Good afternoon and welcome to the fall quarter
This is Math 2B: Calculus and my name is Natalia Komarova
I am a professor here at the math department and I’ll be teaching you this subject
first I want to go over the basics of this class and tell you a little bit about
how it’s organized first of all, I have created a website
with all the information already contained on the website pages
but today I will still spell everything out for you
so first of all, the textbook so this is probably the most confusing part
of the whole thing so it’s called calculus: Early Transcendentals
so it’s 7th edition of calculus
by Stewart so the most important part
is Early Transcendentals there is a 7th edition that does not contain
these words look at your book,
if it doesn’t say that, that’s the wrong book
now there’s various shapes and forms of this book
for instance, what I have only contains single variable calculus
so that’s good for 2A and 2B and so that will be fine for this class
there’s also a bigger book that has, you know, parts that pertain to
other classes like if you’re planning to take 2D or 2E
you should get the full book. The big one. There’s also another online book,
So there are many questions that people ask me over email
Is it okay to use a second hand copy? And the answer is yes. It’s okay.
If you bought your book from a friend, if it’s used,
And your electronic stuff doesn’t work If you cannot create an account, that’s
okay All you need from this book is the chapter
material And the homework problems
Okay? And so I do not require you to buy a new copy, of course
a secondhand copy is okay any questions about the book?
Okay. So now exams.
We’re going to have two midterms On October 18th, and November 8th
And we are going to have a common final exam. Question?
Will we need to bring our book to class? No, absolutely not.
So common final which is held on Saturday, December 7th
from 1 to 3 so I will tell you a little bit about the
common final but first I know that according to the policy
of the mathematics department you are not allowed to use books, notes, or
calculators during any tests. So it is a closed book test, no cell phones,
no calculators are allowed. Now what’s a common final?
If you’ve taken 2A, you know what that is. It means that you come here on Saturday.
And it’s a common final that’s held across all the sections of this class
So there are unique requirements for everybody You have to produce your valid UCI ID card
For the midterms and the final You have to make sure that you have an ID
And you also have to make sure that you are recognizable in the picture
so very often students produce something that looks like this
with just the circle instead of a face so make sure that you get your ID replaced
such that the picture is recognizable so another thing about common final is that
if you cannot make it you should let us know early on
and you don’t let me know, you let the secretaries of the math department
know you have to fill out the form that is contained
online you have to follow the link on the website
there is a special form a standard form that you fill out
and there should be no problems there. You can arrange for a make-up
But of course you have to have a valid reason Not to attend the final
Questions about the final? Question?
Do we need a scantron or a bluebook? No, nothing like this. I will provide a paper
copy of the exam And all you need is a pencil and an eraser
Yes? When you said no calculator, are we allowed
to bring a simple calculator no.
no but we always make sure that you can do all the math in your head
there will be nothing horrible more questions?
Will a calculator be needed for the class? No, no.
You can use it when you do homework, of course But in class, no
Okay. So now some other assignments that you have
apart from the midterms and the final we will have homework, okay?
so the way it works in this class is homework is optional
which means that it’s not graded nonetheless, this is probably the most essential
part of this class because everything that you’re tested on
is based on this optional homework.
So the list of homework problems is provided in the website for the class.
And it goes by section number So for instance section 6.1
and it gives you a list of homework problems so you sit down and do as many as you can
after we have covered the material if you’re fine with all the homework problems
for each section you’ll get an A+
okay? If you’re fine with most of them you’ll get an A. And so on.
So this is your way to study. Do the homework problems
nobody will test them directly, but we will have quizzes
quizzes are held once a week at the discussion sessions
on Thursdays and the quizzes are completely based on the
homework assignments for the previous week
so if you’ve done your homework you will know how to do the quiz problems
it’s either just a homework problem taken from the list or something that is very very
close to it so in order to prepare for the quiz, you have
to do the homework. And the quizzes are graded
okay, so this is graded now another part of assignments is webwork
can you raise your hand if you know what the webwork is?
