homework.htmlAPPLMOSSFi%1 . Wednesday March 28

Schedule For Math 8

Spring Term 2001

Wednesday March 28: Introduction to differential equations, slope fields. Read: Sections 17.1 and 17.5 (through page 995).

Friday March 30: Separable differential equations and applications.

Read before class: Sections 3.4 and 17.2.

Homwork due today:

    1. Show that y=(1/4)t2 +c/(t2) is a solution of the diffrential equation y’+(2/t)y = t, where C is any constant.
    2. Consider the differential equation y’=x(y-2)(y-3). Show that the constant function y=2 is a solution. Can you find any other constant solutions?
    3. Sketch the direction field for the differential equation y’=y2. Can you find any constant solutions to this differential equation?
    4. Match the following differential equations with their direction fields, shown on your class hand-out. Then sketch some solution curves for your chosen direction fields. (You may tear off or xerox the chosen direction fields and sketch your solutions directly onto the picture.)
  1. y’=y+t
  2. y’=tan y
  3. y’=tan t.

Monday April 2: First-order linear equations; mixing problems.

Read before class: Section 17.3 and p. 467, Example 4.

Homework due today: p. 994: For each of problems 1,3,5, and 6, indicate whether the differential equation is separable. Do not solve these equations.

Section 17.2: 1, 5, 8, 14, 21. (In all these problems, be sure to check for any constant solutions.)

Section 3.4: 11, 17, 23.

Also match the differential equation y’=1+(y/t) with its direction field in the hand-out from last time, and sketch a few solution curves on the direction field.

Tuesday April 3: (X-hour). Integration by parts.

Read before class: Section 6.1.

Homework due today: none

 

 

Wednesday April 4: Taylor Polynomials.

Read before class: Section 4.7 and 4.8 (You may skim Big O notation), concentrating on 4.8.

Homework due today: Section 3.4: 12. Section 6.1: 1, 5, 7, 13. Section 17.4: 1, 4, 5, 11. Also do the following problem: A tank contains 1000 liters of brine with 15 kg of dissolved salt. Pure water enters the tank at a rate of 10 liters/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank after t minutes? After 20 minutes?

 

 

Friday April 6: Taylor Polynomials and Taylor Series

Read before class: In Section 4.8, re-read the material from the middle of page 282 on, skipping the proof of Theorem 10. Think about the Big-O notation and how it makes it easier to deal with what the book calls the Lagrange remainder. Look over Section 9.6

Homework due today: Section 4.8 Do problems 1, 2, and 3. Do problem 14, but don't answer the Lagrange Remainder question. Section 7.9, problem 2. Do this two ways: (a) by separating variables and (b) by viewing it as a first-order linear differential equation.)

Use the method we used in class to find a power series solution to the differential equation y'=y. (Class notes should be posted on the web page by midday Thursday at the latest.) Do this problem:

A tank contains 1000 liters of pure water. Brine that contains 0.05 kg of salt per liter of water enters the tank at a rate of 5 liters/min. Brine that contains 0.04 kg of salt per liter of water enters the tank at a rate of 10 liters/min. The solution is kept thoroughly mixed and drains form the tank at 15 liters/min. How much salt is left in the tank

(a) after t minutes?

(b) after 1 hour?

 

 

Monday April 9: Taylor Polynomials and Taylor Series

Read before class: Read Section 9.5, and reread Section 9.6 Note that Section 9.5 through Example 2 is relevant to the homework due today. Don't worry about endpoints of intervals of convergence.

Homework due today: Section 4.8: Do problems 7, 12, 15, and 18. Section 9.5: Do problems 2 and 3, dealing with the open interval of convergence only. (There will be notes on the radius of convergence and the open interval of convergence available from the web page by midday Saturday at the latest, and a longer version of those notes later on.) Section 9.6: Do problems 1, 7, and 15. Do the folllowing problem.

A 2000 liter container initially contains 1000 liters of a brine solution containing 40 kg of salt. A brine solution containing 0.02 kg/liter of slat flows into the container at a rate of 50 liters/min. The solution is kept thoroughly mixed, and the mixture flows out at a rate of 25 liters/min. How much salt is in the container at the moment it overflows? (Caution: when you compute the "rate out", be sure to check how much water is in the tank at time t.)

 

 

Wednesday April 11: Second order linear differential equations

Read before class: Section 17. 1 and Pages 1005 (starting with Second-Order Linear Equations)-1009 (ending with Example 2). Also begin reading supplementary notes.

Homework due today: Section 9.5: Do problem 5 dealing with the open interval of convergence only. Do problem 12 and 17 (open interval only). Section 9.6: Do problems 3, 11, 12, and 37. Section 9.7 Do problems 13 and 14. (They don't look like something we did in class, but they are remarkably close. Just relax and think about what you know.)

 

Friday April 13: Complex numbers; Second order linear differential equations

Read before class: Supplementary notes and pp. 1009-1012, pp. A-1 to top of A-9. (The A pages are the appendix in Adams.)

