A Proposed Revision of the Calculus Sequence

Don Krug and Andy Long October 10, 2003

Abstract

1 Introduction and Philosophy

We propose this revision with the following objectives in mind: to • draw students,

• expose students to topics other than calculus in their freshman year,

• excite students,

• keep students, and

• motivate students to – participate in more contests,

– become more active in the department,

– take more upper level courses, and

– conduct more research and engage in other scholarly activity. We suggest the elimination of five-hour calculus courses. Instead we will offer a sequence of four three-hour courses, called Calculus A-D. We think that this will have several benefits: this will

• allow for more variety in the freshman mathematical experience (provided we have the courses to offer);

• prevent five-hour Fs or withdrawals; • prevent the “a course is a course” accounting problem, by which we mean that five-hour

courses are not given five-thirds the attention that three-hour courses are given;

• make physicists happier;

• make computer scientists happier;

• make statisticians happier; and

• make ourselves happy (of course).

We no doubt will make some people unhappy; but people will be unhappy, unfortunately, no matter what form calculus takes (including if it were to remain the same). We simply seek to maximize the pleasure-to-pain ratio.

Calculus A and B will basically cover the material currently covered in MAT120. Calculus C will serve to cover those topics required especially by our students in CS and statistics. Calculus D builds depth on the topics covered in Calculus A-C, and allows for supplementary topics of the instructor’s choice. Some material currently covered in our calculus sequence has been moved: in particular, the material of chapter 17 has been moved to MAT330 (which implies changes to that course).

The total number of hours of required material in each of our courses has been set a little low intentionally, so as to allow the instructor to supplement the material as he or she sees fit, or to add depth to the mandatory topics. The mandatory material should not be so jam-packed that one cannot stop to smell the four-leaved rose function.

Students who’ve had calculus before should not be allowed to take Calculus A. They should start in Calculus B. Calculus A and B are prerequisites for Calculus C, and Calculus C is a prerequisite for Calculus D.

Calculus A and B should be available concurrently. That way, those interested in calculus (or in “getting calculus over with as quickly as possible”) can take what is effectively a six-hour calculus course. In particular, we anticipate that physics might require that their students take both courses simultaneously, so that they would see both integral and differential calculus early. Students who do so would also have the advantage of seeing two different instructors present the calculus simultaneously. While there is always the risk of some confusion due to different instructor preferences for notation, etc., we imagine that it would be beneficial for the students to see different styles and methods of presentation of the material.

These two courses would, in some sense, end on the same note (the Fundamental Theorem of Calculus): Calculus A students would be fully prepared to appreciate it, and it would be a focal point of Calc B. Physics would also be happy to know that we would suggest as a unifying theme the study of motion (as is frequently the case, anyway). Another focal point would be the importance of units, as an aid in constructing formulas, detecting errors, and locating errors.

Calculus C has been designed for those students who are only required to take 9 hours of calculus, such as the CS students. Statistics faculty have also expressed an interest in cutting back to 9 hours of calculus. This course would serve the needs of both of these groups of students, as well as for other students who choose not to continue in calculus. Topics such as series and an introduction to multivariate and vector calculus were chosen specifically for these students.

Calculus D is an enhancement of Calculus C, with the important addition of the geometry of curves (parametrics, polar, space, curvature). Computer science students interested in graphics would benefit from this course, in particular, as would any student interested in pursuing more advanced mathematics.

2 The Grim Details

2.1 Calculus A

Calculus A is an introduction to the differential calculus, with a review of function, an introduction to limits in an intuitive way, an introduction to the derivative as a limit, drill in the usual shortcuts to differentiation, applications to graphing, linear approximation and tangents, and optimization.

Students who’ve had calculus before should not be allowed to take this course. They should start in Calculus B.

2.1.1 Mandatory coverage (sections from Stewart)

1.1, 1.3; 2.2, 2.3, 2.5; 3.1-3.3, 3.5, 3.6, 3.8, 3.10; 7.1-7.3; 4.1, 4.3, 4.5 and 4.7 (lightly).

