Reformed Math 112: A Syllabus
Principal Textbook is Calculus -- Early Transcendentals, by James Stewart. Brooks/Cole, 1995, 3rd Ed. ISBN: 0-534-25158-7.
Note on Usage
PART I: Integration Review, Techniques, Applications; Diff. Equations.
• Week 1: Definite Integral, Numerical Integration, FTC. (Stewart §§ 5.3, 5.4, 7.8)
•  Day 1:
• Handout about Math 112, office hours, etc; Flyer on Special Sections of 112 [[TeX version]
• Flyer: "Some Suggestions on Working in Groups"   (Praise for this Handout )
• Nuts/Bolts information on Special sections including Course Structure;
• Activity: Time and Speed, then brief lecture: how Activity relates to week's §§.
• Day 2:
• Lecture on 7.8
• Activity: Kidney-shaped Pool (Stewart);
• Day 3:
• Activities: Sigma Detectives & Barehands Integration: one in class, one for HW.
• Day 4:
• Remaining lecturing from Stewart
• Activity: Fundamnental Theorem of Calculus.

• Week 2: Area and Volume. (Stewart §§ 6.1,6.2, 6.3; review u-substitution (5.5))
• Activity: Series, Integrals, and Hidden Paterns.
• Activity: Graphical Integration (Modified from IC book)
• Lecture: How to Compute Volume Integrals (Lecture notes).
• Films from MAA Calculus Film Series: Disks/Washers & Shells;
followed by examples on blackboard afterwards.
•  Last day of Week 2:
• Project #1 assigned: "Houdini's Escape"
• Handout with guidelines on how to work on projects
• Schedule Preliminary Meeting with each Project group.
• Quick review of u-substitution; chain rule.

• Week 3: Start Techniques of Integration. (Stewart § § 7.1, 7.2, 7.3, start 7.4)
(Only (polynomial)/(Ax2 + Bx + C) for 7.4)
• Mostly lectures with examples
• Deliberate timing to contract these "nuts & bolts" computational techniqutes with Project.
• Hold Preliminary Meeting with each Project group.

• Week 4: More Techniques of Integration. (Finish §§ 7.4; 7.6 ("Strategy"); 8.2 (Arc length))
•  Days 1 & 2:
• Finish lecturing on Techniques of Integration.
•  Day 2:
• Activity: Work on old prelim integration problems (Announce Integration Olympiads)
•  Day 3:
• Activity: The Shorter Path (IC Book)
•  Day 4:
• Last day of Week 4 (or first of Week 5):
• Project #1 is due including Project Evalution Questions to be turned in by each student.

• Week 5: Differential Equations. (§ 8.1 & Activities; Medical (§ 8.6) if time permits).
• Lecture on 8.1.
• Second Day of Week 5: "Nuts & Bolts" Exam held
• Illustrate Separation of Variables with Examples;
• Activity: Save the Perch (IC);
• Examples of "Water-Solution" problems on board
• Student continue/finish "Save the Perch"
• Last Day of Week 5:
• Project #2 Assigned:Tripple Trouble, due in 3 and a half weeks.

[Fall Break here, if Fall Semester; if so, delay the following by half a week]
PART II: Sequences, Series, Taylor Series
• Week 6: Introduce Infinite Sequences & Series. (Stewart §§ 10.1, 10.2, and Activities).
• Lecture: Infinite Sequences (limits of sequences, least "monotone" sequences, "bounded" sequences, etc); Geometric Series.
• Activity: The Race.
• Handout: "If Only Every Function were a Polynomial "(Marshall Cohen)

• Week 7:
• Lecture: Telescoping series, careful definition of limit of a sequence being , etc. Difference and relation between infinite Sequences and Series -- repeating this message. The Harmonic Series.
• Handout: How Do Your Favorite Sequences Stack Up? (Marshall Cohen).

• Lecture/Discussion: What is wrong with Guido Ubaldo's "Proof" that 0=1:
 0 = 0 + 0 + 0 + 0 + ... = (1-1)+(1-1)+(1-1)+... = 1+(-1+1)+(-1+1)+... = 1 + (0) + (0) + (0) + (0) + ... = 1
A discussion often follows; ask students to decide, discuss (a friendly debate often ensues): which equal sign(s) is the source of the problem? Why?

• Lecture: Test for Divergence, and the difference between the converse and the contrapositive. (A different which "It cannot be emphasized too often to instructros, that it cannot be emphasized too often to the students...")

• Week 8:
•  Days 1 and 2:
• Converting infinite decimals (e.g. 1.21888...)to fractions. Final discussion of The Race.
• State as Fact: p-series Test. ("Be able to use it"; improper integrals covered later).
• Lecture: Comparison Test (CT)and Limit Comparison Test (LCT).
• Examples: E.g. writing (ln(n))/(n3) as..? as [(ln(n))/(n)] times [1/(n2)]
• What about (ln(n))/((n2)? Not so simple? Notice log is "beaten out" by even root powers.
•  Days 3 and 4:
• Examples with CT and LCT. Series of 1/((2n)? Series of 1/((2p + 1)? (two ways).
• Examples like series of (3n2 + 2n) / [square-root(n7 + 5)].
• Rule of Thumb: "Look at top power in numerator, and denominator" -- then check rigorously with LCT.
• Activity: Medical Dosage (Marshall Cohen's writeup).
• Project #2 is due around this time, depending on the semester.

• Week 9:
• Midterm is held this week -- Midterm includes Activity-type problem(s) (30-35%)
• Project #2 is due.
• Solutions to Medical Dosage, Stack Up, and Nonbook Problems.
• Review for Midterm.
• Lecture: Finite Absolute Convergence and Ratio & Root Tests; Cover "Strategy".

• Week 10:
• Activity: Students work in groups to solve the Midterm's Activity-type problem and more advanced add-ons. Typically this goes very well, very appreciated by the students.
• Project #3 is assigned: Pi in the Sky (due Week 13).
• [under construction: Clarifications about languate of Pi in Sky project][ALSO: student evals for Pi et al]
• Lecture: Improper Integrals; The Integral Test; Power Series.

• Week 11:
• Lecture: Taylor Series; Taylor Series Approximations
• Activity: Taylor Series Cookbook.
• Taylor Series Cookbook
• Student Evaluations of "Taylor Series Cookbook" (F95).

• Activity/HW: Estimating (e). [If done, Puzzler needs some of Week 12 to finish].
• Activity: The Calculator Puzzler.

• Student Presentations: Students to choose groups
and select => Topics for Week 13 Presentations. <=
PART III: Polar and Parametric Equations; Vectors
• Week 12: Parametric and Polar Equations (Stewart §§ 9.1, 9.2, 9.4)
• Lecture: Polar & Parametric Equations. Why these are natural (Motion; Airplane/Radar;..).
• Activity: (Henderson's other polar coordinates Activity: 19th cent. style sewing machine).
• Air Traffic Controller.

• Week 13: Introducing Vectors (Stewart §§ 11.1, 11.2)
• Handout: "Galilean Relativity" pp 123-127, from Why Math? by R.D. Driver.
• Activity: Steamboat Captains.
• Activity: Airline Passenger.
• Project #3 is due this week or next depending on semester/scheduling.
• Student Presentations Night is held.

• Week 14: Dot Product and its Applications; Mention Cross Product etc.
• Handout: "Vectors, their Properties, and Pictures (Need to HTMLize these lecture notes).
• Activity: Sail Away [under construction : A "transparent" background for sailboat.gif]
• Student Evaluations of "Sail Away" (F95)
• Handout: "Components of Vectors (§ 8.3 from Why Math? by R.D. Driver).