Do you know what it is? Oh, so I see some of you are not familiar
with what it is so this is an online homework assignment
so on the website for the class I created a link that takes you to the webwork
homepage this page is not active as of now
they will activate it in about two weeks when the first assignment is posted
so the first assignment will be posted the tenth of October
okay so until then you don’t have to worry about it
so you go there, you log in, and you do your problems.
They will be 8 webwork assignments during the fall quarter
There will be every week, posted on Thursday and due the next Friday
The first webwork assignment is based on the problems from the beginning of the class
And they are cumulative such that later on webwork problems could test your
knowledge from, you know, far long ago. So everything that you have studied up to
that date can be tested with webwork
each assignment has a varying number of problems and I posted the full schedule of all the
webwork assignments that is already known for class
so you will know the due dates and the dates when these things are posted
you have an extension for thanksgiving for thanksgiving week, you have a little bit
more time to finish so there are a few things that you need to
know about webwork
so among the class files I posted a PDF file
that tells you about webwork.
So one new thing that they told us about this year
is that somehow it doesn’t work very well from
so campus. It works very well from campus if you’re away from campus, you have to
use VPN to connect, and the instructions that will
help you set it up are given on the website for the class.
And there are various quirks associated with webwork
Because sometimes you may type something in the wrong font
And it thinks that you made a mistake all these things
please read what I posted it gives you a lot of information
and so if you experience technical problems with webworks,
please do not email me because I will not be able to help you
I can only confuse you if you ask me technical questions
About the website setup and stuff email them
there is an email address provided on the website of webwork
and they will be able to resolve your technical issues
if you have a problem with your mathematics then I’m your guide. Okay?
Email me or come to the office hours and I will be able to help you with any math
problems but not with technical problems and website
problems like that questions?
Okay. Grade consists of 40% final
20% each of each of the two midterms 10% webwork
And 10% quizzes So the lowest quiz and the lowest webwork
are dropped So one lowest dropped
And here the same because of this policy, there is no opportunity
to make-up for quizzes or webwork
there’s one seat here if you need to sit down
and there is also one over here there are also two seats over there
in the middle so if you miss a quiz
or if you miss a webwork don’t worry about it, because
because of this you have a chance to miss one
question so is the webwork like an online quiz, or
is it like an online homework assignment? it is like an online homework assignment
I think you have an infinite number of attempts More questions?
Will we be able to use a calculator on the webwork?
oh, yeah. You’ll be doing it at home but not during quizzes
so quizzes are like exams more questions?
Okay. Now I think the last thing is this so Early Transcendentals
so what does it mean? It means that if you took calculus 2A
Prior to fall 2012 You will have some catching up to do
So the mathematics department has changed the syllabus
We used to teach things like logarithm Or arcsin
or e^(x) we used to teach that in 2B
now these things are taught in 2A and you are supposed to know them
you are supposed to know these things and you are supposed to know their derivatives
you should know what they look like, you should be able to plot them
so how many of you have not seen those before okay, very good. So that’s good
however if you want to refresh your memory by
these functions there is some online video material
you can watch the videos and refresh your memory about these things
such that you are well prepared for what we have here
instead of these things, we will be studying sequences and series
which are much more fun. I’ve taught both, this is better
Questions? Okay
question are the grades going to be curved?
no, so there is no curve there’s going to be a standard grade
so the finals are graded across the whole you know, all the sections
then if they think that it was too hard or too easy
then they are going to uniformly add some points or subtract some points
like multiply by a factor so everybody gets the same treatment
and then after that there’s no curving. Also there is no extra credit
There’s nothing you can do But do the homework
Okay? So everything is very straightforward in this class
Old information is in the homework If you have questions, if you don’t understand
something Come to my office hours
which I will announce next week I don’t know yet
when my office hours are and that will be fine. Question?
You said that if the average is low, is the final curved or the class is curved
Like overall so if they think that the common final was
too hard, they are going to kind of boost that grade
just for the final though? just for the final.
