Daily Agendas and Class Notes

Week 12

Monday:  Today I'll go over the "additional problems" list posted by our course supervisor, and appearing on the Tests page.  Based on comments from those at last night's review, I think we need some extra work on the earlier sections.

Tuesday:   Today I thought we would (finally) look at the end-of-chapter challenges called "Problems Plus".   We'll all work on the folding problem #14, and we'll set up another folding problem for you to submit next week.

Week 11

Monday:  Today we begin a never-ending effort to improve our ability to solve "word problems"

Tuesday:   No class today.  

Wednesday:  Today I'll go over a 2-3 more problems from Section 4.7, and then start the next section 4.8, a real departure from our other applications of differential calculus.   Many applied scientists devote their professional careers to what we see in Section 4.8.  Perhaps you will be one such person, and perhaps you will enjoy the short video from our text on what they call Newton's Method.  You may also be interested in some of the history of such methods.  The early recorded history takes us to northern India where great algebraic strides were made during the early Middle Ages (6th - 13th centuries) by astronomers such as al-Biruna (973-1048) and al-Kashi (1380-1429).  Early European mathematicians such as Vieta and Oughtred from the late 1500s and early 1600s rediscovered and improved on these methods, and Newton, Raphson and Simpson rediscovered and improved on their methods of solving polynomial equations using tangents and x-intercepts.

Friday:  

Week 10

Week 9

Monday: 

Tuesday:   Today I stole much of your lab time going over Exam 2, and doing an example like those on today's Tuesday assignment.  Click here the write-up of that example, and click here for the assignment.

Wednesday:  Today I'll share the more "resistant" indeterminate forms:  products, differences and powers.  In these cases we seek an equivalent indeterminate ratio that permits use of L'Hopital's Rule. Time permitting, we'll discuss some new vocabulary, e.g., absolute and local extrema, and critical points.  This section also contains the "Extreme Value Theorem", also known as the Weierstrass Extreme Value Theorem.  I'll share a few stories about Weierstrass, and his famous student, Sonya Kovalevskaya, the first woman to get a PhD in Mathematics.

Friday:  Let's return to stories of how the Weierstrass theorem is used to answer questions such as "If things go wrong, how wrong can they go?"  Click here for a short video about suspension bridges, and one that went wrong.  In this section we work mostly from graphs, to establish the relationship between critical points and the minima and maxima over an interval (see page 278).   In class, we start with the graph problems #3, 4, and #6, and look at a case where the graph has details hard to catch by graph alone, e.g.,   f(x) = .2 x^5 +x^4 - 4x^3 +3.  A big take-away is that not all critical points correspond to mins or maxs.  Our last example, time permitting, will be Exercise #70 page 281.  It's a great example of a function with parameters that we can't study easily from graphs.  The interval given doesn't include values of theta outside the function or derivative domain.  Note that the typical coefficient of friction values are smallish.  Teflon has a value of approximately 0.4.  This means that the model predicts a critical value of the dragging force for some small angle of the rope with the drag surface.

Week 8

Monday:  Today I extended the due date on Friday's HW, and went over the 2nd problem a bit.  I worked out Exercise 33 on page 236 with the intention of illustrating the power of the verbal analysis of derivative statements like that for A(x) here.  We don't need to understand in any way the productivity function p(x)  to realize that A'(x) > 0 is related to p'(x).   The rest of our time was spent going through the rest of the Section 3.9 solved examples and discussing a few of the 11 WebAssign problems.

Tuesday:  Before starting work on today's problems -- a subset of the exercises in the end-of-Chapter-3 review problems -- I reviewed last week's problem about Newton's Law.  Your assignment is to do #72-76, 80, 83, 97 and 100 for next Tuesday.  The take-away lesson for us is the importance of taking derivatives before we simplify rather than after.  I hope it's also a take-away lesson to think about the simple laws we learn, and what might be the broader theory behind them.

Wednesday:  

Friday:

Week 7

Monday:  

Tuesday:

Wednesday:

Friday:  

Week 6

Monday:  

Tuesday:

Wednesday:

Friday:

Week 5

Monday:  I shared exams but they won't be yours to keep until Wed.  Message of the day:  if we continue with this level of effort --- class average = 60% -- we'll get a C.  If we ramp up, we can be a class with a B average.  Click here for example.  Sub-message is the need to review exponents!  I shared some serious mistakes from Exam 1.  Turning to the future and Ch 3, we did a few examples from 3.2 and started 3.3 on trig derivatives.  Since too few appreciate these tools, I shared a few a little history noting their presence in Egyptian papyrus, that there are 2 "fathers of trig" (one Greek/Turkish fellow named Hipparchus and a German printer named Regiomontanus), and shared the Greek vision as lengths associated with a circle, an angle and that angle's complement.

Tuesday:  Today we practiced derivatives using the short-cuts from sections 3.1 and 3.2.  Click here for the worksheet and here for the answers.

Wednesday:  Today, after my review of the issues we faced on Exam 1 and what we might do to improve, I showed a few more complex examples from Sections 3.1 and 3.2.  With the remaining time, I shared heuristic reasons to believe that  (cos x)'=- sin x, using the sketching strategies from last week Tues.  Similarly, we checked the graphs to confirm that (sin x)' = + cos x, and (tan x)' = [1/cos x] ^2. Thanks to an attentive young woman in our class, I realized that I have sketched the tangent function incorrectly for years:  thanks to you I'll make the following change

For those struggling with how to see ALEKS progress in terms of %s, click here for a short video.

