## Syllabus: M408K

DIFFERENTIAL CALCULUS

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

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

**Core curriculum**

This course may be used to fulfill the mathematics component of the university core curriculum and addresses core objectives established by the Texas Higher Education Coordinating Board: communication skills, critical thinking skills, and empirical and quantitative skills.

Calculus is the theory of things that change, and so is essential for understanding a changing world. Students are expected to use calculus to compute optimal strategies in a variety of settings (Chapter 3, max/min), as well as to apply derivatives to understand changing quantities in physics, economics and biology.

Students improve their number sense through qualitative reasoning and by comparing the results of formulas to those guiding principles.

Student activities include creating logically ordered, clearly written solutions to problems, and communicating with the instructor and their peers during lecture by asking and responding to questions and discussion in lecture.

**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: M403K, M408C, M408K, M408N.

Calculus is offered in two equivalent sequences: a two-semester sequence, M 408C/408D, which is recommended only for students who score at least 600 on the mathematics Level I or IC Test, and a three-semester sequence, M 408K/408L/408M.

For some degrees, the two-semester sequence M 408K/408L satisfies the calculus requirement . This sequence is also a valid prerequisite for some upper-division mathematics courses, including M325K, 427K, 340L, and 362K.

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.

**Course description:** M408K is one of two first-year calculus courses. It is directed at students in the natural and social sciences and at engineering students. In comparison with M408C, it covers fewer chapters of the text. However, some material is covered in greater depth, and extra time is devoted the development of skills in algebra and problem solving. This is not a course in the theory of calculus.

The syllabus for M 408K 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, as well as definite integrals and the Fundamental Theorem of Calculus.

### Overview and Course Goals

The following pages comprise the syllabus for M 408K, 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 408K/L/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 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.

### Forty Class Days As:

- 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 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
- 3.8 Exponential Growth and Decay (
*optional*) - 3.9 Related Rates
- 3.10 Linear Approximations and Differentials
- 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
- 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