Prof. Allen Knutson's Math 16b Course Homepage

Class: 8-9:30 AM Tuesday-Thursday, 145 Dwinelle

Professor: Allen Knutson, Evans 1033, (510) 642-4319, office hours 9:45-11 AM T-Th or by appointment

Book: Applied Calculus, Hughes-Hallett et al. (It's dark blue.)

Head GSI: Joseph Steever, Evans 845; office hours Tues 3:30-5:00, Thurs 12:00-1:30

TAs: Gizem Karaali, Nicolas Henckes

Final: Saturday, May 20, 8-11 AM, in 1 Pimentel.

allenk@math.berkeley.edu, jsteever@math.berkeley.edu, gizem@math.berkeley.edu, henckes@uclink.berkeley.edu

So how did things turn out?

You can look at the final (with answers) here (your browser will probably be happiest with the .pdf files).

The approximate grade scale from before the final actually looked just right with the final included. So combine your homework (scaled to 44%), your midterms (each scaled to 8%), and your final (scaled to 32%), and compare to

  • 44 or more: D
  • 57 or more: C-
  • 62 or more: C
  • 67 or more: C+
  • 71 or more: B-
  • 75 or more: B
  • 80 or more: B+
  • 84 or more: A-
  • 86 or more: A
  • 90 or more: A+
    • It turned out that nobody's grade was improved by counting their final 36% and their homeworks 40%, so we didn't do this.
    What this web page used to say...

    We're up to the final exam!

    The final is going to be like the last midterm - no calculator, no book, no notes. Note that it is not in the usual room, but in 1 Pimentel.

    The emphasis will be on the whole term, pretty evenly spread. Obviously three hours isn't really enough for this, and certain topics will fall by the wayside. This is as it should be - certain things we did were really only important as examples, and not as a fundamental idea one should take from the course.

    More specifically, here's what WILL be important:

  • Chapter 5, concentrate on:
  • 1,2: the maximum and minimum stuff, local vs. global, the importance of ends of intervals, what the second derivative really tells you.
  • 3,5: thinking about revenue and profit.
  • but you can forget:
  • elasticity, and sections 4,6,7. To repeat: this stuff was good for examples, but is not really worth remembering for its own sake.
  • Chapter 6 (returning to things you _should_ know)
  • 1: average value, both in the formula and graphically
  • 3: present and future value, comparing values from time t_1 to t_2. (If t_1 is zero and t_2 is positive, this is present to future; if t_2 is zero and t_1 is positive, this is future to present; etc.) Income streams is just mixing this with integrals.
    NOTE: this section requires fluency with exponentials and logarithms. I hope noone says "e^(a+b) = e^a + e^b"!
  • 4: relative vs. absolute population growth
  • 5-7: mainly you should be fluent with the fundamental theorem of calculus. You will NOT need to know any particular integrals for the final that I don't provide you (which will themselves be just be exponentials or something close to that).
  • 8-9: probability stuff. It matters a lot more that you can coherently think about estimating probabilities than that you know the name "cumulative distribution function" and silly things like that.
  • 10: median and mean
  • Junk 2 and the formulae from 5-7. The only formulae worth really thinking about in this chapter (and this book) are the present vs. future value vs. income stream value formulae.
  • Chapter 7
  • 1,3: mostly about the importance of reducing to the 1-variable case. Try to think beyond functions of two variables to functions of many (with all but one held constant).
  • 4: you should be able to calculate partial derivatives as routinely as you calculate ordinary ones. It's just a matter of remembering "in this calculation, that's a constant!"
  • 2: contours
  • Chapter 8
  • 1,2: mainly the idea of a function "satisfying" a differential equation, and the graphical version, following slope fields.
    • Reviews for the final: Nicolas will be holding sessions on both the 10th and the 17th, 2-4 PM, in 4 Evans - open to everybody who emails him which day they want to attend, and what they want to review, specifically.

    My remaining office hours: Wed/Th 10 AM-noon. You can try to find me Wednesday afternoon (call 642-4319 to see if I'm in) but I'm much less likely to be in Thursday afternoon.

    GSI office hours - the usual Mondays 2-4 + after the sessions for Nicolas, and Gizem's extra ones, the Thursday and Friday before the test, 12-5, to which Nicolas' can also go.

    And the old stuff, much of it no longer relevant...

    This class:

  • Chapters 5-8 in the Book
  • A little more detail about minima and maxima than is usually covered in calculus courses

    • (Approximate) grades

    See the FAQ below - there is now a formula to guess your grade, based on HW#1-9 and the midterms.

    The homework

    Homework will be assigned on Fridays and due the next Friday, turned in to your TA. If you can't make it to section, the best thing to do is to give it to someone who can; otherwise arrange with your TA how to turn it in. Write the time of your section on your homework!

    Late homework to be accompanied by doctor's note. Translation: don't be late, just don't, really.

    The midterms

    These happened on February 10, March 9, April 25, and are each 8% of your grade.

    I am trying to pick up the habit of always answering "yes" to the question "will this be on the test?" so thanks in advance for testing me on this.

    Unless I explicitly say otherwise, the only topics on the tests will be those already touched on in the homework, or in other sample book problems I will specify before the test. That does not mean the questions will be exactly of the same type; it only means for sure that you won't have to know any new definitions.

    What's allowed on this (third) midterm?

    Nothing. No books, no notes, no calculator. The only relevant formulae will be mean and median, which will be on there, but you should have a more visceral understanding of these than a formula can provide.

    Before anyone asks: the rules on getting incompletes are simple. You need to both have a doctor's note, and be currently getting a C or better.

