DIFFERENTIAL AND INTEGRAL CALCULUS
Text: Stewart, Calculus, Fifth Edition
Responsible Parties: Kathy Davis, John Gilbert, Gary Hamrick June 19 2003
Prerequisite and degree relevance:
Either a 560 on the Mathematics IC Test or 560 on Mathematics Level IIC or a grade of at least C in M304E or M305G. Note: Students who score less than 600 on the Mathematics Level IC Test should be aware that studies show taking M305G first is likely to improve their grade in M408C.
Only one of the following may be counted: M 403K, 408C, 408K, 308K.
M408C and M408D (or the equivalent sequence M408K, M408L, M408M) are required for mathematics majors, and mathematics majors are required to make grades of C or better in these courses.
M408C is our standard first-semester calculus course. It is directed at students in the natural and social sciences and at engineering students. The emphasis in this course is on problem solving, not on the presentation of theoretical considerations. While the course necessarily includes some discussion of theoretical notions, its primary objective is not the production of theorem-provers.
The syllabus for M408C includes most of the elementary topics in the theory of real-valued functions of a real variable: limits, continuity, derivatives, maxima and minima, integration, area under a curve, volumes of revolution, trigonometric, logarithmic and exponential functions and techniques of integration.
Overview and Course Goals
The following pages comprise the syllabus for M408C, and advice on teaching it. Calculus is a service course, and the material in it was chosen after interdepartmental discussions. Please do not make drastic changes (for example, skipping techniques of integration). You will do your students a disservice and leave them ill equipped for subsequent courses.
This is not a course in the theory of calculus; the majority of the proofs in the text should not be covered in class. At the other extreme, some of our brightest math majors found their first passion in calculus; one ought not to bore them. Remember that 408C/D is the ‘fast’ sequence for students with good algebra skills; students who cannot maintain the pace are encouraged to take the M408KLM sequence.
Resources for Students
Some of our students have weak study skills. The Learning Skills Center in Jester has a wide variety of material ( drills, video-taped lectures, computer programs, counseling, math anxiety workshops, algebra and trig review, calculus review) as well as tutoring options, all designed to help students through calculus. On request, (471-3614) they’ll come to your classroom and explain their services.
You can help your students by informing them of LSC services.
Timing and Optional Sections
A ‘typical’ semester has 43 MWF days; a day or so will be lost to course-instructor evaluations, etc. The syllabus contains material for 39 days; you cannot afford to lose class periods. If you plan to give exams in lecture rather than in TA section, you will surely have to cut time somewhere. We have added some flexibility to the syllabus by designating all the application sections as optional. It is expected that you will choose three or four of these and cover them well; it is also expected that you will not have time to do all of them.
Those teaching on TTh should adjust the syllabus; a MWF lecture lasts 50 min; a TTh therefore 75.
Forty Class Days As:
Appendixes (for reference and reading by student)
A Numbers, Inequalities, and Absolute Values
B Coordinate Geometry and Lines
C Graphs of Second-Degree Equations
E Sigma Notation
1 Functions and Models (for reference and reading by student)
1.1 Four Ways to Represent a Function
1.2 Mathematical Models: A Catalog of Essential Functions
1.3 New Functions from Old Functions
Principles of Problem Solving
2 Limits and Rates of Change (Five Days)
2.1 The Tangent and Velocity Problems
2.2 The Limit of a Function
2.3 Calculating Limits Using the Limit Laws
2.4 The Precise Definition of a Limit (briefly)
2.6 Tangents, Velocities, and Other Rates of Change
3 Derivatives (Seven Days)
3.2 The Derivative as a Function
3.2 Differentiation Formulas
3.4 Rates of Change in the Natural and Social Sciences
3.5 Derivatives of Trigonometric Functions
3.6 The Chain Rule
3.7 Implicit Differentiation
3.8 Higher Derivatives
3.9 Related Rates
4 Applications of Differentiation (Seven Days)
4.1 Maximum and Minimum Values
4.2 The Mean Value Theorem
4.3 How Derivatives Affect the Shape of a Graph
4.4 Limits at Infinity; Horizontal Asymptotes
4.5 Summary of Curve Sketching
4.7 Optimization Problems
5 Integrals (Six Days)
5.1 Areas and Distances
5.2 The Definite Integral
5.3 The Fundamental Theorem of Calculus
5.4 Indefinite Integrals and the Net Change Theorem
5.5 The Substitution Rule
6 Applications of Integration (Two Days)
6.1 Areas between Curves
7 Inverse Functions: Exp Log and Inverse Trig (Six Days)
7.1 Inverse Functions
Sections 7.2-7.4 or Sections 7.2*-7.4* at discretion of instructor
7.2 Exponential Functions Their Derivatives
7.3 Logarithmic Functions
7.4 Derivatives of Logarithmic Functions
7.5 Inverse Trigonometric Functions
8 Techniques of Integration (Seven Days)
8.1 Integration by Parts
8.2 Trigonometric Integrals
8.3 Trigonometric Substitution
8.4 Integration of Rational Functions by Partial Fractions
8.5 Strategy for Integration