Week 4

Monday:  Housekeeping:  due to WebAssign outages, this assignment was extended to end-of-day Tuesday.  Today I will discuss the three remaining notions in Chapter 2:  (1) points where derivatives  don't exit; (2) derivatives of derivatives; (3) matching derivatives to symbols and graphs.  Time permitting, we'll look at the easy shortcuts to find derivatives in Section 3.1

Tuesday:  As promised, today's group work offers suggestions and practice sketching derivative graphs AND learning how to use your graphing calculator to evaluate slopes and derivatives, and draw both derivative curves and tangent lines.  There is also a little practice on using the short-cuts of section 3.1.

Wednesday:  Knowing the shorter "short-cuts" for finding derivative formulas might help you check yourself when applying the difference quotient method from Ch 2.  We spend the day practicing problems from Section 3.1, and since people asked, I also give examples of both the product and quotient "short-cuts" from Section 3.2.  

Thursday:  Since many at last night's review session have not practiced all the review materials from the text, from the sample test, and from the Fall 2012 exam, there was general interest in limit problems which require harder algebraic work.  Click here for the list I made.  I plan to do a few examples from this list.

Friday: 

Week 3

Monday:  

Tuesday:

Wednesday:

Thursday:

Friday:

Week 2

Tuesday:  Housekeeping:  You might want to spend some of today's class time finishing Tues #1 from last week.  Kudos on doing so well on WebAssign!  Not so many kudos on ALEKS.   Heads-up on the rest of this week:  I'll leave town after class Wed, so there will be no office hours Wed afternoon, Thursday or Friday.  I won't hold our Thursday session in HS 105, but my colleague Hakima Bessaih will lecture on Section 2.6 on Friday. As we started our work today, I made a few remarks about each problem from last week, and  I then did the same for today's four questions, Tues #2. 

Wednesday: Housekeeping:  We took a survey and I'll change the weekly due day from Saturdays to Sundays.  I also urged everyone to make some "green bar" headway on ALEKS so that I can better see how I might help you reach our 3-week goal of being 50% done.  Take a look at the Fall 2012 exam to better understand what to expect, e.g. #1 shows you the graph to use to answer a graphical question and #2 shows you the table to use.  You won't make your own graphs/tables.  Today's lecture was on Section 2.6, continuity.  Most functions we study in HS are continuous, so it's no surprise that you don't yet know the power of continuity.  I shared two stories where one can PROVE the answer to a seemingly complex riddle:   (1) a person's age and weight must equal at least once in a lifetime that extends beyond the primary grades and (2) exercise #69 page 130 about the journey of a monk.   The power is in the Intermediate Value Theorem (bottom page 125).   I worked through 3 examples (click here to see them), and shared the text's definition of continuity (page 118), and its message about continuity:

• All polynomials with real number coefficients are continuous functions.
• Functions such as rational functions, root functions, trig functions, inverse trig functions, and log functions, even ones with "breaks" (vertical asymptotes) are everywhere continuous on their domains.  For such functions, finding the domain is tantamount to finding the points of continuity.
• Sums/differences/products of continuous functions are also continuous (on their domains).
• Compositions of continuous functions are continuous.

Thursday:  No study session today.

Friday:

Week 1

Monday:  Today we started with a review of the syllabus and a round of introductions.  As our textbook author notes on page 1, calculus is unlike any other math content you may have studied.  It is built on ideas about infinity,  and is extremely powerful in modeling in STEM and humanistic disciplines.  Calculus requires tools we often forget to mention, such as advanced reading of graphs and making sense of numerical data.    Even though we skip Ch 1, perhaps you should spend some time reviewing it.  I selected a couple of exercises to illustrate data and graphical problems.

Tuesday:  Housekeeping matters:  we have presentation by Ben Steer about SI, and we'll debrief on our success with ALEKS and WebAssign accounts.  It's only right that you journey through limits start with insights from Vi Hart.  We'll then break up into work groups and start your first Tuesday assignment.  It's due next Tuesday, by the end of the day.  You may submit alone, or in a group of 2-4 people.  Give me a nice write-up, so I won't accept anything during class.

Wednesday:  Housekeeping:  Be aware that:   (1)   the first assessment within ALEKS will determine how much additional time you'll need practice, so for all who have not yet started, take care to start with care; (2) the first HWs on WebAssign are due THIS WEEK SATURDAY at midnight.  They cover sections 2.2 and 2.3.   Today's class was all about the section 2.2

Friday:  Housekeeping:  HW 2.3 is now due next Monday rather than Saturday.  On Tuesday 9/3 I extended the due date through 9/4.  Also, it's possible to "restart" ALEKS if you haven't already paid your account.  Finally, it might be a good thing to go to our Exams page and download old exams, and in this way, track your progress with Ch 2.  The general message today is that limits work like we might wish that roots worked:  it's completely legit to change the orders of operations (see rules pages 99-101).  There are also 4 tools we have:  charts, numerical (calculator) values, graphs, and symbolic evidence.  Whenever possible, confirm what one tool says using another.  Note the perhaps surprising limits that result when the variables are not coordinated, e.g. limit of 3n as x goes to a is simply 3n:  x is the declared variable and 3n looks like a constant in terms of x.  We went through the solved examples in Section 2.4, noting the tool(s) used.  I shared some extra factoring shortcuts including:  (1) a trick related to both infinite and finite geometric series that says, for example, that 1+x^4=(1+x)(1-x+x^2-x^3) and 1= (1+x)(1-x^2+x^3-x^4+....) and (2) How to use Pascal's Triangle to foil out large powers.   I noted that the least integer function is related to the (more modern) floor and ceiling functions, and finally, discussed the squeeze theorem using its more colorful name, 2 Policemen and a Drunk.  This is sure to be on our first exam.