## Math 285 Quizzes, Spring 2010

There will be up to 10 quizzes during the semester. Your lowest 2 quiz scores will be dropped. Quizzes are designed to test basic competence. (Homework and test problems will be more difficult.)

A score of 4 or 5 on a quiz is acceptable. A score of 0,1,2 or 3 is not, and you need to take action to catch up.

The median scores on the Quizzes so far are: 4, 4, 3.5, 4.5, 4, 4, 5, 3.5, 4, 4.5 (all out of 5)

Quiz problems will be taken from the material listed below, sometimes with the numbers changed in the problem. Prepare for each quiz by working through the problems, writing out each answer several times so that you know the problems thoroughly. After the quiz, check the solutions on this webpage.

Answers to many problems are in the back of the textbook. And the Student Solutions Manual has complete solutions for most odd-numbered problems.

Quiz 10: Wednesday 28 April.
9.5
Use Separation of Variables to find the solution of the heat equation with Neumann boundary conditions, on the interval 0 < x < L.
Notes. The PDE, BC and IC are stated in equations (32a),(32b) and (32c) on page 623. The solution is stated in Theorem 2 on page 625.
Your task is to carry out the method of separation of variables like in equations (16) through (20) on page 620 (or like we did in class). Then follow the treatment for Neumann boundary conditions in equations (33) through (39) on pages 624-625.
You can state (with no explanation) what the Neumann eigenvalues and eigenfunctions are, using equations (34) and (35). (I won't ask you to do it on this Quiz because you showed how to find those eigenvalues and eigenfunctions already in Quiz 7, for L=pi). But you do need to explain how to get the functions T0(t) and Tn(t) that go along with those eigenfunctions.

Quiz 9: Wednesday 21 April.
9.2 #3 (the point here is that the period is not 2pi, and so you have to use the formula on page 590 - what is the value of L, in this problem?)
9.3 #13 (first ask yourself: should you use a sine series or a cosine series? Then use the appropriate formulas from page 600.)
9.4 #7,9

Quiz 8: Wednesday 14 April.
9.1 #17,19,21 (answer for #19 at back of book has a typo: it should say a0=pi/2)
- The instructions are given before Problem 11.
- Formulas for integrals can be found on the inside cover of the textbook, or by using Wolfram Alpha (for example enter: integral of u^2 cos(u)). You will be given these indefinite integrals, on the quiz.
- If you can show the function is odd or even, then you can save yourself a lot of work because half the Fourier coefficients will be zero automatically, as we explained in class (or see formulas (5b) and (7a) on page 598).
- To "find the Fourier series of f(t)" means that you should evaluate a0, evaluate an and evaluate bn. Then write the series as
a0 + a1 cos(t) + b1 sin(t) + a2 cos(2t) + b2 sin(2t) + ... Write out enough terms in the series to make the pattern clear.
9.1 #19 continued: substitute t=0 into the Fourier series, and then find the value of that series (using the Convergence Theorem on page 592 to justify your answer). Answer: (pi+0)/2=pi/2, because the function jumps at t=0 with value pi from the left and value 0 from the right.
Aside. By simplifying that last series you find a nice formula for pi2=8(1+1/32+1/52+...). Once more we see that pi can be evaluated in terms of integers! Many beautiful formulas of this kind are derived at the end of Section 9.2 (page 594, and exercises 24 and 25).

Quiz 7: Wednesday 7 April.
3.8
Consider the eigenvalue problem X''(x) + lambda X(x) = 0 for 0 < x < pi, under the Neumann boundary conditions X'(0) = 0, X'(pi)=0. Show that the eigenvalues are lambdan = n2 for n = 0,1,2,3,..., with corresponding eigenfunctions X0(x) = 1 for n=0 and Xn(x) = cos(nx) for n = 1,2,3,...
Hint. Divide into three cases: lambda = 0, lambda < 0, lambda > 0 (because the general solution of the DE is different in each case).
Solution: see #7(a) in the Practice Test 2 Solutions.

