Exam 1 Take-Home Test Problems

Exam 1 Take-Home Test Problems

PROBLEM 1: EULER'S METHOD PROBLEM (Take-Home Test Problem 1): Solve the differential equation given below analytically, finding the particular solution satisfying the given condition. Approximate the solution using Euler's Method over the interval [0,4pi] with a step size of pi/8. Discuss the nature of the accuracy of Euler's Method as applied to this problem. If the Euler solutions are too high or too low at some sub-interval and then seem to become more accurate explain why this happened. Construct a chart of values and draw a graph showing both the analytical solution and the discrete points depicting the Euler's Method solution over the interval [0,4pi]. In the 2147 problem x_o= 0, y_o = 2, and h = pi/8. Show the computations involved in finding y₁ and y₂. Click here to see a development of Euler's Method and some examples. Here is a very nice Euler's Method applet by David Protas of California State University that will both draw a graph of the Euler solution and generate a table of values.

2147 PROBLEM: dy/dx = x/3 + 4cos(x) y(0) = 2

Bonus: Compute the table of values and draw the graphs using a step size of h = pi/16.

Bonus: Compute the table of values and draw the graphs using the Improved Euler Method shown in class and a step size of h = pi/8.

Exact and Euler Solution Graphs using Excel for the Term 2041 take-home problem (dy/dx = 6x² - 24x + 22 y(0) = 1) can be found by following the link.

PROBLEM 2: Exam I Parachute Take-Home Test Problem

Click here for a sample test problem on parachuting including the development of the velocity function and a few hints and helpful graphs and a link to the solution.

PROBLEM 3:

EXAM I MIXTURE TEST PROBLEM: A tank contains 100 gallons of salt water with 40 pounds of salt in the water (total). Pure water begins entering at the rate of 2 gallons per minute and the well-stirred mixture leaves at the same rate. How long until the amount of salt in the tank is 20 pounds?

PROBLEM 4: EXAM I HEAT SEEKING PARTICLE TAKE-HOME TEST PROBLEM--Find the path followed by a heat-seeking particle in an xy-coordinate plane with the temperature at any point in the plane given by the temperature function T given below if the particle starts at the point (2.5,2.5) and also if it starts at the point (-1,2.5). Show the steps involved in solving the two differential equations whose solutions lead to a parametric representation of the path. I will explain the Maple worksheet in class. You only need to derive the parametric representation of each path shown in the analytical solutions on the Maple worksheet.

T(x,y) = 100 - 10x² - y⁴ Maple Worksheet Powerpoint Presentation (Different Temperature Function)

Extra Credit Problem--Link

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Lane Vosbury, Mathematics, Seminole State College email: vosburyl@seminolestate.edu