DIFFERENTIAL AND INTEGRAL CALCULUS
Text: Stewart, Calculus, Early Transcendentals, Eighth Edition
Responsible Party: Ray Heitmann June 2014
Prerequisite and degree relevance: An appropriate score on the mathematics placement exam or Mathematics 305G with a grade of at least B-.
Math majors are required to take both M408C and M408D (or either the equivalent sequence M408K, M408L, M408M; or the equivalent sequence M408N, M408S, M408M). Mathematics majors are required to make grades of C- or better in each of these courses.
408C may not be counted by students with credit for any of Mathematics 403K, 408K, 408N, or 408L.
Course description: M408C is the standard first-semester calculus course. It is directed at students in the natural sciences and engineering. The emphasis in this course is on problem solving, not the theory of analysis. There should be some understanding of analysis, but the majority of the proofs in the text should not be covered in class.
The syllabus for M408C includes most of the basic topics in the theory of functions of a real variable: algebraic, trigonometric, logarithmic and exponential functions and their limits, continuity, derivatives, maxima and minima, integration, area under a curve, and volumes of revolution.
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. You will do your students a disservice and leave them ill equipped for subsequent courses.
For those instructors who have taught M408C previously, some changes should be noted. Chapter 7 has been moved to M408D, allowing a slightly less hectic pace and more importantly the coverage of some topics which have been omitted or optional in the past. The formal definition of a limit should be covered, although you still shouldn’t expect delta-epsilon proofs. Sections 3.8, 3.9, 3.10, 4.7 are no longer optional. Sections 6.3, 6.4, 6.5 have been included as optional sections and some, but not all, of these topics should be covered. These adjustments give more attention to applications of both differentiation and integration.
Remember that 408C/D is the fast sequence for students with good algebra skills; students who cannot maintain the pace are encouraged to take either the 408NSM or the 408KLM sequence.
Resources for Students
Many students find the study skills from high school are not sufficient for UT. 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 will 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 37 days, allowing some time for testing and review. Those teaching on TTh should adjust the syllabus; a MWF lecture lasts 50 min; a TTh 75 min. The purpose of Chapter 6 is to provide applications showing students what integration really means. It does not matter which optional sections you cover, but it is crucial that you cover some of them or provide alternative examples.
37 Class Days As:
1 Functions and Models (Three Days )
- 1.4 Exponential Functions
- 1.5 Inverse Functions and Logarithms
2 Limits and Derivatives (Six 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
- 2.5 Continuity
- 2.6 Limits at Infinity; Horizontal Asymptotes
- 2.7 Derivatives and Rates of Change
- 2.8 The Derivative of a Function
3 Differentiation Rules (Eleven Days)
- 3.1 Derivatives of Polynomials and Exponential Functions
- 3.2 The Product and Quotient Rules
- 3.3 Derivatives of Trigonometric Functions
- 3.4 The Chain Rule
- 3.5 Implicit Differentiation
- 3.6 Derivatives of Logarithmic Functions
- 3.7 Rates of Change in the Natural and Social Sciences (optional)
- 3.8 Exponential Growth and Decay
- 3.9 Related Rates
- 3.10 Linear Approximations and Differentials
- 3.11 Hyperbolic Functions (quickly)
4 Applications of Differentiation (Eight 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 Indeterminate Forms and L'Hospital's Rule
- 4.5 Summary of Curve Sketching
- 4.7 Optimization Problems
- 4.9 Antiderivatives
5 Integrals (Five 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 (Four Days)
- 6.1 Areas between Curves
- 6.2 Volume
- 6.3 Volumes by Cylindrical shells (optional)
- 6.4 Work (optional)
- 6.5 Average value of function (optional)