Calculus 1 -- Fall 2017

Information about the first exam

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.

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.