So you don’t curve this class? No. And they are not supposed to.
What’s the website? I think you can find it on EEE
there’s a link also you can go to my homepage and there’s
a link there but it’s easier to go from the EEE
just to make sure, you said the webwork is due Friday? That’s the following week, correct?
yes, it’s posted on a Thursday it’s due the Friday next week
so it’s the following week, yes so you have a week and one day to finish it
more questions? Okay.
and you can always ask me questions during the class
just raise your hand, okay? so now we do some review
of some essential stuff that you learned in 2A
if you cannot see something, please let me know
so I avoid some parts of the whiteboard So today we cover section 4.9
antiderivative okay ready?
So let’s suppose that the derivative of function F
is given by lower case f for all values of x in an integral
then F capital is called the antiderivative of f.
so for example let’s suppose that f(x) is equal to x^(2)
what’s the antiderivative of this function? So there are two opposite operations
you can take the derivative and you can go back and take the anti-derivative
question? So this happens to be one of the antiderivatives
Of this function In fact, how many are there?
Infinitely many Plus C
So by adding a constant C We create an infinite family of functions
each of which is an antiderivative of this how can we check?
So this is antiderivative To check, we say what is
F’ (x) we have to differentiate
so we have (3x^(2) /3)+0 is x^(2) check
because it coincides with my original function question
I cannot see that you cannot see this?
That’s very bad. Okay. so are you telling me that you can only see
above this line? okay.
so we have a theorem if F is an antiderivative
of f then F+C is the most general antiderivative
and here C is a constant okay
so let’s practice and do some examples so let’s suppose that
f(x) is cosx. Find F(x)
so basically we are looking for a function whose derivative is equal to cosine
so what is F(x)? very good
sinx+C another example
f(x) is given by x^(n) where n is not equal to minus 1
so may I ask you please do not talk during this class
if you have questions ask me, but do not talk. So we have a rule to evaluate the antiderivative
of this It’s called the power rule
power rule okay so it tells us
it tells us that the antiderivative is given by
this function plus C
and in fact this is the rule that we used here
to calculate the derivative in our very first example
so I have a question why is it that we have to require n
is not equal to -1 what happens to this formula when n equals
-1? Yes. You divide by zero
So this formula is obviously not applicable Something else goes on there
so that’s my next example a very special case
where n is equal to -1 f is 1/x
what’s the antiderivative? Natural log of x
In fact, it’s like this Natural log of the absolute value of x
plus C and we actually have to say that this formula
holds for any interval that does not contain 0
so if I give you if I give you an interval from one to five
first on this interval
any such function with a constant C will serve as an antiderivative of the log
if I want to write down the most general antiderivative on the whole
real line, it’s something slightly more complicated
so f(x) is ln(x) +C for positive values of x
and this logarithm of –x oh—I’m sorry
this is noted C¬1 + C2
When x is negative and the antiderivative is not defined for
x=0 because the original function is not defined
for x=0 question?
can C1 and C2 be equal? they can be equal,
but in general they don’t have to be so this is the most general form of the antiderivative
of the function 1/x so this function can have a discontinuity
it goes like lnx, ln(-x), but here, when you equal x=0,
you can have a different constant. So it’s in general a discontinuous function.
We also have something like this For negative values of n
So let’s suppose That f is x^(-4)
So now we have a very similar situation So by using the power rule
My power is -4 So I have to
So x^(-4+1) -4+1
It’s x^(-3) Divided by –x
So the most general antiderivative is given by the following
discontinuous function so the antiderivative again is not defined
for x=0 because the function itself is not defined
there and the experience
it’s discontinuating, as it goes to 0 because we’re allowed to take different
constants for negative and positive values of x
so what we need to know for this class is a list of antiderivatives
it’s best to know those by heart so we have a table
in the textbook that goes like this. We have a function
and we have a particular antiderivative so what are the most common
functions that you should know by heart? So first I list
a rule if you have a function f, any function f multiplied
by a constant C so the antiderivative also gets multiplied
by C you know that, right?
