Math 221: Calculus and Analytic GeometryTEXTBOOK: Calculus and Analytic Geometry, Thomas & Finney, 5th Ed.Math 221 covers the first 6 chapters of Thomas and Finney, 5th. Roughly speaking, 1000 students takeMath 221, both in the Fall and in the Spring semester. Most of these students will become users of mathe-matics, not mathematicians, and are taking the 221-222-234 sequence because it is required for their degreeand subsequent classes in other departments. It is therefore important that lecturers follow the syllabusclosely. Substantial departure from the syllabus requires advance approval from the Calculuscommittee.Notes and recommendations for InstructorsWhen covering section 3-10, emphasize that the “extended mean value theorem” is known as “Taylor’sformula” and use Taylor’s formula as often as possible (as an alternative to l’Hˆopital’s Rule for instance).Taylor’s formula is very important for applications and as preparation for treatment of series in 222. Taylor’sformula leads to deeper understanding of the concepts while l’Hˆopital’s rule leads to thoughtless applicationsof formulas. L’Hˆopital’s rule is very useful for limits involving logs and exponentials however.When covering integrals, discuss the various incarnations of the fundamental theorem and the notationsubtleties including the concept of “dummy variables”:Zf(x)dx,Zxaf(s)ds,ZdFdxdx,ZdF,Zf0(x)dx,Zxaf0(s)ds,ddxZxaf(s)ds, . . .An example from linear differential equations which confuses most students isddxZxaf(x)g(s)ds. Whenusing substitution, students should know how to change the limits of integration on definite integrals, thisis important fot applications and is often easier computationally. Note that the book does not mention themethod of substitution explicitly. It is good to correct that.Section 4-6 is conceptually useful to help explain dummy variables and as an introduction to the fundamentaltheorem if telescoping sums are employed (e.g.Pnj=1j =Pnj=1[(j +1)2− j2]/2), but it may also be confusingto students who might think that they must compute elementary integrals as limits of sums. In practice, theconnection between sums and integrals is used in the opposite manner: using integrals to estimate sums.Make sure you leave enough time to cover logs and exponentials properly at the end of the semester. Notethat the book uses the exnotation right away. It is good to use also the notation exp(x) and it is nice todeduce that exp(x) ≡ ex.Math 221 Syllabus: Calculus and Analytic Geometry(assumes 15 weeks of instruction)Week 11-1,2,3,4,5 Coordinates and lines, 1-6 Functions and graphs,1-7,8,9 Tangent and Velocity.Week 21-10, 2-11 Limits and continuity, 2-1,2 The derivativeWeek 32-3 Differentiation formulas,2-4,5 Inverse and Implicit functions, (6-1,2,3 Inverse trig functions?)Week 42-8 Composite functions, chain rule, 2-9,10; 6-2,3 Differentiating trig functions.Week 52-6, 2-12,13 Increments, differentials and higher derivativesWeek 63-2 Related Rates,3-1,3,4 Curve plottingWeek 73-5,6 Max Min,3-7,8 Mean Value TheoremWeek 83-10 Taylor’s formula,3-9 Indeterminate forms (l’Hˆopital’s Rule)Week 94-2,3,4 Indefinite integrals (antiderivatives) and differential equationsWeek 104-5,(6),7 The definite integral (area under a curve),4-8 The Fundamental TheoremWeek 114-9,10 Trapezoidal rule, notation and summary, 5-1,2 Plane Areas,5-3 Distance, 5-9 Average value.Week 125-4,5 VolumesWeek 136-1,2,3 Inverse Trig Functions6-4,5,6,7 The natural logWeek 146-8,9,10 Log and exponential functions6-11 Exponential growth (the equation y0= y).Week 15Review and/or further applications of integration: (5-7,8,10,11 Arclength, Surfaces, Center of
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