Syllabus: M408D


Text: Stewart, Calculus, Early Transcendentals, 7th Edition

Responsible Parties : Ray Heitmann and Lorenzo Sadun, June 2014


Prerequisite and degree relevance: A grade of C- or better in M 408C, M 408S, M 408L or the equivalent.  M 408D may not be counted by students with credit for both of M 408S and M 408M, nor for students with credit for both of M 408L and M 408M. The two courses M 408C and M 408D are required for mathematics majors, and mathematics majors are required to make grades of C- or better in these courses.  (Majors may also complete the equivalent sequences M 408N/S/M; or M 408K/L/M, with grades of C- or better.)

Certain sections of this course are reserved as advanced placement or are honors sections; they are restricted to students who have scored well on the advanced placement AP exams, or are honors students, or who have the approval of the faculty mathematics advisor. Such sections and their restrictions are listed in the Course Schedule each semester.

Course description: M 408C, M 408D is our standard first-year calculus sequence. It is designed for students in the natural and social sciences and engineering students. The emphasis in this course is on problem solving, not on theory. While the course necessarily includes some discussion of theoretical notions, its primary objective is not the production of theorem-provers. M 408D contains a treatment of infinite series, and an introduction to vectors and vector calculus in 2-space and 3-space, including parametric equations, partial derivatives, gradients and multiple integrals.

Overview and Course Goals

The following pages comprise the syllabus for M 408D, 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 first found their passion in calculus; one ought not to bore them. In general it is fair to say that M 408D students will do better than M 408C students; on the other hand M 408D is a more difficult course. Please keep in mind that students who pass this course meet the prerequisite for M 427K, where it assumed they have good calculus skills.  The M 408C/D sequence is the fast sequence for students with good algebra skills; students who cannot maintain the pace are encouraged to take either the M 408N/S/M or the M 408K/L/M sequence.

Resources for Students

Some of our students have weak study skills. The Sanger Learning 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 SLC services.

Timing and Optional Sections

A typical semester has 42-44 MWF days. The syllabus contains material for 38 days; this allows some time for testing, reviews, and optional material.  In the spring semester, you will have more time to cover optional material.  Those teaching on TTh should adjust the syllabus; a MWF lecture lasts 50 min; a TTh lasts 75 minutes.


38 Class Days As:



  • 7 Techniques of Integration (eight days)
    • Substitution Review
    • 7.1 Integration by Parts
    • 7.2 Trigonometric Integrals
    • 7.3 Trigonometric Substitution
    • 7.4 Integration of Rational Functions by Partial Fractions
    • 7.5 Strategy for Integration (use as reference with good problem set)
    • 7.8 Improper Integrals


  • 9 Differential Equations (six days)
    • 9.1 Modeling with Differential Equations
    • 9.2 Direction Fields and Euler’s Method
    • 9.3 Separable Equations
    • 9.4 Models for Population Growth
    • 9.5 Linear Equations
    • 9.6 Predator-prey Systems (optional)
  • 10 Parametric Equations and Polar Coordinates (four days)
    • 10.1 Curves Defined by Parametric Equations
    • 10.2 Calculus with Parametric Curves
    • 10.3 Polar Coordinates
    • 10.4 Areas and Lengths in Polar Coordinates
    • 10.5 Conic Sections (optional)
    • 10.6 Conic Sections in Polar Coordinates (optional)
  • 11 Infinite Sequences and Series (twelve days)
    • 11.1 Sequences
    • 11.2 Series
    • 11.3 The Integral Test and Estimates of Sums
    • 11.4 The Comparison Tests
    • 11.5 Alternating Series
    • 11.6 Absolute Convergence and the Ratio and Root Tests
    • 11.7 Strategy for Testing Series
    • 11.8 Power Series
    • 11.9 Representations of Functions as Power Series
    • 11.10 Taylor and Maclaurin Series
    • 11.11 Applications of Taylor Polynomials
  • 14 Partial Derivatives (three days)
    • 14.1 Functions of Several Variables
    • 14.2 Limits and Continuity
    • 14.3 Partial Derivatives
    • 14.5 The Chain Rule
  • 15 Multiple Integrals (five days)
    • 15.1 Double Integrals over Rectangles
    • 15.2 Iterated Integrals
    • 15.3 Double Integrals over General Regions
    • 15.4 Double Integrals in Polar Coordinates
    • 15.5 Applications of Double Integrals (optional)
    • 15.10 Change of Variables in Multiple Integrals (if time permits)
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