And similarly with the summation
the antiderivative of a sum is the sum of two antiderivatives
now particular examples of functions
we’ll list some of them here I do this for completeness so I’ll continue this table here
let’s do cosine x gives me sinx
so this is f, this is F so this is sinx
-cosx secant squared
gives me tangent secant tangent
gives me secant these two follow from the definition of the
derivatives of secant and tangent we know if the derivative of this is equal
to this then the antiderivative of this
is this this table works both ways
to go from here to here you have to take a derivative
to go from here to here, you have to take the antiderivative
what else? here
we have something that is associated with arcsin
and arctangent so this is the best one
why is it the best one? It’s equal to itself, right?
it's the easiest one to remember, the exponent is equal to its own derivative
also its equal to its own antiderivative questions?
So let’s practice So a simple common problem
That we encounter is find the antiderivatives of functions
so find all functions g such that
g prime is 5+cosx
+ 3x^(2) So look at this function
So we go to the table and of course we don’t find that function in the table
However, if we simplify this function we’ll find its components
in the table, right? so the first thing we do here is simplify
and then we’ll be able to use the table so I’m going to say that this is 5
plus cosx and here I divide through by x, so I have
3x + x -1/2
Now each of the components here And we found them in the table
and I’m going to use the table so G
so I’m sorry G the derivative of G is this
so therefore I have to find the derivative the antiderivative
so it’s 5x + sinx
how did I get the first term? we will learn how to integrate in this class
but we have to find an antiderivative of a constant. So
where can you find that? for example,
here if n equals 0 that’s a constant 1, right?
So n equals 0 gives me x to the power of 1 Divided by 1, so that’s x
so the antiderivative of 1 is x and here this tells me that multiplication
by a constant jus carries through so this number 5
appears in front of the antiderivative sine is the antiderivative of cosine
just pulls right from the table this one is easy
again, it’s a power function, so it’s 3x^(2) over 2
this one is also easy because you use the same rule. Power rule.
Now we don’t have an integer power, the power is equal to -1/2
so we have x^(1/2) divided by ½
and then don’t forget +C
this usually costs one point on any test
questions? do we have to simplify the ½ to a radical
or can we leave it when it’s easy, like this
probably I wouldn’t take a point off for this
but different graders are different okay the next question
is a little bit more sophisticated we can talk about differential equations
in itself it’s a huge topic and there are whole courses taught on this
but I will just show you what it is so the problem is like this: find f
if f’ is equal to x^6 and f(1) is equal to 3
so I need to find the function f given this information
two pieces of information the first piece of information pertains to
its derivative and the second one tells me what the value
of the function f is at one point
x is equal to 1 so that’s what I need to find. So from this
equation I can find f
By looking at the most general antiderivative So general antiderivative
I take the antiderivative of x^6 Which is x^7 over 7 +C
So I found a whole bunch of functions f they all differ by this constant
and that is why I’m given this second condition this condition will help eliminate most of
these and 0 on the relative one
so use f(1) equals 3 how do I use them?
I plug it in. Exactly. So I go f(1) is 1^7 over 7 +C
and that’s supposed to be equal to 3 so I can say that 1/7 +C equals 3
where C equals 3-(1/7) which is 20/7
therefore my function f not the most general one, but
the actual one that solves both of these, okay,
that’s given by x^7 over 7 plus 20/7 or ((x^7)+20)/7
so out of all of these functions, I identified the one that satisfies not only the first
equation, but the second one, too questions?
okay so now we will refresh our memory with regards
to graphic antiderivatives and we will talk about the notion of velocity
so let’s suppose that the function f
is given graphically something like this. One second.
So it starts off here, goes negative,
like this, and like this
okay, something like this let us sketch the graph of the antiderivative
so no formula there given and I want to draw F capital
so how do I do this in principle? This is the derivative of this function
Now remember, what is an antider— what is a derivative?
