1. Consider the ODE model for Newton's Law of Cooling. We have some object that has a temperature of T degrees F after t minutes of being placed into an ambient environment that has a constant To degrees F (such as a room). We know that the rate of temperature change is negatively proportional to the difference between the temperature of the object and the ambient temperature. More specifically, dT dt -k(T-To) Suppose that k = 0.1 for an uninsulated mug and To = 78 degrees F in a room. Assume that the initial temperature of a hot cup of coffee placed into this room is 165 degrees F. a. Solve the equation using separation of variables and find the arbitrary constant. b. Create a table of temperature over the first 5 minutes of cooling showing the exact temperature of the cup of coffee in At = 1 minute increments (using the exact solution you came up with). c. Use Euler's Method (the original one) to estimate the temperature of the cup of coffee over the first 5 minutes using At = 5. d. Use Euler's Method (the original one) to estimate the temperature of the cup of coffee over the first 5 minutes using At = 1. e. Use Euler's Method (the original one) to estimate the temperature of the cup of coffee over the first 5 minutes using At = 0.5. f. Using the y estimates at t = 5 for Euler's at At = 5, At = 1, At = 0.5, compute the relative error of the values (as percentage error), as compared to the exact value of y at time t = 5 (from your table in b.). What do you notice about the precision of these estimates as step size decreases? g. Using Desmos, create a graph of the exact solution, and create tables of data for each of the step sizes (click the "+", insert a table, type in the values of the inputs and outputs, and they should be plotted on the graph.). Does the graph also support the idea that smaller step sizes get you closer to the exact solution? h. What do you notice are the pros and cons of using smaller step sizes? What is bound to happen "in the long run" regardless of the choice of step size?