Homework due today: Section 9.5: 6. Section 17.1: 2, 4, 6, 9, 16. (Be sure to read the instructions for 2, 4, 6, and 9. You don't have t o solve the equations!) Section 17.7: 2, 4, 5, 8, 13. Also the following problem:

Solve the differential equation y''+ 4y' =0 in three ways:

  1. by writing (D2 + 4D)y = 0, factoring to obtain D(D+4)y=0 and solving each of Dy=0 and (D+4)y=0
  2. By letting v=Dy=y', rewriting the original differential equation as an equation for v, solving it to find v, and then solving y'=v.
  3. By looking for solutions of the form ert. (What happens if r=0?)

 

Monday April 16: Harmonic motion and resonance. Complex numbers (time permitting)/

Read before class: pp. 1016-1019

Homework due today: Section 9.6: 17, 33. Section 17.7: 3, 6 (note: you may notice that your solution doesn't make sense for a vibrating spring. The equation of motion for a vibrating string would not have a negative y' coefficient.), 9, 14, 15. Chapter review exercises p. 1026: 3, 18 (on 18, also find the solution satisfying y(0)=0.)

 

Wednesday April 18: Reading and lecture material will be posted later

Homework due today: Section 17.8: 3, 5, 7.

 

 

Friday April 20: Vectors

Read before class: Section 10.2

Homework due today: Section 10.1: p603 #1, 2, 8,10, 12, 15, 19, 26

 

 

Monday April 23: Vectors and cross products

Read before class: Section 10.2, focusing on the phrases basis, linear combination, and projection.

Section 10.3 through p 619

Homework due today: Section 10.2: p613 #1a,b,c, 4, 5, 10 (hint, see example 3), 20, 27, 28

 

 

Wednesday April 25: Cross products and Equations of planes

Read before class: Read 10.4 up through and including Example 4; also read Example 7.

Homework due today: Section 10.2: p613 #17, 29, 30. Section 10.3 p622 #1, 2, 4, 5, 8 (Hint: what do you know about the cross product of u+v with itself?)

 

Friday April 27: Equations of lines and planes; distance from a point to a plane

Read before class: Complete section 10.4.

Homework due today: Section 10.2: 2(b-g). Section 10.3: 3, 6 (also find the area of the parallelogram spanned by the two vectors in problem 6). Section 10.4: 2, 4, 6, 7, 10, 12.

 

Monday April 30 Matrix Arithmetic.

Read before class: Read Section 1 of the notes on Matrix Algebra posted on the web under supplementary notes.

Homework due today: Section 10.2: 3(b-g). Section 10.4: 16, 18, 20, 21, 24, 25, 28 (First part only; i.e., show that the line is parallel to the plane.) Also do the following problems:

A. Compute the products (i x i) x j and i x (i x j). What do the results say about the

possibility of the cross product satisfying the associative law?.

B. Consider the line given parametrically by x=1+2t, y= 3+4t, z=2+7t.

Part (i): Find the equation of the plane through the origin perpendicular to this line.

Part (ii) : Find the equation of the plane containing both this line and the origin.

 

Wednesday May 2 Solving systems of equations.

Read before class: Read Section 2 of the notes on Matrix Algebra posted on the web under supplementary notes.

Homework due today: Section 1 of the supplementary notes. Do Exercises 9, 22, 25, 27-32 (one point each), 33, 35, 45 (Note, these exercises are meant just to be practice and shouldn't be a lot of work) then do Problems 4, 5, 12 (Note, problems are meant to make you think; these three won't be so straightforward.)

A. Find the area of the triangle with vertices (1,0,1), (2,1,4) and (1,1,1). Also find the equation of the plane containing this triangle.

 

 

Friday May 4 Inverse matrices.

Read before class: Read Section 3 of the notes on Matrix Algebra posted on the web under supplementary notes.

Homework due today: Section 2 of the supplementary notes. On pages 29-32 in Section 2, do Exercises 5, 14, 16, 20, 23a, 25a, 27, 29, 37

A. Find the equation of the plane containing the parallel lines given parametrically by

x=1+t, y=2-t, z=3+2t

and x=3+t, y=2-t, z=4+2t 

 

 

Friday May 4 To be announced

Read before class: To be announced

Homework due today: Sectione of the supplementary notes. On pages 29-32 in Section 2, do Exercises 5, 14, 16, 20, 23a, 25a, 27, 29, 37

A. Find the equation of the plane containing the parallel lines given parametrically by

x=1+t, y=2-t, z=3+2t

and x=3+t, y=2-t, z=4+2t 

 

Monday May 7 Graphs and level sets.

Read before class: Section 12.1.

Homework due today: Read Section 3 of the supplementary notes. Don't panic about the number of exercises and problems. All of the exercises up to about 53 should go very quickly (41 will take a few minutes, though). On pages 41-43 in Section 3, do Exercises 1, 4, 7, 13, 16, 19, 25, 28, 31 (in the last 3 exercises, write down the actual inverse matrix), 41, 43 (you do not need to write down the actual matrices, only their shorthand description), 53, 57, 59. Also do problem 1 on p 32 and problem 2 on p 17.

