MATH 17C (Spring 2011).

Course materials

Discussion session materials (courtesy of Prof. D. A. Kouba):
  • Discussion Sheet 1 (not on exams: problems 1, 7, 9), Spheres in three dimensions.
  • Discussion Sheet 2 (not on exams: problems 1, 9, 10, 11).
  • Discussion Sheet 3.
    Midterm 1 will be on Wed., Apr. 20, 2010, in class. It covers sections 10.1-10.6 in the book, and first two homework assignments. Topics: functions of two variables, domains, level curves, spheres, partial derivatives, tangent planes, linearization, chain rule (section 10.5.1), directional derivatives, gradients, local extrema (incl. second derivative test), global extrema, applications. No limits, no continuity, no proofs of any kind. For practice (i.e., as a practice exam), solve the following problems on your own.
    1. Discussion sheet 1, problem 8a. Also sketch the level curve f(x,y)=0.
    2. Discussion sheet 3, problem 3.
    3. Discussion sheet 3, problem 6.
    4. Discussion sheet 3, problem 7.
    5. Discussion sheet 3, problem 10.
    6. Discussion sheet 3, problem 11d.
    7. Textbook, p. 570, 22.
    Brief solutions to the practice exam above; please check.
    Mon., 7:45 pm. Solutions to 5, 6, and 7 corrected. Thanks to Juliette Zerick. Juliette also has some notes on Discussion sheet 3.
    Another worked-out example of a global extremum problem.
    Solutions to midterm 1.
  • Notes on double integrals.
  • Discussion Sheet 4 (not on exams: problems 8abce, 9).
  • Notes on parametric equations.
  • Discussion Sheet 5 (not on exams: use of graphing calculator in problem 2).
  • Notes on complex eigenvalues.
  • A recommended Java-based program to draw trajectories. No, the knowledge of this program is not required for tests.
  • A mixture example and its solution.
  • Discussion Sheet 6 (not on exams: 3, 4, 5, 6, 7, 8, 10, 13; for problems 14-17, use only linearization at fixed points --- no geometric methods will be required; zero-isoclines are the same as nullclines)
    Midterm 2 will be on Wed., May. 11, 2011, in class. It covers sections 11.1 and 11.3.1 in the book, plus double integrals and parametric curves; and homework assignments 3, 4, and 5. Topics: double integrals, parametric equations (speed and direction of motion), linear systems of differential equations (direction field, general solutions, exact solutions with initial values, drawing trajectories, classification of the fixed point (0,0)), nonlinear systems of differential equations (finding fixed points, and their classification by linearization). No proofs of any kind, no applications. The formula for the general solution in case of complex eigenvalues will be given; that is, the expression after "The general solution is given by" in the notes will be printed on the front page without comment.

    For practice (i.e., as a practice exam), solve the following problems on your own.
    1. Consider the region R described in Discussion Sheet 4, problem 3. Compute the volume of the solid above R whose top is given by the surface z=2+x+y+xy.
    2. Consider the curve given in Discussion Sheet 5, problem 1f. (a) Compute its speed and the unit vector in the direction of motion when t=1. (b) Find all t at which the direction of motion is horizontal. (c) Find all t at which the direction of motion is vertical.
    3. Discussion sheet 5, problem 8b. (a) Find the general solution (b) Find the solution that is at (1,2) at time t=0. Sketch its trajectory. (c) Classify the fixed point (0,0).
    4. Problem 1b on this worksheet. (a) Find the general solution (b) Find the solution that is at (1,1) at time t=0. Sketch its trajectory. (c) Classify the fixed point (0,0).
    5. Discussion sheet 6, problem 14a.

    Advice: The content of this midterm is entirely predictable, but this is not easy stuff; be very well-prepared and on top of your game on Wednesday.

    Juliette Zerick has posted solutions to the practice midterm and many other notes on her page. Thank her when you get the chance! Another short guideline to solving Problem 1 of the practice midterm.

    Solutions to midterm 2.
  • Slides for a short introduction to combinatorial probability I gave to high school teachers a while ago.
  • Discussion Sheet 7. Only problems 3 and 5; interpret 3(b)-(e) and 5 as probability problems. For example 3(d): What is the probability that a randomly chosen committee of 5 people consists entirely of women; 5: What is the probability that, if you pick 4 out of 18 classes at random, you will pick one of each subject.
  • Discussion Sheet 8. Only the following problems, interpreted as probability problems. 3: arrange the 8 letters at random; what is the probability that ECLECTIC is spelled. 4(b)-(f): e.g., for (b), if 6 balls are selected at random, what is the probability that none is green. 5: what is the probability that there is at least 1 ball of each color. 8(a)-(c): e.g. for (a) if the 15 bass are caught at random, what is the probability that exactly 4 are tagged?
    Midterm 3 will be on Wed., May. 25, 2011, in class. It covers sections 12.1, 12.2 and 12.3.1; and homework assignments 6 and 7. Topics: counting (multiplication principle, permutations, combinations), probabilities involving permutations, combinations, selection without and with replacement (be familiar with cards, dice, and coins), conditional probability and independence, probabilities of a sequence of events in multistage experiments. No genetic applications, no mark-recapture computations.

