This Knowledge Check must be done with YOUR OWN WORK! SPRING 2023 According to Newton's Law of Warming and Cooling, if an object at temperature T is immersed in a medium having the constant temperature M, then the rate of change of T is proportional to the difference of temperature, T - M, provided that this difference is not too large. A cup of hot tea initially at 98"C while sitting in a room temperature 18"C cools at a rate of 1"C per minute when its temperature is 68C. Note that you need to use this specific case in the following questions. (A) Find the sign of the proportionality constant and explain why. (solution) (B) Find the proportionality constant and its units. You are supposed to use your proportionality constant in your answers afterwards. (solution) (C) Let T'(t) be the temperature of the tea at time t. Write (Do NOT solve) an IVP that describes what happens to the tea's temperature and that can be solved to find T(t). (solution) (D) Find an equilibrium solution(s) if there is any. Show your work. (solution) (E) What is the limiting value of the tea's temperature? Explain why it makes sense to you. (solution) (F) Can T(t) be 18C? Explain why or why not. (solution) (G) Circle the one(s) in the following list that describes the DE you wrote in (C) above. (solution) Normal Form, Standard Form, General Form, Differential Form, Autonomous, Second-order, Separable, Linear, ODE, PDE This is an assessment that must be done with YOUR OWN WORK! Page 2 of 5This Knowledge Check must be done with YOUR OWN WORK! SPRING 2023 (H) If there is any, find a region in the tT-plane where a solution T' = (t) of the DE decreases at an increasing rate. You are not supposed to find the solution of the DE to answer this question. Show your work and answer the question without using the solution. (solution) (1) Use the isoclines method with at least 5 isoclines to sketch a direction field for the differential equation. (solution) (J) Use the phase line method to sketch a direction field for the differential equation and sketch the solution curve for the initial value problem on the direction field. (solution) graph of 7" on the TT'-plane phase line direction field and solution curve (K) Determine whether the equilibrium solution you found in (D) is stable or not. (solution) (L) Discuss about Existence and Uniqueness Theorem of the IVP. (solution) This is an assessment that must be done with YOUR OWN WORK! Page 3 of 5