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Section 02 meets Mon, Tues, Thurs, Fri 8:10am-9:00am in Morton 322.
Section 04 meets Mon, Tues, Thurs, Fri 9:10am-10:00am in Morton 222.
My current office hours are shown on my web page.
Students from all sections of Math 163A, not just sections 02 and 04, are welcome to come to my office hours.
The information on this page mirrors the information posted on the course's "Blackboard" site.
Course Information Sheet (link)
Syllabus (link)
Homework List (link)
On the second day of class, Thursday Sept 8, we discussed the shift in attitude about functions that we would see in calculus, a shift from thinking of them as equations to thinking of them as "mathematical machines", with input and output. I distributed the one-page handout entitled "Functions in Precalculus and Beyond", which was recycled from my fall 2004 Math 115 Precalculus course.
Functions in Precalculus and Beyond (link)
''Sign chart'' was the name we gave to a tool for determining when a function is positive, negative, or zero. It used concepts from algebra and precalculus. Students who took those courses only in high school might not have learned about sign charts. Students who took those courses at the college level would have encountered the sign chart there, although it may have had a different name or no name at all. Furthermore, in many books, the approach taken to sign charts involves plugging "sample numbers" into the function and seeing whether the result is positive or negative. In class on Tuesday Sept 13 or Thurs Sept 15, we discussed why this approach is both too hard and not hard enough. The method is too hard because one is often called upon to plug a fraction into a high degree polynomial. Computing the function value in such a case is a nuisance, especially without a calculator. The method is not hard enough because it does not make one think about why the function had the sign that it did. The document entitled "Roots, Linear Factors, and Sign Charts" is meant to be complete enough to give a student who had never seen sign charts an understanding sufficient for the needs of Math 163A. The approach taken is to not use sample numbers, using instead only the symbols +, -, and 0. The handout ends with a list of six exercises. Two of these problems are on Homework #2. The others will show up on later homework sets. This is the second mathematical handout of the quarter.
Roots, Linear Factors, and Sign Charts (link)
The textbook's approach to graphing rational functions is rather ad-hoc. In class, I stressed the importance of establishing a plan and and sticking to it, both to keep one's own thoughts organized when solving a problem, and to help give a clear structure to the written solution. In class on Friday, Sept 16, we discussed a six-step method for graphing rational functions. I distributed a handout entitled "Six Step Method for Graphing Rational Functions without Calculus" which was recycled from my fall 2004 Math 115 course. It is the third mathematical handout of the quarter.
Six Step Method for Graphing Rational Functions without Calculus (link)
Later in the course, we revisited the topic of graphing rational functions, this time with the additional tools of calculus available. The Six Step Method for Graphing Rational Functions grew into the Ten Step Method. The additional four steps involved analysis of the first and second derivative.
Ten Step Method for Graphing Rational Functions with Calculus (link)
By the time we finished section 5.3 of the textbook, it was hard to keep track of all of the properties that we have learned, and to remember the different significance of a derivative f ' having a property, as opposed to the function f having it. We used a small "Table of Equivalent Statements" to help organize our thoughts when discussing properties of a function f, its derivative f ', or its second derivative f ''. This table is certainly useful when using the 10-step method for graphing rational functions. But in fact, the table will pop up in discussions of almost every type of problem during the rest of the course.
Table of Equivalent Statements (link)
Four of the exercises on the assigned & suggested homework problem list are exercises from section 5.4 of the textbook that deal with graphing rational functions. These are difficult problems, made even more difficult by the fact that the derivatives involved are messy and the factorizations of those derivatives is tricky. Our goal is to use the 10-step method to graph these functions. This handout presents the derivatives needed for the problems, along with their factorizations.
Derivatives (and their factorizations) for four exercises from Chapter 5 (link)
In Section 9.3 of the textbook, we learned one method for finding relative maxs, mins, and saddle points in a function of two variables. We called the method the "D-test". To use the method, we followed the step in the handout below.
Worksheet for the "D-Test" (link)
On Tuesday, Sept 27, we did our first group work exercise. Entitled "Representations of Slopes", it consisted of drills that test the reader's recognition of mathematical expressions that represent the slopes of secant lines and tangent lines for a given graph.
Group Work 1: Representations of Slopes (link)
On Friday, September 20, we did our second group work exercise. In it, the students are given a ruler and the graph of a function. They used the ruler to draw tangent lines on the graph and to make measurements for computing the slopes of those tangent lines. With the information about the slopes of the tangent lines, they sketched a graph of the derivative.
Group Work 2: Graphical Differentiation (link)
On Thursday, October 20, we did our third group work exercise. In it, students are asked to sketch the graph of a function f given certain information about the sign of f, f ', and f ''.
Group Work 3: Sketching graph of f from given information about sign of f, f ', and f '' (link)
On Friday, October 21, we did our third group work exercise. In it, students are asked to sketch the graph of a polynomial function using the 10-step method for graphing rational functions.
Group Work 4 (Friday October 28): Given a graph of a function f, make a sign chart for f, a sign chart for f ', and a sign chart for f ''. (link)
Group Work 5 (Friday October 28): In this group work, the students are asked to draw the graph of a function a function f. However they are not given the function f. Instead, they are given the results of the analysis of f from the first nine steps of the 10-step method. Their job is to do step 10. (link)
Group Work 6 (Friday October 28): Graph a polynomial function using the 10-step method. (link)
Group Work 7 (Thursday November 10): Using the "D-Test" to Match Functions with Graphs. (link)
On Monday, Sept 12, we began discussing "Transformations of Graphs." Many of graphs that we deal with in 163A are simple transformations of basic graphs. The techniques of shifting, stretching, and flipping graphs - and the corresponding changes in the functions that generate the graphs - are taught in precalculus and reviewed in calculus. Some students need more review than our textbook or our class time could provide. (For example, most will not rememember or can not explain why the graph of f(2x) is shrunk and not stretched. This transformation was not even mentioned in the textook.) The document entitled "Transformation of Graphs" is meant to provide a thorough review. Many worked examples are provided, and a list of exercises was also suggested, all drawn from the homework list for Math 163A. The document is recycled from my fall 2004 Math 115 Precalculus course. It was not distributed in class.
Transformations of Graphs (link)
Graphical Differentiation Examples similar to Group Work #2 (see above).
Graphical Differentiation Examples (link)
Midterm #1 was given in class on Friday, September 23, 2005. It covered through section 3.2 of the textbook. ("Continuity").
Midterm #2 was given in class on Tuesday, October 11, 2005. It covered through section 4.3 of the textbook. ("The Chain Rule").
Midterm #3 was given in class on Monday, October 31, 2005. It covered through section 6.2 of the textbook. (Max/min problems)
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