    For practice (i.e., as a practice exam), solve the following problems on your own. (The exam will have four questions of this type.)
    1. A bag contains 15 yellow, 20 red, and 25 green golf balls. Select 30 balls at random without replacement. (a) Compute the probability that you select 5 yellow, 10 red, and 15 green balls. (b) Compute the probability that you select 10 balls of each color. (c) Compute the probability that your selection contains two different colors of 15 balls each.
    2. Roll a fair die 10 times. (a) Compute the probability that every number you roll is 6. (b) Compute the probability that you roll no 6. (c) Compute the probability that you roll exactly three 6's. (d) Compute the probability that you roll exactly two 6's and exactly three 5's. (d) Given that you roll exactly three 6's, what is the probability that you roll no 5's?
    3. Toss a coin 20 times in a row. (a) Compute the probability that you toss exactly 7 Heads in all 20 tosses. (b) Compute the probability that you toss 3 Heads in the first 10 tosses and 6 heads in the second 10 tosses.
    4. Again, a bag contains 15 yellow, 20 red, and 25 green golf balls. Select balls from the bag one by one without replacement. (a) Compute the probability that first ball selected is yellow, the second is red and the third is again yellow. (b) Compute the same probability if the selection is done with replacement.
    5. A group of 20 people consists of 8 Californians, 7 Nevadans and 5 Oregonians. They are seated at random on a row of 20 chairs. (a) Compute the probability that all three groups end up siting together (i.e., 8 Californians occupy adjacent chairs, and so do 7 Nevadans and 6 Oregonians). (b) Compute the probability that all 8 Californians end up sitting together and so do all 7 Nevadans. (c) Compute the probability that all 8 Californians end up sitting together.
    6. You are dealt 6 cards at random from a full deck of 52 cards (recall the cards have four different suits and 13 different values). (a) Compute the probability that all 6 cards are of the same suit. (b) Compute the probability that the 6 cards have different values. (c) Compute the probability that you get exactly 3 Aces. (d) Compute the probability that you get at least two Aces.

    Advice: As all problems on this exam will be word problems, be prepared to make common sense interpretations during the exam. The proctors will be instructed not to discuss interpretation issues to prevent disruptions. Stay concentrated and make sure you understand what you are doing.

    Brief solutions to the practice exam above; please check.
    May 22, 5:30 pm. Solution to 1b corrected. Thanks to Bassem.
    May 24, 1pm. Solution to 6b corrected. Thanks to Juliette.

    Solutions to midterm 3.
  • Solution to the Russian roulette problem from class. (Not required for exams.)
    Finals week information.
    Here are some practice exam problems on conditional probability (i.e., on material covered since Midterm 3); you need to know how to solve these kind of problems on the final.
    1. There are three bags: A (contains 2 white and 4 red balls), B (8 white, 4 red) and C (1 white 3 red). You select one ball at random from each bag, observe that exactly two are white, but forget which ball came from which bag. What is the probability that you selected a white ball from bag A?
    2. There are two bags: A (contains 2 white and 4 red balls), B (1 white, 1 red). Select a ball at random from A, then put it into B. Then select a final ball at random from B. Compute (a) the probability that the final ball is white and (b) the probability that the transferred ball is white given that the final is white.
    3. A test for certain disease is not perfect: if you have a disease it gives a positive result (i.e., indicates you have the disease) with probability 0.9, and if you do not have a disease it gives a positive result with probability 0.2. Assume different tests are independent. It is known that 5% of the population has the disease. (a) Assume that you do not have the disease, and are tested three times. What is the probability that none of the tests are positive? (b) Under the same assumption as in (a), what is the probability that exactly 1 test is positive? (c) A random person is chosen from the population, tested once, and the test is positive. What is the probability that he/she has the disease? (d) A random person is chosen from the population, tested twice, and both tests are positive. What is the probability that he/she has the disease?
    4. A group of 30 people includes 10 Californians (among them 5 men and 5 women), 10 Nevadans (3 men and 7 women) and 10 Oregonians (6 men and 4 women). First choose a group at random (so that the three groups are equally likely), then choose a committee of three people from the chosen group at random. (a) What is the probability that all members of the resulting committee are women? (b) The described procedure in performed in a closed room and after it's over you are told that the committee consists of three women. What is the probability that it consists of three Californian women?
    5. A bag initially contains 3 white and 5 black balls. Each time you select a ball at random and return it to the bag together with an additional ball of the same color. (a) Compute the probability that the first two selected balls are black and the next two are white. (b) Compute the probability that the second selected ball is white. (c) You have made a second selection and it is a white ball but you have no memory of your first selection. What is the probability that your first selection was also white?
    Brief solutions to the above 5 problems; please check.

    Next two are job interview questions. They are for your amusement and will not be featured on an exam, although you are able to solve them.
    6. There are ants at three corners of a triangle. Each ant starts moving on an edge towards another vertex, chosen at random. What is the probability that no ants collide?
    7. You find yourself in a shooting match with two other desperadoes, Bob and Alice. You shoot with 1/3 accuracy, while Bob and Alice shoot with 2/3 and 100% accuracy, respectively. There is one shot per person per round, and each round goes from worst to best shooter, until only one remains standing. What is your strategy?