Quiz 6: Wednesday 17 March.
3.3 Let D=d/dx.
(a) Write the differential equation (D2-3D-4)y=sin(x) in more familiar notation using y' etc. [Answer: y''-3y'-4y=sin(x)]
(b) Write the differential equation (D2-3D-4)y=0 in factored form. Find the general solution. [Answer: (D-4)(D+1)y=0 or (D+1)(D-4)y=0. General solution y=c1e-x+c2e4x.]
3.5 For problems # 1, 2, 3, 4, first find the complementary solution yc of the homogeneous DE, and then make a guess for the particular solution yp of the nonhomogeneous DE, and then determine the constants in your guess. Finally write the general solution y=yc+yp. [Answers: See back of the book. Or check your answers with Wolfram Alpha.]
3.5 58 (and also: show how to find yc, using the equidimensional method). Answer: yp=x3(ln(x)-1)

[Quiz 6 Solutions: Versions 1 and 2.]

Quiz 5: Wednesday 10 March.
3.4 # 16, 17, 21. Your task is to classify the DE as overdamped, critically damped or underdamped, and then to find the general solution and sketch a typical graph (including the envelope curves, in the underdamped case); also state the pseudo-frequency and pseudo-period if the DE is underdamped; you should ignore initial conditions; answers are in back of book.
3.3 # 39, 41

[Quiz 5 Solutions: Versions 1 and 2.]

Quiz 4: Wednesday 3 March.
3.1 # 41
3.3 # 1, 5, 9, 23
3.4
(a) Suppose -3cos(t)+4sin(t)=Ccos(t-gamma). Show how to find C and gamma. Your solution must include a sketch showing the angle gamma and the values of cos(gamma) and sin(gamma), labeled on a suitable triangle. Hint. The angle gamma lies in the second quadrant. Answer: C=5, gamma = pi - arcsin(4/5)
(b) Suppose y = -3cos(2x)+4sin(2x). Sketch y(x) by hand, and evaluate the amplitude. Show that the location of the first maximum point is x=gamma/2.

[Quiz 4 Solutions: Versions 1 and 2.]

Quiz 3: Wednesday 10 February.
1.5 # 15, 19, 21
1.6 # 17 (linear substitution), 21 (Bernoulli substitution)
2.2 Draw the phase lines for problems 1, 9, and label the equilibrium points as either stable or unstable. For #9, you should begin by factorizing the quadratic on the right side of the DE. [Answers:
#1 - arrow up for x > 4, arrow down for x < 4, equilibrium at x=4 is unstable.
#9 - arrow up for x > 4, arrow down for 1 < x < 4, arrow up for x < 1; equilibrium at x=4 is unstable, equilibrium at x=1 is stable.
Notice how up arrows on the phase line correspond to solution curves in the textbook answer that are increasing, and down arrows correspond to solution curves that are decreasing.]

[Quiz 3 Solutions: Versions 1 and 2.]

Quiz 2: Wednesday 3 February.
1.2 # 17 (acceleration is nonconstant, so eq. (11) does not apply),
27 (to simplify the arithmetic, take the gravitational acceleration to be g = -10 m/s2)
1.3 - sketch the direction field of the differential equation dy/dx=x+y in the first quadrant (where x>0 and y>0), and then sketch two solution curves.
1.4 # 5, 23, 25
1.5 # 3

[Quiz 2 Solutions: Version 1 and Version 2.]

Quiz 1: Friday 22 January.
Do the Algebra and Calculus review handout, and do the following problems from the textbook:
1.1 # 3, 7, 19

[Quiz 1 Solutions: Version 1 and Version 2.
Feedback.
1(a). To "verify" a solution of a differential equation means to start with the given solution, plug it into the DE, and check that the DE is satisfied. If instead you are asked to "solve" the DE, you should start with the DE and then find the solution (by the separable method, the first order linear method, etc).
1(b). The exponential curve approaches a horizontal asymptote as x approaches -Infinity. It does not approach the x-axis.
2(a). 1/(x+2) is NOT equal to (1/x) + (1/2). ln(x+1) is NOT equal to ln(x)+ln(1).
2(b). Indefinite integrals should include a constant of integration: "+C".]