Calc 1A Assignment Two
One of the things
mathematicians wish to do is study functions. As we approach a higher level of
sophistication, we'll start to study behaviors of functions in a very 'local'
way. Imagine magnifying the graph of a function about a particular point. We
are studying the function's behavior then in what we call a 'neighborhood' of a
point in its domain. We wonder what the function is doing. Is it approaching a
particular value? (If so, that value is called 'the limit of the function at
the point'.) Is it increasing in absolute value without bound? Does it exhibit
'pathological' behavior? Such study is particularly interesting when we use
analytic tools (i.e. algebra and our brains) instead of merely looking at
graphs. Historically this is what was done, since mathematicians of yore didn't
carry TI-83s nor did they have easy access to graphs of functions. We call such
study: determining if a function has a limit or not. In the case a limit
exists, there are many instances in which we can actually determine the limit.
We will also
consider 'long run' behavior (also known as 'extreme' behavior) of functions.
This means we will ask the question: If I go really far out to extreme values
of the domain (assuming the domain includes points which are, in absolute
value, very very large), how will the function behave? Again: Is it approaching
a particular value or not?
Along the way we
will hit some bumps. These will result in studying some interesting oddities
which may occur. For example, we will look at functions with holes. We will
look at piecewise functions.
Once we
understand how to determine limits, we are ready to be introduced to an
important property of some functions: Continuity. Initially, continuity of a
function will be determined on a point by point basis, but with some theorems
in hand, we can describe whole functions as 'continuous'. Continuity is THE
property which permits practically all important theorems in calculus to hold.
Among the various
theorems we will be introduced to, we will encounter one known as the
Intermediate Value Theorem, which at a theoretical level is quite deep, even
though to a student at your level may feel more like a "yeah, so
what?!". We will also discuss how continuity at a point may effect our ability
to analytically determine limits in a composition.
We will be
reviewing categories of functions throughout. In particular, we will be
reviewing polynomial functions, rational functions and root functions.
This Assignment
will by far feel the weirdest to you. You will be given strict definitions,
which is something you may not be used to. Additionally, we'll be using
theorems to demonstrate things in a formal way which may seem downright obvious
on an intuitive level. But the rules of the game will hopefully, with time, be
appreciated since they form the foundation of much that will follow.
If all goes
according to plan, we will be following the daily schedule below:
|
Day Number |
Topic |
Homework |
|
One |
Overview of Limits: Definition, notation, using graphs and tables |
Pg 124 #1,3,9,11,17,19 |
|
Two |
Formal properties Quiz, class after next, through today's lesson. May include absolute value problems. |
Pg 124 # 5,7,13,21, Pg 137 #1,3,49
|
|
Three
|
Rational Functions: polynomial review, Holy vs. Unholy functions, Limit of 1/x^n at 0 and infinity |
Study for Quiz Pg 137 #5-11 odd, 51 Pg 124 #27, 29 |
|
Four |
Quiz and Rational functions at infinity (Horizontal Asymptotes) |
Pg 137 #25-29 odd, 13-17 odd,
37,39,53,63 |
|
Five |
Continuity: At a point, Left vs. Right cty, Using properties of limits Quiz, class after next, through today's lesson |
Pg 137 #33,35, 43 Pg 156 #1-9 odd |
|
Six |
Continuity cont'd: Continuity of a functions, Classes of cts functions, Removable discontinuities |
Pg 156 #11-17 odd, 21,25,27 odd
, Pg 137 #31 |
|
Seven |
Quiz and Theorem: Composition Limit Theorem |
Pg 156 #23,29-33 odd,19 |
|
Eight |
Trig functions and continuity |
Pg 163 #1,3,9, 11a)-c) |
|
Nine
|
Sinx over x at 0 limit |
Pg 163 #7, 11d)-f),17, 19, 21, 25,27,33 |
|
Ten |
Intermediate Value Theorem |
Pg 156 # 41,37,42 |
|
Eleven |
Inverse Functions |
Pg 233 # 1a),1c), 3, 5 d)-f),7,9,15,33,37 |
|
Twelve |
Review and Practice. |
Study |
|
Thirteen |
Exam (includes Assignment One Material which has been incorporated in this Assignment) |
Format: 5 or 6 multiple choice problems |