Notes from Class
The sandwich theorem can sometimes let us calculate limits of complicated functions by comparing to simpler functions.
Lecture notes from class will appear here each evening after class. The slides in class may contain animations which will only play in Adobe Reader, and will not play in another PDF viewer.
The printable version of the notes contains the same material, but without "incomplete" slides; e.g., a bulleted list where more bullets are revealed on consecutive slides appears as only a single slide containing the complete list in the printable notes. If you are printing the notes, you will save a lot of paper by using the printable version.
Quizzes:- Lecture 1 - Introduction to Math 1060, and review of exponentiation
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An introduction to Math 1060 and coordinated courses in general,
as well as some specifics of Dr. Johnson's sections. Also
contains a review of exponential functions.
Printable version. - Lecture 2 - Inverse functions and logarithms
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A quick review of the basics of functions and their inverses, in
particular describing how to tell if an inverse exists and how to
algebraically calculate an inverse. These ideas are then applied
to two special types of examples: inverse trig functions and
logarithms.
Printable version. - Lecture 3 - Limits
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In this lecture we introduce limits from a naïve point of
view, looking at the graphs of several different types of
functions.
Printable version. - Lecture 4 - Limit laws
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We show how to apply various "limit laws" to algebraically
calculate the limits of several different types of functions which
may be built from simpler functions. In particular, we discuss
the limits of constant functions and the identity function, then
apply the limit laws to determine the limits of polynomials and
rational functions. We also discuss the squeeze theorem (aka the
sandwich theorem) which can sometimes let us calculate limits of
complicated functions by comparing to simpler functions.
Printable version. - Lecture 5 - The precise definition of a limit
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We replace the earlier "hand-wavy" definition of limit with the
rigorous $\epsilon$-$\delta$ definition. The notes contain
specific examples of finding the right $\delta$ for a chosen
$\epsilon$, and give proofs of some of the limit laws we used in
the previous lecture.
Printable version. - Lecture 6 - Continuity
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In this lecture we define continuity, make a few simple
observations about continuous functions, state the intermediate
value theorem and some interesting applications, and finally
consider a few "pathological" examples to show that continuity is
not quite as intuitive an idea as it may seem at first glance.
Printable version. - Lecture 7 - Limits at infinity
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Limits at infinity and horizontal asymptotes are introduced and
several examples are worked out. In particular, we consider some
examples which show that arithmetic with infinity does not follow
the same rules as arithmetic with real numbers.
Printable version. - Lecture 8 - Derivatives
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We introduce derivatives in this lecture, first considering three
motivational problems: calculating tangent lines, calculating
instantaneous velocities, and calculating instantaneous rates of
change. We then note that the solutions to all of these problems
follow a common theme: approximate difficult-seeming problems with
easier problems we know how to solve, and then try to improve our
approximation. In the limit, this gives us
the derivative of the function. We define derivatives as
special types of limits, show a few simple calculations, and then
mention applications to other areas of mathematics, physics,
computer science, and engineering.
Printable version. - Lecture 9 - The power rule, polynomials, and exponentials
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In this lecture we begin our study of "shortcuts" which make
calculating derivatives much easier. In particular, we discuss
derivatives of polynomials and exponential functions. Several
simple examples appear in the notes.
Printable version