Preparing for the Final
Make sure that you get a copy of the Final Exam Review Sheet. This
review sheet should serve as a guide to studying for the final exam. You
are
strongly encouraged to construct a review packet covering the topics
listed on the review sheet. This packet should include definition,
theorems,
your explanations of the key concepts/ideas, as well as sample problems
and solutions. Your class has decided that you will turnin a copy of
this review
packet the day of the final exam and it will be considered as part of
your final exam grade (roughly 10%). You are also allowed to use this
packet on the
final exam itself.
To help you prepare for the final, I outlined the general feel of the
final exam in class on Friday, December 3. Your classmates voted on
certain topics
that we would cover in class during the review sessions scheduled the
week of December 6. Listed below are topics for each day and a list
of problems that
you should look at before coming to class. Note that some of the
problems may be homework problems that you have already completed. Try
them again and
check your answers with your solutions or the the solutions posted online
(see homework page).
Monday, December 6
Topics Include: Derivatives (Definition, Existence), Linear
Approximations/Tangent Lines, Optimization, and Curve Sketching
Derivatives: Review the Definition of the Derivative.
(pages 107, 108)
Think about when a derivative
exists (i.e. what conditions must we place on f(x) so that we know we can
find f'(c) for some x=c?).
Think about f(x)=
|x| and try computing f'(0) using the definition. Why does f'(0)
fail to exist...think about what is happening at f(0)?
Think about f(x)
= 1/x and try to computing f'(0) using the definition. Why does f'(0)
fail to exist?
Linear Approximation/Tangent
Lines (pages 100, 101, 153, 154)
Remember
that Tangent Lines provide a good approximation to the function for values
of x close to c
That is,
if g(x) is the line tangent to f(x) at x=c, g(x) is approximately equal
to f(x) for x values close to c. Try some out!
Examples
1,2 on Page 100, 101
Page 105:
9, 10, 11, 12
Page 119:
49, 50
Page 155,
156: 36, 37
Page 196:
34, 35
You do not need to do all problems; just pick a few and work through
them. Do enough so that you are comfortable with the
idea and computing them.
Optimization (pages 174-177
and Section 4.4)
Remember that
the goal is to find maximum and minimum values of a function!
We will cover this later in the week, but think about
why we might want to prove the existence of a value x such that f'(x)=0.
To do this, one would most likely use the Intermediate
Value Theorem.
Review the First
Derivative Test and the Second Derivative Test
Check out Examples
1, 2, 3, 4, 5 on Pages 175, 176
Section 4.3: 7-14,
20, 23-27
Section 4.4: 1,
4, 7, 8, 9
Curve Sketching (pages 192-195...review
steps provided on page 195 to help with #1 below)
The goal is to
be able to use information about f'(x) and f''(x) to figure out the general
"shape" of the f(x) and then try your hand at providing
a graph of f(x).
Review the Monotinicity
Theorem and the Concavity Theorem.
Review Stationary
Points, Singular Points, and Inflection Points
Check out Examples
1, 2, 3 on Pages 193, 194, and 195
Try problems 1,
28, 29, 30, 31, 32, 37, 38, 39 from Section 4.6
I encourage you to do #1, one or two from 29-32 and
at least one from 37-39
Tuesday, December 7
Topics Include: Implicit Differentiation, Related Rates, Antidifferentiation
Implicit Differentiation: The goal here is to find a way to find the
derivative of a function y = f(x) that is defined implicitly. Implicit
equations frequently show up
in many research
settings and so we must find a way to deal with them. Once we find
the derivative of an implicit equation, we can do the
same things with
them as we did with derivatives of explicit equations (Linear Approximation/Tangent
Lines, Optimization, etc.)
Review what an implicit equation
is. (Page 139)
Review how one can show an implicity
equation is a function y=f(x). (Page 139)
Review the technique of implicit
differentiation and convince yourself why we need to use the chain rule to
find the derivative of y with
respect to x. That is why is the derivative
of y^2 with respect to x given by 2y*(dy/dx) instead of just 2y? (Page
139)
Check out Examples 1, 2, 3 on
Pages 140, 142
Some good problems to try include
1-12 in Section 3.8
Again, no need to do them all. Just pick a few we
did in the homework or that have solutions in the back of the book. Try
them
until you are comfortable with them.