The derivative is the rate of change It’s the rate of change
It's the rate at which the function changes. if we think of the independent variable
as time, the derivative is how quickly that function changes.
it tells you the slope, or the rate of change and the rate of change can be interpreted
as a velocity so let’s suppose that this is velocity of
motion and as you can see, as time goes by, it changes
sometimes it will go faster, then it will slow down, okay,
at this point, the velocity is equal to zero and here it becomes negative. Which means
that we go backward
then again at this point, we turn and start going forward
so by using this information, I am going to draw the position, given the velocity
okay? I have to recreate the position given the velocity
I start with some arbitrary point okay, let’s suppose that we know we start
at 1 and now, so look
the velocity here is positive which means that I am going forward
I go forward means that my position, the coordinate of my position, increases
So for a while between time equals 0 and time equals 1 I
go forward slower and slower and slower, but I move forward
this is my positive direction, according to increases
at this point I stop at this point my derivative is equal to zero
which means that I’m going to have a maximum here, right?
and now my velocity becomes negative at this point I start going backward
and that’s exactly what I’m drawing here I start going backward
Faster and faster and faster at point 2, my velocity is the fastest
negative and then it becomes slower and slower and
slower. so at point 2 I get something like this
and I stop at point 3 because my velocity again is 0
after 3, I continue to go forward so my coordinate increases
and then the velocity decreases so somehow I level off
I start going slower and slower and slower And eventually almost stop but I don’t quite
stop Questions?
so you should be able to take the graph of a function
and draw its antiderivative but I want you to think about velocity
I want you to think if this is positive, This increases
If this is negative, this decreases
if this is 0, it means I experience either a maximum or a minimum
I don’t change at that point Questions?
very good so now in the last problem I think it’s the hardest of all
we will talk not only about velocity but also acceleration
because they’re both connected to derivatives and antiderivatives
so the problem is like this suppose that
the acceleration of a particle is given by this function
so here is my vocabulary a is acceleration
v is velocity and s is position
these are the common notations and you know
that the velocity is the derivative of the position
and the acceleration is the derivative of the velocity
do you know this? Okay. So what is given is the acceleration
and also some information about the position at the beginning, the position is 2
and the velocity at the beginning is -1 find the position as a function of time
given this information I start by saying that acceleration
is a derivative of the velocity so v’
okay is t+1
if I know the derivative of the velocity I can find the velocity
by taking the antiderivative so v(t) is found by calculating the antiderivative
of this function which is really easy
t^2 +t +C
so given the acceleration so let me just
I don’t want to misstep so this is a
a is the same as v’ and it’s given by t+1
so if I know v’ I know v and it’s given by this
unfortunately I have this unknown constant here
but I can fix that by saying that oh, look
v(0) is equal to -1 that tells me find the appropriate C
so we use v(0) is -1
so what is v(0)? it is 0/2 + 0 + C
and it’s supposed to be equal to -1 therefore C is -1
therefore v is (t^2)/2 + t -1
so I completed the first step using the information about the acceleration
I found the velocity. and I also used this condition about the initial
velocity that helped me identify this particular constant,
C okay, so that’s step number one
step number 2, I know the velocity now but I need to know the position
so step number 2 I go here
The velocity is the derivative of the position So v is the same as s’
and that’s given by this formula that I just derived
t^2 over 2 + t - 1
From this, I can find the position by taking the antiderivative so if I know the derivative of s,
I can find the antiderivative
Again I have an unknown constant called A So what’s this constant? That’s the last
step I’m going to use this piece of information,
the initial position S(0) is given by 0/6 + 0/2 – 0 + A is supposed
to be equal to 2 Which means that A equals 2
So I can write down the answer s(t) is given by t^3 over 6
plus t^2 over 2 minus t plus 2
so this is the formula that defines the position of that particle as a function of time
questions? Okay, thank you very much