**M 408N, DIFFERENTIAL CALCULUS FOR SCIENCE**

**Text: Stewart, Calculus, Early Transcendentals, Eighth Edition**

** Responsible Parties:** Jane Arledge, Kathy Davis, Ray Heitmann, June 2011

**Prerequisite and degree relevance:** An appropriate score on the mathematics placement exam or Mathematics 305G with a grade of at least B-.

Only one of the following may be counted: M 403K, 408K, 408C, 408L or 408N.

Calculus is offered in three equivalent sequences at UT: an accelerated two-semester sequence, M 408C/D, and two three-semester sequences, M 408K/L/M and M 408N/S/M. The latter is restricted to students in the College of Natural Sciences.

Completion (with grades of C- or better) of one of these calculus sequences is required for a mathematics major. For some degrees, the two-semester sequence M 408N/S satisfies the calculus requirement. These two courses are also a valid prerequisite for some upper-division mathematics courses, including M 325K, 427K, 340L, and 362K.

M 408N may not be counted by students with credit for any of Mathematics 403K, 408K, 408C, or 408L.

**Course description:** M 408N is the first-semester calculus course of the three course calculus sequence. It is directed at students in the natural sciences, and is restricted to College of Natural Science Students. The emphasis in this course is on problem solving, not on the presentation of theoretical considerations. While the course includes some discussion of theoretical notions, these are supporting rather than primary.

The syllabus for M 408N includes most of the basic topics in the theory of differential calculus of functions of a real variable: algebraic, trigonometric, logarithmic and exponential functions and their limits, continuity, derivatives, maxima and minima, as well as definite integrals and the Fundamental Theorem of Calculus.

**Overview and Course Goals**

The following pages comprise the syllabus for M 408N, and advice on teaching it. Calculus is a service course, and the material in it was chosen after interdepartmental discussions. Please cover the material that is not deemed "optional." 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 408N/S/M is the sequence designed for students who may not have taken calculus previously. Students who have seen calculus and have done well might be better placed in the faster M 408C/408D sequence.

**Resources for Students**

Many students find the study skills from high school are not sufficient for UT. The Sanger Learning Center (http://lifelearning.utexas.edu/) 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 they will come to your classroom and explain their services.

You can help your students by informing them of these services.

**Timing and Optional Sections**

A typical fall semester has 42 hours of lecture, 42 MWF and 28 TTh days, while the spring has 45 hours, 45 MWF and 30 TTh days (here, by one hour we mean 50 minutes -- thus in both cases there are three "hours" of lecture time per week). The followimg syllabus contains suggestions as to timing, and includes approximately 35 hours. Even after including time for exams, etc., there will be some time for the optional topics, reviews, and/or additional depth in some areas.

**Syllabus**

- 1 Functions and Models (3 hours)
- 1.4 Exponential Functions
- 1.5 Inverse Functions and Logarithms

- 2 Limits and Derivatives (9 hours)
- 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 (optional)
- 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 (10 hours)
- 3.1 Derivatives of Polynomials and Exonential 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 (
*optional*) - 3.9 Related Rates
- 3.10 Linear Approximations and Differentials (
*optional*) - 3.11 Hyperbolic Functions (
*optional*)

- 4 Applications of Differentiation (9 hours)
- 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 (optional)
- 4.7 Optimization Problems
- 4.9 Antiderivatives

- 5 Integrals (4 hours)
- 5.1 Areas and Distances
- 5.2 The Definite Integral
- 5.3 The Fundamental Theorem of Calculus