2.1.2 Supplementary Topics

The instructor would have roughly 12 hours for exams and his/her choice of additional material. One might consider adding, for example,

• more on differentials

• related rates

• hyperbolic trig functions

• the calculus of inverse functions

• implicit differentiation

• higher derivatives

2.1.3 Concerns

We’re introducing the derivative in the fifth week: that may be too late for physics....

Table 1: Calculus A – Differential Calculus

50-minute sections Stewart Sections

Description

6 {1.1,1.3, parts of 7.1- \atop 7.3} Functions Review: polynomials, rationals, trigs, logs, exponentials; composition and inverse; periodicity and symmetry

6 2.2, 2.3, 2.5 Limits and Continuity. Intuitive idea of limit approaching a value. Using graphs and tables. Continuous functions versus non-continuous. Limits are easy for continuous functions. Continuity fails outside of the domain.

9 {3.1-3.3, 3.5, 3.6, 3.8, \atop 3.10; parts of 7.1-7.3} Derivative.

• As limit of difference quotients.

• As the velocity of a moving object.

• Shortcut formulas for computing derivatives

– “polynomial rules”

– product and quotient rules

– trig rules

– chain rule

12 4.1, 4.3, 4.5 and 4.7 (lightly)

Applications:

• Acceleration

• Concavity and second derivatives (links with graphs)

• Linear Approximations.

• Optimization problems

33  Total hours

2.2 Calculus B

Calculus B begins with the concept of area, a review of classical area formulas, and then proceeds to an introduction to the idea of a Riemann sum. One of the novel features of

Table 2: Calculus B – Integral Calculus

50-minute sections Stewart Sections

Description

6 Area:

• Review of basic area (length, volume) formulas • Area defined as number of squares contained (for

units!)

• The area function: A(x) = area under y = f(x)

from 0 to t.

9 5.2, 5.4, 8.7 (light)

The Integral:

• Σ-notation

• Lefthand/Righthand endpoint schemes

• Area as a limit of LH/RH approximations

• Midpoint (trapezoidal) rule for integration

4 Applications, prior to Fundamental Theorem; focus on Riemann Sum

3 4.10

Review of derivatives, and anti-derivatives

7 5.3, 5.5

Fundamental Theorem of Calculus

5 6.1, 6.2, 9.1(possibly) Applications, post-FT, including distance travelled in motion problems, time elapsed, etc.

34  Total hours

the course is treating applications before the FT, in order to reinforce the ideas of Riemann sums, alternative interpretations of the integral, units, etc. Integrals at this point would be calculated using technology (e.g. TIs), or approximation techniques such as the midpoint rule. Later the students will study applications after having seen the FT. 2.2.1 Mandatory coverage (from Stewart) 5.2, 5.4, 8.7 (light); 4.10; 5.3, 5.5; 6.1, 6.2, 9.1(possibly)

2.2.2 Supplementary Topics

The instructor would have roughly 11 hours for exams and his/her choice of additional material. Specific focus might be given to

• work and other physics applications

• additional applications in business, biology, etc.

• volumes by shells

• standard integration methods (e.g. partial fractions, trig substitution)

• Simpson’s rule

2.3 Calculus C

Calculus C is the third and final calculus course for many (especially the stats and CS folks). Integration by parts provides a review of differentiation and integration; improper integration at ∞ previews the issue of convergence (and serves the statisticians well). Series are introduced with an emphasis on applications of power series to represent functions. Students learn about the relative growth rates of functions. Vector calculus will be particularly useful for students interested in graphics. Partial derivatives and multiple integration are introduced, but not developed deeply (Calculus D serves to augment these topics for students who need more detail). A quick introduction to those topics paves the way for students to study other areas of mathematics.

