I will give an overview of Section 3.2, making connections to the introduction to differential equations that was given in Section 3.2.

·        Then, you are to work through the Activities and Checkpoints of Sections 3.1 and 3.2 in your groups.

In class on Friday, [day 18]:

·        Your individual paper for the Project is due.  See the separate handout on the Project for particulars about the project and how it will be graded.

·        Section 3.3: Another Initial Value Problem: A Falling Body

I will give an overview of Section 3.3.  Then you will have time in your groups to continue working through the Activities and Checkpoints of Chapter 3. 

·        We will go back through Chapter 3 – reviewing the two-step process for solving differential equations, and looking at the two families of differential equations presented in Chapter 3.

·        The Benchmark will be given during the last 30 minutes of class.  The Study Guide is now available.   (Look in the Study Guides folder under Course Materials.)