Calculus 1 -- Fall 2017

Our first exam will be Friday, 9/29.  This will be a closed note and closed book exam.  In particular, students will not be permitted to use any notecards or other references during the exam.  A calculator is permitted, but not a calculator app on a smart phone.

If you are keeping a portfolio, bring it to the exam for me to review.  You should also remember to pick it up after the exam as you leave.  For more information about requirements for your portfolio, click here.

The exam will cover these sections of the text:  1.7, 1.8, 2.1-2.6, 3.1-3.3.

Overview of Material.   In sections 1.7 and1.8 we learned about limits and continuity.  Most of  chapter 2 focussed on definitons, interpretations, and properties of derivatives.  As a result,our studies so far have been relatively light on procedural knowledge: we have not covered a large number of techniques, although the start of chapter 3 does emphasize procedures.  Most of our time has been spent in consideration of conceptual understanding of the ideas of limits and slopes and derivatives.  So you should expect to see a strong emphasis on this sort of conceptual understanding on the exam.

As a general rule, I try to make each of my exams cover a variety of different kinds of knowledge.  You certainly need to understand and know how to apply procedures.  So you will see some questions of that sort on the exam.  But you also need to have a factual understanding of what is true.  That means knowing what the definitions and theorems say.  Ideally, you should know the correct mathematical definitions of limit, continuous function, slope of tangent line, and derivative. Otherwise, if you have a fuzzy understanding of the specific meanings of these things, you will be in  no position to understand the theorems and techniques that make reference to them.  Similarly, if you don't know the statements of the theorems, you cannot say for sure, in any particular situation, whether or not the theorems can be applied.  For example, one theorem states that under certain circumstances you can take a limit by decomposing a function into simpler parts.  But what are the circumstances?  Are there situations where this sort of decomposition can lead to a wrong answer?  Unless you know what the statements of the theorems are, you simply cannot tell.  So, there may be a few items on the exam that will ask you to give mathematically correct statements of important definitions and theorems.

Another kind of exam question tries to get at conceptual understanding beyond the exact statements of theorems and definitions.  For example, we saw in class a theorem that says that all polynomials are continuous at every real number.  This has a meaning in terms of the correct mathematical definitions of polynomials and continuous.  But there is also an understanding that goes beyond the definition, that being continuous means that the graph is an unbroken curve.  You should know both the correct mathematical statements of definitions and theorems, but you should also be able to explain, in your own words, what these things mean.

As a kind of topical breakdown, here is a brief outline of what we have covered.  First, you should understand and be able to work with the concepts of average velocity, speed, or rate of change, and the slope of a secant line (which is a line connecting two points on the graph of a function).  You should know how these are related to each other.  Given a function in the form of a graph, an equation, or a table of data, you should be able to compute and interpret slopes and average speeds.  You should understand how we use secant slopes and average speeds to determine slopes of tangent lines and instantaneous speeds, both in terms of successive approximation and also in terms of limits.  You should understand the limit concept, and be able to make informed guesses about the values of limits using numerical data and/or graphs, or using equations for functions.  You should also know the limitations of such informed guessing, and how to compute limits exactly using algebraic methods, and the rules for finding exact limits.  You should know what those rules are, and how they get applied.  This includes rules about simple functions (f(x) = x or a contant), and rules about breaking functions down into simpler parts (like lim (f +g) = lim f + lim g).   Similar comments apply to derivatives. You should know what is meant by continuity or differentiability of a function at a point, and of course, how to recognize what functions are continuous or differentiable and when and where they are not.  One of the consequences of continuity is the intermediate value theorem; know that.  Know the definition of slope of tangent line, derivative of a function at a point, and what it means when we say the derivative of a function is another function.  In addition, be familiar with the notation f ' (x) and the Leibniz notation expressed as a fraction dy / dx.  From chapter 3, know the rules for derivatives of polynomials, exponentials, and sums and differences of functions, products and quotients of functions, and constant multiples of functions.  Know the power rule, and that the derivative of a constant is 0.

To help you study, I am providing a sample exam: CLICK HERE.  Be advised that this is not a comprehensive review.   Do not limit yourself to studying the sample exam.  To do well, you need to study all of the material we have covered, by reviewing the text, your notes, class worksheets, and assigned homework.  The sample exam serves merely to illustrate a mix of question styles (procedures, conceptual understanding, definitions and theorems) and the approximate length you can expect.  Actually this sample exam was assembled from parts of two different exams used in prior semesters, and is probably a little longer than the actual exam will be.

In general, an exam on a body of material cannot directly test every single topic or idea covered.  There is not enough time.  So, in making up an exam, I have to choose certain things to include and others to leave out.  The same is true of the sample exam.  But do not assume that the real exam will include and leave out exactly the same topics as the sample exam.  There will almost certainly be things on the sample exam that do not appear on the real exam, and vice versa.