Wednesday May 9 Limits, continuity, directional derivatives.

Read before class: Section 12.2 and 12.3.

Homework due today: Section 12.1: 14, 16, 20, 27, 28, 38.

 

Friday May 11 Tangent planes, higher order derivatives.

Read before class: Section 12.3 , 12.4 and 12.6, pp. 731-732.

Homework due today: Supplementary notes on matrices, pp 32-33, problems 3 and 5. Adams Section 12.1: 22. Section 12.2: 5, 7, 10, 12 (Hint: Consider the line y=x.), Section 12.3: 2, 6, 7, 26.

And

A. Consider the surface z=x2y2. The plane x=5 intersects this surface in a curve. Find the slope of this curve at the point (5, 1, 25). Then write down a 3-dimensional vector parallel to this curve (note that x is not changing) and write down the tangent line to this curve.

 

 Monday May 14. Chain Rule.

Read before class: Section 12.5

Homework due today: Section 12.3: 14 (tangent plane only), 18 (tangent plane only), 36 (Hint: When you're computing partial derivatives at the origin, you only need to look at the values of the function on the axes See the example at the beginning of class today). Section 12.4: 1, 5. Section 12.6: 1, 6. Supplementary notes on matrices, p. 44: 10(a,b).

And:

  1. Consider the equation of a plane in the form z-z0=a(x-x0)+b(y-y0). Write the right-hand-side of this equation as a product of a row matrix involving the coefficients and a column matrix involving x-x0 and y-y0.
  2. Follow the procedure in part A to express the tangent plane to a surface z=f(x,y) through a point (x0,y0,z0) on the surface in the form z-z0 = product of a row matrix involving partials derivatives and a column matrix involving involving x-x0 and y-y0.

 

Wednesday May 16. Chain Rule and gradients

Read before class: Section 12.6 through Example 3, Section 12.7 through Example 3.

Homework due today: Read Section 12.5, and p736 (Functions from n-space to m-space)-738 Problems: p 706#4, p714 # 23, p730 #1, 6, 7, 19 p739 # 13 (Note that Jacobian matrix means the same thing as derivative matrix).

 

Friday May 18. Gradients; we do have class on Friday; it is not a College Holiday, and it is too close to final exams to move the class to an X-hour.

Read before class: Section 12.7.

Homework due today: Read Section 12.6 and and 12.7 through Theorem 6. Problems: p730 # 3, 10 (the answer should be in terms of partial derivatives of f). p738 #1 (the linearization is another word for the tangent plane approximation), # 14, p749 # 1, 2, 7 (remember, the gradient is normal to the level surface.) p613 #14 (to start reviewing for the final).

 

Monday May 21. Extreme Values

Read before class: Section 13.1.

Homework due today: Read Section 12.7. Problems: p749 #4, 9, 11, 14, 17, 21, 26, 31. P287 #8.

 

Wednesday May 23. Classification of critical points.

Read before class: Go over Section 13.1 with emphasis on the second derivative test and 13.2 up to page 782.

Homework due today: Section 3.4: 9 (for review). Section 12.7: 19. Section 13.1: In the following problems, just find the critical points. Don't classify them except as noted in the parenthetical comment concerning the first problem. 1 (complete the squares in f in order to determine the behavior of f at the critical point), 4, 7, 9. Also do the following problem.

  1. Define a function f by f(x,y)=0 when x=0 or y=0 (i.e., on both axes) and f(x,y) =x+y everywhere else. Show that the partial derivatives of f at (0,0) with respect to x and with respect to y are both zero. Then use the definition of directional derivative (not the gradient formula but the derivative of a certain function of t) to show that the directional derivative of f at (0,0) in the direction of any unit vector (u,v) which does not point along either axis is given by u+v. (Hint: note that f(0+tu,0+tv)=tu + tv in this case.) Can f be differentiable at the origin? (Hint: what does the gradient formula give you for the directional derivative?)

 

Friday May 25. Lagrange Multipliers.

Read before class: Section 13.3.

Homework due today: Section 13.1: In problems #4, #7, and #9, classify the critical points that you found in the previous assn. (You can refer to the homework solutions for the critical points.) Next do problem #6. (First try the second derivative test. If that fails, try to understand the behavior of the function in order to classify the critical points.) Also do problem #21. (We will learn an easier way to do this problem on Friday. For now, express the surface area first as a function of three variables, the length, width and height. Then use the given information to express one variable in terms of the other two so that the surface area becomes a function of two variables. Then find its minimum.) Section 12.7: #22. Section 17.7: #4.

(Note: 13.2 #5 is not included in Friday's assn. This syllabus takes precedence over my earlier blitz message.)

 

Tuesday May 29. X-hour. This time will be used either for review or for answering questions.

Homework due today: Section 13.2: # 5. Section 13.3: #1, #3(a,c), #6, #8, #21 and the following problem:

A. Find the maximum and minimum values of x2 + 3y2 + 2z2 on the sphere x2+y2+z2=1.

 

Wednesday May 30. This time will be used either for review or for answering questions.

Homework due today: none