A great application of implicit
differentiation is related rates. Related rates are just what they
say them are. The rate at which two or three
things are changing may be related
to one another. For instance, the rate at which volume of a bubble
increases is directly related to the rate
at which air is entering the
bubble. Another example might be the rate at which a boat is entering
the dock is directly related to the rate at
which the boat is being pulled
by a person standing on the dock. These were actually a couple of examples
we did in class, but in practice the
examples are virtually endless
since much of the world around us has interacting (hence related) variables.
Check out Examples 1, 2, 3 on
Pages 145, 146, 147.
Try some problems: 1,
2, 5, 6, 7 on Page 149.
Don't spend too much time on these problems. Make
sure you understand the general procedure and the underlying idea of the
problem.
Antidifferentiation: The goal of this course was to understand what
a function is, what kind of functions are commonly used in calculus, what
a continuous function is,
what a derivative is, and last, but certainly not least,
what a definite integral is. To answer the last question, we must be
familiar with the process
called Antidifferentiation. Recall the general idea
is I give you f'(x) and you try to figure out what f(x) is.....we work backwards
from finding the
derivative (hence anti-differentiation).
Review the definition and make sure you understand
the idea of family of functions. (Page 209)
Review the rules for antidifferentiation. (Pages
210, 211, 212, 213)
Make sure you understand u-substitution (many of
you need some extra help here...so check out the book and come with questions.)
Check out Examples 2, 3, 4, 5, 6 on Pages 210, 211,
212, 213
Some good problems include 1-39 in Section 5.1 as
well as 1-26 in Section 5.8
Also try your hand at solving some ODE (Ordinary Differential
Equations using the Separation of Variables Technique)
Check out Examples 1,2 on Pages 215, 216, 217
Some good problems include 5-14 in Section 5.2
Wednesday, December 8
Topics Include: Definite Integrals, Intermediate Value Theorem,
Limits (as needed)
Definite Integrals: One of the reasons we study Antidifferentiation
is so that we can compute Definite Integrals using the Fundamental Theorem
of Calculus. Remember
the general idea of the definite integral is that we are simply "adding
stuff up." We generally think of definite integrals as computing the
"area" of the
region bounded between the function and the x-axis (where x is the
independent variable) on some domain [a,b].
Review the Definition of the Definite Integral. (Page 236)
Review the types of functions that we can compute Definite Integrals for
(the integrability theorem!). (Page 237)
Review the Fundamental Theorem of Calculus I and II. Make sure you
note that the conditions we must impose on f(x) to use these theorems!
Check out Example 3 on Section 5.5; Examples 1, 2, 3, 4 on Section 5.6; Examples
1, 4, 5, 6, 7, 8, 9 on Section 5.7; Examples 1, 2 Section 5.8;
Try your hand at problems 7, 8, 9, 10 on Section 5.5, problems 13 - 22 on
Section 5.6, 1-32 and 47-50 on Section 5.7, 27-33 on Section 5.8
Get comfortable with recognizing
definite integrals in the reimann sum form and using the Second Fundamental
Theorem of Calculus to
compute them. Also,
make sure you are comfortable with the First Fundamental Theorem of Calculus.
Applications
we covered include finding Areas, Volumes, and Arc Length.
Check out Examples 1, 2, 3, 5 (make sure you think about why the Intermediate
Value Theorem could be important here as well) on Pages 273, 274, 276
Check out Examples 1, 2 on Pages 281, 282 and Examples 1, 2, 3, 4 on Pages
294, 295, 296
Look over the problem I suggested in class!
Intermediate Value Theorem: Proving the existence of solutions of a
function on some interval. Make sure you understand why this theorem
is useful in the context of
Optimization and Definite Integrals (we could also discuss the importance
of this theorem for Differential Equations). Look over the
theorem and review the example problems. Make sure you
also understand the conditions we must impose on f(x) to use this theorem.
Limits: Understand what limits attempt to do. What is the goal?
Think about how we are using limits to find the derivative or definite
integral. Look over the main
limit theorems
as dictated by the handout. Remember that these theorems are important
in many applications!