1. Solve the following initial value problems (a) y'ysin0. y(0)1/2 2 2. Suppose y(r) is the solution to the initial value problem Use Euler's method (step size 0.1) to approximate y(0.5). estimate In(2) and use Taylor's inequality to give bounds on the error proportional to the difference in temperature between the object and its surroundings 3. Use the third degree Taylor polynomial (centered at zero) for f(x)-ln(1 +ã¨) to 4. Recall Newton's law of cooling: the rate of change in temperature of an object is dT=k(T-T.) dt where T(t) is temperature as a function of time, k is the proportionality constant, and T, is the constant surrounding temperature Suppose a cup of coffee is 200°F when it is poured and has cooled to 190°F after one minute in a room at 70FF. When will the coffee reach 150°F? What will the temperature of the coffee be after it sits for 30 minutes? 5. The following variation on the logistic equation models logistic growth with constant harvesting: dP dt = kP(1-PM)-c. For this problem consider the specific instance -008P( 1-P/1000) _ 15. dt modeling fish population in a pond where 15 fish per week are caught (time t in weeks) (a) What are the equilibrium solutions to the differential equation in part (i.e. what are the constant solutions)? (b) Find the general solution of the differential equation. Integrate using partial fractions. You should get P(t)- where C is an arbitrary constant (c) Find and interpret the solutions with initial conditions P(0) = 200.300.