MA241-012, Fall 2008Test 3 review sheetDisclaimer: “This review sheet is only intended to give you some guidance, and thus should not be taken asan exhaustive listing of possible test question topics. Do not expect this every time.”Calculators will not be allowed on this test. No notes or other aids will be allowed on this test.The test will cover §§7.3-4, 7.7-9 (from the supplement), and 8.1.General Comments: Since I am not allowing calculators on this test, there will [obviously] not be anyproblems where a calculator is actually needed. Thus, in a mixing or Newton’s Law of Co oling problem, youwon’t have to plug any values of t into the solution or solve for an explicit value for t. However, for Newton’sLaw of Cooling, remember that it is still possible to solve for k and get a solution to the problem as long asI give you enough information about the rate at which the object is cooling (similar to problem 7.4.16).In problems where you have to use the quadratic equation, any square root will evaluate to an integer(or an integer times i, in the b2− 4ac < 0 case). So, your final answers should not have any square rootsin them; however, the coefficients (c1and c2) may still involve e or π, not just integers or fractions. Finally,when trying to find the roots of the auxiliary equation, keep in mind that they may not be something thatyou can easily just look at and see, so if you try to factor without using the quadratic equation, double checkto be sure you’ve factored correctly.§7.3You should be able to recognize and solve a separable DE and also apply any given initial or boundaryconditions. You need to be able to find the orthogonal trajectories to a given family of curves (unless toldotherwise, an implicit solution is sufficient). You need to be able to set up and solve a mixing problem.§7.4You need to be able to set up and solve a Newton’s Law of Cooling problem. Even though the book goesthrough and solve s for the solution in general, I still want you to set up and solve the problem using thedifferential equation (as I did in class).§7.7You need to be able to solve second order linear constant coefficient homogeneous DEs using the methoddiscussed in class (factor the auxiliary polynomial, and plug the roots into the appropriate form for the caseit falls into). If given initial or boundary conditions, then you also need to be able to use them to solve forc1and c2.1§7.8You need to be able to solve the non-homogeneous DEs discussed in this section. You need to be ableto form the correct final “guess” for ypbased on the form of G(x) and any overlap with yc. You shouldexpect at least one problem where you have to plug your guess for ypinto the DE and actually solve forthe unknown coefficients (i.e., A, B, C, . . .). There will also be a problem (multiple parts) which asks youto simply set up the correct, final guess for yp(again, based on the form of G(x) and any overlap with yc)but not actually solve for the unknown coefficients (similar to the examples I did toward the end of class on10/22). Remember, you can’t be sure your final guess for ypis correct unless you first find ycand checkfor overlap! I won’t ask you to solve a non-homogeneous DE with initial or boundary conditions on the testsince these problems can already be rather involved without them.§7.9You need to be able to set up and solve the motion-for-a-mass-on-a-spring (both undamped and damped)typ e of application/word problem we looked at in this section. You will not be asked a question about theelectrical circuit type of problem on the test (or on the final).§8.1Given the defining equation or recursion, you should be able to write out the first few terms of a sequence.You should also be able to go the other way: given the first few terms of a sequence, find the definingequation. You need to be able to determine whether a sequence converges or diverges (by looking at its limitor the limit of the absolute value of the terms, or by using the squeeze theorem). You need to know whatit means for a sequence to be increasing, decreasing, bounded above or below, and bounded, and be able todetermine which of those properties a particular sequence