2.3.1 Mandatory coverage (from Stewart)

8.1, 8.8; 12.2, 12.6, 12.8-12.10, 12.11 (light), 12.12; 13.1-13.5; 15.1-15.4; 16.1, 16.2

2.3.2 Supplementary Topics

The instructor would have roughly 9 hours for exams and his/her choice of additional material. Specific additional focus might be given to

• sequences

• error bounds for numerical integration (an application of series)

• further topics in series

2.4 Calculus D

Calculus D is a course for those who want to see more calculus, in particular, any student interested in pursuing advanced mathematics. It is an enhancement of Calculus C, with the important addition of the geometry of curves (parametrics, polar, space, curvature). Computer science students interested in graphics would benefit from this course.

Table 3: Calculus C - series, vectors, partial derivatives, multiple integration

50-minute sections Stewart Sections

Description

3 8.1, 8.8

Integration by parts, and improper integration at ∞

12 12.2, 12.6, 12.8- 12.10, 12.11 (light), 12.12 Series:

• (2) Review Σ-notation and geometric series. Factorials.

• (1) Relative growth rates of functions

• (1) Ratio test (comparing size to geometric)

• (2) Power series and interval of convergence

• (2) Manipulation of series

• (2) Taylor’s formula

• (1) Binomial Series (for integral powers)

• (1) Applications

9 13.1-13.5

Vectors and vector operations

9 15.1-15.4

Partial Derivatives

3 16.1, 16.2

Multiple Integration and apps

36  Total hours

2.4.1 Mandatory coverage (from Stewart) 12.3, 12.5; 11.1-11.5; 14.1-14.3; 15.5-15.7; 16.1-.5 2.4.2 Supplementary Topics The instructor would have roughly 14 hours for exams and his/her choice of additional material. Specific additional focus might be given to

• conic sections

• proofs, perhaps induction

• a closer look at limits

• more on improper integrals (e.g. L’Hôpital’s rule)

Table 4: Calculus D

50-minute sections Stewart Sections

Description

6 12.3, 12.5 Series:

• (3) More on convergence: p-series (reviews improper integral) and integral test

• (2) alternating series

• (1) other topics/apps

12

• (7) 11.1-11.5 (parametrics and polar)

• (3) 14.1, 14.2 (space curves)

• (2) 14.3 An intro to curvature

Geometry of curves

6 Review, 15.5-15.7

Partial derivatives

7

• (2) 16.1-2 re- ; view ; • (5) 16.3-.5

Multiple Integration

31  Total Hours

3 Other Issues

We believe that a placement exam should be administered to all students, so that we can identify those students who are weak (in spite of their backgrounds), and who would need remedial help in order to succeed in calculus. This implies that remedial help should be made available. We should have a computer-based curriculum for helping the student get up-to-speed on areas of weakness, and the student should be compelled to pass the necessary section(s) before being allowed to continue in calculus. If, after a probationary period, a student is not able to pass the deficiencies, the student should be allowed the option of dropping back into MAT119.

We could also consider a sort of “running start” in calculus: this would be good for those needing assistance, as well as for our majors (or other strong students), who would serve as assistants in the classroom or as section leaders.

4 Conclusion

What we have presented is a shortened version of the calculus, as measured by total hours: 12, in contrast to the 13 currently required (well, 15, actually, if we count the five-hour sections honestly). We do not reduce the number of hours because we believe that the material is unimportant: rather, it is our (possibly vain) hope that students will actually be inspired to take \underline{more} calculus, in the form of real and complex analysis, or a (future) course in advanced calculus.

Some material that we may have covered in the past will be moved to other courses. For example, chapter 17 would be covered in the Math/Physics course MAT330; DEs would be relegated to MAT325.

Our goal is to deliver calculus to our core constituencies in the best possible way, so that their students will get what they need (and not too much more) from our calculus sequence. This includes our math majors. We hope that our majors will be enticed (read “advised”) to take courses in addition to calculus in their freshman and sophomore years. It would be nice if our majors were introduced to discrete math early on, for example, or probability and statistics.

We realize that rearranging topics as we suggest puts a greater burden on the professor: there is a greater danger that problems taken from the text will require material not yet covered in our sequence, so that the instructor will need to take care in assigning problems that are actually completely within the scope of the course. We feel, however, that the proposed arrangement of topics will be more appropriate for our department’s needs, and our university’s needs.