DATA observations experiments and measurements Real world phenomenon

DATA observations, experiments, and measurements. Real-world phenomenon Qualitative predictions Quantitative predictions Re fi ne model Determine dependent and independent variables Assign symbols to variables, Choose sensible units of measurement for variables Apply principles, laws and assumptions Mathematical inferences Analysis Computing Graphics Mathematical representation A diagram of the Modeling Process Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Portraits - 1D — (5/50) Mathematical Modeling Introduction to MatLab Qualitative Behavior of Differential Equations More Examples Maple - Direction Fields Solution of Linear Growth and Decay Models Mathematical Modeling Newton’s Law of Cooling Murder Investigation Linear Differential Equation Newton’s Law of Cooling 1 Newton’s Law of Cooling: After a murder (or death by other causes), the forensic scientist takes the temperature of the body Later the temperature of the body is taken again to find the rate at which the body is cooling Two (or more) data points are used to extrapolate back to when the murder occurred This property is known as Newton’s Law of Cooling Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Port — (6/50) Mathematical Modeling Introduction to MatLab Qualitative Behavior of Differential Equations More Examples Maple - Direction Fields Solution of Linear Growth and Decay Models Mathematical Modeling Newton’s Law of Cooling Murder Investigation Linear Differential Equation Newton’s Law of Cooling 2 Newton’s Law of Cooling states that the rate of change in temperature of a cooling body is proportional to the difference between the temperature of the body and the surrounding environmental temperature If T ( t ) is the temperature of the body, then it satisfies the differential equation dT dt = - k ( T ( t ) - T e ) with T (0) = T 0 The parameter k is dependent on the specific properties of the particular object (body in this case) T e is the environmental temperature T 0 is the initial temperature of the object Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Portraits - 1D — (7/50) Mathematical Modeling Introduction to MatLab Qualitative Behavior of Differential Equations More Examples Maple - Direction Fields Solution of Linear Growth and Decay Models Mathematical Modeling Newton’s Law of Cooling Murder Investigation Linear Differential Equation Murder Example 1 Murder Example Suppose that a murder victim is found at 8:30 am The temperature of the body at that time is 30 ◦ C Assume that the room in which the murder victim lay was a constant 22 ◦ C Suppose that an hour later the temperature of the body is 28 ◦ C Normal temperature of a human body when it is alive is 37 ◦ C Use this information to determine the approximate time that the murder occurred Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Port — (8/50)