1. a. Check that y = ½x2 + x + 3 satisfies the differential equation dy/dx = x + 1.
b. Check that y = sin t satisfies the differential equation d2y/dx2 = -y.
2. Sketch a slope field corresponding to the equation dy/dt = 2t - y. Then sketch the solution that passes through the point (1,2).
3. Sketch a curve y = f(x) that satisfies: f(0) = 3, and dy/dx = -2 Then determine f(x).
4. A curve y = f(x) has the property that the slope of tangent to the curve at every point is reciprocal to the y value of the curve. Write down a differential equation whose solution is y = f(x). Then show verification that y = √(2x + C) and y = - √(2x + C) satisfy this differential equation.
5. Determine the time it takes an amount of money to triple if compounded continuously at an annual rate of 15%.
6. I got a Guinness can whose temperature is 21o C. To enjoy it cold at the perfect temperature of 9oC, I set my refrigerator to 0oC. The temperature T(t) of the can at time t measured in seconds satisfies Newton's Cooling Law equation:
dT/dt = -.000iT.
How long will it take for the can to cool off to 9oC?
7. A cup of tea with initial temperature of I = 85oC is left in the room a constant temperature A = 21oC. In one minute the tea cools down to 84Cdegree. What will be the temperature of the tea after an hour? When will the temperature of the tea be 22o?