    PLEASE check out Frequently Asked Questions about Math 16b before emailing one of us! Questions currently answered there:

  • "Do I need the other book, the study guide?"
  • "How will grades be computed?"
  • "Will this class be curved?"
  • "What's allowed on the tests?"
  • "How come there are, or may be, graphing questions on the test despite our not using calculators?"
  • "In what way are the homeworks and midterms related? They seem different to me."
  • "Still, why are they different?"
  • "Why are the scores on the tests so low?"
  • "How is the class graded, exactly?"
  • "How is the class graded, approximately, now that I have my third midterm grade?"
  • "I'd really like to sleep in during/leave town before the scheduled final exam. Is that okay?"
    • Class topics so far
  • 1/18: First pass through 6.1 and 6.2. Was informed after class that not everyone had done chapter 5.
  • 1/20: Retreat to 5.1, with extra discussion of minima/maxima in the case the second derivative, too, equals zero.
  • 1/25: More about the need to test higher derivatives when the first two are both zero, and the corresponding problems with inflection points. 5.3 & 5.5.
  • 1/27: Rederivation of elasticity of demand - it just pops right out of the maximize-revenue calculation. Worked through the book problems 262#24,25,27; #24 was straightforward, #25 a little tricky to set up, and #27 couldn't happen with differentiable functions (so we had to use functions that were only continuous). Major emphasis on the fact that maxima/minima can happen at places without zero derivative - places where calculus breaks down, namely the endpoints and nondifferentiable places.
  • 2/8: Reviewing for the test, just doing problems
  • 2/10: First midterm, answer key here
  • 2/15: Some discussion of the midterm, returned to 6.1, did #3,4,6,14 in class. There IS homework this week, though it's short (see below).
  • 2/17: Did 6.2 again, and started 6.3. See homework below. Some of the 6.3 problems are tricky to set up, but that's a fact of life. The only tricky aspect to the calculus is knowing that
  • integral(a^x) = a^x/log(a), and
  • integral(x a^x) = x a^x/log(a) - a^x/log(a)^2. (Check you've got them right by taking the derivative of both sides.)
  • 02/22/2000: Finished 6.3. Remember, there IS homework this week!
  • 2/24: Basically talked about techniques of integration, which is almost totally absent from the book. Change of variable, integration by parts. These will not be assumed for the tests; you just have to know things from the short list at the back of the book, plus x a^x. Emphasized the importance of not leaving out the dx, plus writing the limits of integration as "x=a" to "x=b" (in case you want to change variable). Also started 6.4.
  • 3/7: review for second midterm
  • 3/9: second midterm, answer key here
  • 3/16: more chapter 6, talked a little bit about information theory
  • 3/21: finishing up chapter 6.
  • 3/23: Special lecture on mathematics of juggling!
  • 4/4: Started 7.1, discussed examples 3&4, and problems 5,22,23.
  • 4/6: Section 7.2, emphasized the discrete x&y property of tables, vs. the discrete z property of contour plots.
    • Plans for the future: plow through chapter 7.

    The homework so far

  • HW#1, due Friday 1/28: 5.1 #3,4,5,8,11,13,15,19,20, and one not from the book: Let f(x) = x raised to the power n, where n is some positive integer. For which n does f have a local minimum at zero? Explain why WITHOUT using derivatives - just definition of minimum. For which n is f''(0) > 0? Answer key
  • HW #2, due Friday 2/4: 5.2 #15,16,20,26, 5.3 #4,6,20,23, 5.5 #10,11,16,17. Answer key
  • Suggested review problems for midterm #1 (not to be turned in):
  • Overall: p289 #3,6,7,9,10,16,17,20,21,23,25,26,29,30,37,38
  • 5.1 #6,17,18,
  • 5.2 #2,4,5,7,12,13,14,16,18,21,22,24,
  • 5.3 #4,6,7,13,16,21,26,29
  • 5.4 nothing - we didn't go through this section
  • 5.5 #1,2,3,4,6,7,13,17,18,21,22
  • 5.6 #3,5,6,9,12,14,15,16,17,18
  • 5.7 #2,8
  • Answers to a few select exercises
  • HW #3, due Friday 2/18: 6.1 #1,10,12,15,16.Answer key
  • HW #4, due Friday 2/25: 6.2 #1,2,4; 6.3 #2,3,4,6,8,16.Answer key
  • HW #5, due Friday 3/3:
  • 6.4 #4,5,7,16
  • 6.5 #9,10,11,16,23,31,32 (these are very short)
  • 6.6 #5,13,18,21.Answer key
  • Suggested review problems for the midterm (not to be turned in):
  • 6.1 #2,3,5,8,17
  • 6.2 #3,6,10,11
  • 6.3 #5,10,12,13
  • 6.4 #3,9,11,13,14,15
  • 6.5 #1-24,33
  • 6.6 #8-15, 23,26,29 note that this is in rabbits/day so is NOT relative growth rate
  • 6.7 #1,5,9,10,11,12,13
  • HW #6, due Friday 3/17: 6.8 #2,3,6,8,10,12,13,16,17Answer key
  • HW #7, due Friday 3/24:
  • 6.9 #3,4,5,6,8,16
  • 6.10 #3,6,8,9 Answer key
  • HW #8, due Friday 4/7: 7.1 #2,3,7,9,11,13Answer key
  • HW #9, due Friday 4/14: 7.2 #8,14,19,20,25,27,31Answer key
  • Midterm review (not to be turned in):
  • 7.3 #2,8,17,25
  • p350 #37,43,44,46
  • p404 #2,3,4,5,6,7,9,11,15,19,25,26,27
  • HW #10 (last one!), due Friday 5/5:
  • 7.4 #16,20,28,35
  • 7.5 #9,12
  • 8.1 #3,13
  • 8.2 #1,4Answer key