Clark University
                        Math 120 Calculus I
Fall 2014
Prof. D. Joyce, BP 322, 793-7421
Department of Mathematics and Computer Science
Clark University

Veritas et Falsitas
   
Zeno of Eleashows Youths the Doors to Truth and False
   
click for source Zeno of Elea (ca. 490–ca. 430) shows Youths the Doors to Truth and False.
Fresco in the Library of El Escorial, Madrid.
Date 1588–1595, by Bartolomeo Carducci or Pellegrino Tibaldi

[This course page is under construction.]

Please bookmark this page, http://aleph0.clarku.edu/~djoyce/ma120/, so you can readily access it.

  • Syllabus. We will follow the order of topics in the text, University Calculus by Hass, Weir and Thomas, but we’ll stress some topics and pass over some others.

    Preview. Calculus is about the relation between a quantity and its rate of change. For an example, if the quantity is the distance (from the initial position) on a line at a given time, then its rate of change is velocity. If the velocity is constant, then calculus is not required: the distance travelled is the product of the elapsed time and the velocity. But when the velocity is not constant, then this formula doesn’t apply. Nonetheless, the distance and velocity are intimately related. If the distance at all times is known, then the velocity at any given time can be determined; and if the velocity at all times is known, then the distance at any given time can be determined. These two operations are called differentiation and integration.

    Much of calculus involves analyzing and developing these concepts and their applications.

    Proofs are going to used throughout the course. When we first meet a concept, we’ll discuss it intuitively. Then we’ll formalize it with a formal definition. We’ll use that definition to prove the things we expect to be true actually are true.

    Notes:

    Chapter 1. Review of functions. There are a slew of things in the first chapter that you should already know. We aren’t going to cover them in class. We’ll mention a couple of them, but chapter 1 is primarily there to show you some of the things that you’re supposed to know already.

    Besides what’s in chapter 1, there are all the things on the page Mathematics background needed for calculus at http://aleph0.clarku.edu/~djoyce/ma120/background.html too.

    If you’re trigonometry is rusty, you can look at Dave's short course in Trig for a review of trigonometry.

    Chapter 2. Limits and Continuity. After the introduction explaining where we’re going, this is where the subject matter starts.

    We first must clarify the concept of derivative. In some ways it is intuitively clear that a travelling body has a velocity, or more generally, any changing quantity has a rate of change. But just what is the rate of change? The answer is that the rate of change at an instant is the limit of the average rates of change near that instant. That is, the derivative (instantaneous rate of change) of a function  f(x)  is the limit of the average rate of change of the function over an interval  [x, x+h]  as the length h of that interval approaches 0. The average rate of change over the interval is how much  f(x)  changes over that interval divided by the length of the interval h. We want the limit of that average as the length h approaches 0.

    The concept of limit is much more subtle than it first appears. We will discuss it in some detail and develop a formal defintion of a limit and a formal notation to go along with it.

    Key concepts associated to the concept of limit are tangent lines, limit laws, continuity, the sandwich theorem (also known as the pinching theorem), the formal definition for limits, discontinuities, asymptotes, continuity, the intermediate value theorem (IVT), and the definition of a derivative (which is why we study limits in the first place).

    Notes:

    Chapter 2 Limits and Continuity

    Chapter 3. Derivatives. Now that we’ve got a solid definition of limit (that was section 2.7), we can start to study derivatives. There are a number of rules for differentiation (finding derivatives), mostly easily learned, although the chain rule, for some reason, seems to be more difficult to master. There are a couple of different notations for derivatives that everyone uses.

    It is assumed that you know the trig functions, sine, cosine, etc., and we will find and use their derivatives. Further topics in differentiation include higher derivatives, implicit differentiation, differentiation of inverse functions, and differentiation of logarithms. We’ll finish this chapter studying situations when several quantities are changing; we’ll see how their derivatives are related.

    Notes:

    Chain rule

    Chapter 3 Differentiation

    Chapter 4. The Mean Value Theorem and Applications. We’ll verify some intuitive ideas with formal proofs. For instance, if the derivative is positive, then the function is increasing; at a maximum or a minimum of a function, the derivative is zero. We will prove these (obvious) statements using a theorem called the mean value theorem. We’ll also do some curve sketching. The purpose of curve sketching is no so much to draw the graph of the function, but to get a better understanding of the relation between a function and its derivative. We’ll also see what second derivatives have to do with the graph of a function.

    The applications of derivatives are numerous. Besides classical applications in physics and the natural sciences, there are applications in the social sciences, for instance, marginal profits are just derivatives of profits.

    Sample graph of a function Notes:

    Chapter 4. The Mean-Value Theorem and applications

    Chapter 5. Integration and the Fundamental Theorem of Calculus (FTC). Derivatives are only half of calculus. We studied them first because they’re the easier half. The other half is integration. For a positive function, its integral is just the area under the graph of the function. Although for many purposes, the intuitive concept of area is enough to understand and use integrals, we’ll look at a more formal algebraic definition in terms of limits. It gives a way of approaching that area that can be used in practice and in theoretical proofs. We’ll close with a formal proof of the FTC which ties together the main topics of the course.

    Notes:

    Chapter 5 Integration and the FTC

  • Tutors If you’re having any difficulties in the course, there are tutoring groups which will be set up and coordinated by Professor Satz.

  • Time and study

    Besides the time for classes, you’ll spend time on reading the text, doing the assignments, and studying of for quizzes and tests. That comes to about five to nine hours outside of class on average per week, the actual amount varying from week to week. For more detail about how to study mathematics, see About studying mathematics in general, and Calculus in particular.

  • Tests and final exam There will be two tests during the semester and a final examination during finals week in December. The final is cumulative, but post-midterm material will be emphasized. The tests are closed book, no calculators, and no notes allowed. There may also be short quizzes given periodically throughout the semester.

    [Dates for the midterms and finals to be determined.]

  • Course grade [Tentative] The course grade will be based on 20% for homework assignments and quizzes, 25% for each of the two midterms, and 30% for the final.
    Class attendance and class participation are obligatory. During the class meeting the text will be supplemented with more rigorous theory and special topics.

  • Course pages and other associated pages

  • Last year's tests

  • Web pages for related courses

    This page is located on the web at
    http://aleph0.clarku.edu/~djoyce/ma120/
    David E. Joyce