Second Midterm-Practice Test
You know the drill: Work these problems by yourself, and at some point start discussing the solutions with other students. In case of doubt, feel free to drop a comment and I will guide you with hints in the right direction. If you feel like it, drop also in the comments the answers that you obtained, so other students can check. Good luck!
- Plot a slope field for the following differential equations, and use it to indicate the stability or instability of their equilibria.
- The time rate of change of an alligator population in a swamp is proportional to the square of The swamp contained a dozen alligators in 1988, two dozen in 1998. When will there be four dozen alligators in the swamp? What happens thereafter?
- Suppose that the fish population in a lake is attacked by a disease at time with the result that the fish cease to reproduce and the death rate is thereafter proportional to If there were initially 900 fish in the lake and 441 were left after 6 weeks, how long did it take all the fish in the lake to die?
- Consider an animal population with constant death rate (deaths per animal per month) and with birth rate proportional to Suppose that and When is When does doomsday occur?
- The skid marks made by an automobile indicated that its brakes were fully applied for a distance of before it came to a stop. The car in question is known to have a constant deceleration of under these conditions. How fast (in was the car traveling when the brakes were first applied?
- At noon a car starts from rest at a point A and proceeds with constant acceleration along a straight road toward point C, 35 miles away. If the constantly accelerated car arrives at C with a velocity of at what time does it arrive at C?
- A ball is dropped from the top of a building hight. How long does it take to reach the ground? With what speed does the ball strike the ground?
- Francisco bails out of an airplane at an altitude of falls freely for 20 seconds, then opens his parachute. How long will it take him to reach the ground? Assume linear air resistance taking without the parachute and with the parachute.
- Apply both Euler and Improved Euler methods to solving numerically the differential equation with initial condition in the interval Use a time-step Prepare a table showing four-decimal-place values of the approximate solution and the actual solution at the points
- [And the famous punch-line: If you are able to finalize in about 45 minutes, and are able to explain it to someone else, I consider that you have mastered this part of the course.] A tumor may be regarded as a population of multiplying cells. The “birth rate” of the cells in a tumor decreases exponentially with time, so that where and are positive constants.
- [Find the differential equation] What is the corresponding population model?
- [Let’s find the values of and for a particular example] Suppose that at time there are cells and that is then increasing at the rate of cells per month. After 6 months the tumor has doubled (in size and in number of cells).
- [Use a slope field for this part, to support your claims] Find the limiting population of the tumor in the previous part.
- [Numerical Approximation] Use the improved Euler’s method to approximate the solution for this particular problem for the interval Use a time step