We were talking about graphing last time so we will do more on graphing. As I said,
Professor Sutherland and I realized that a lot people are uncomfortable with graphing. Which is understandable,
because schools use graphing calculators for graphing, and a graph is just some picture that comes out from your calculator.
I kind of want you to think about graphing a little more. So remember I talked about the absolute value of sine,
the absolute value of cosine, their graphs, we can make a graph library. We talked about what is sin(1/x). sin(x) remember looks like this.
Then, we have sine of 1 over x. that gets a little tricky because 1/x behaves badly as x gets close to zero. First of all,
what happens when x is very big? When x is very big, 1 over x is going to be close to zero.
So when x is a million, 1 over x is 1 over a million. Close to zero.
So way out here, we really almost taking the sine of zero and the sin(0)=0. So, far out is just
how the value of zero and this is kind switch the sine. It sort approaches the axis like this.
The end behavior will approach zero. Now what happens as you get close to x? close to
the x axis. So now when x is small, when x is one this is the sine one 1, 1 radian.
When x is 1/2 the sine of 2, when x is 1/3 the sine of 3, when x is 1 over 1000
the sine of one thousand. So to get the sine of a bigger and bigger number, what happens to the sine?
It just goes up and down. The sine of 1000 is not necesarely bigger than sine of 999. It depends on the radians.
Since we keep switching up and down as we get different numbers and as
x gets closer and closer to zero and 1/x is getting bigger and bigger so this is squeezing on the side.
It starts doing this. What happens is it wiggles like crazy as it goes into zero. It is the same on the other side
and then it goes like that. Pretty looking graph. Maybe it it was the same in both directions.
The sin(1/x) is very strange looking and that is because 1 over x behaves kind in a
funny way. So, why does it do this? When x is very large, 1 over x approaches zero so
you are looking at basically sine of zero ok? So when x is 10 you are looking
at sin(1/10), when x is 1000 you are looking at sin(1/1000), so for x big it is close
to sine of zero, but it is going to go up and down so sometimes you get negative values
and sometimes you get positive values, approaches zero. When x gets very close to 0,
1 over x gets very large. So, the sine will start alternating back and forth as you
come up with other numbers. Sine of, let me do it in the calculator sin(100), sin(101),
equals... I dunno. So you cannot use a calculator but I can use a calculator so
you know, I am in front of the room. Kind of weird I have half on my audience today.
So, sin(100)= -.5 and sin(101)= +.45, sin(102), sin(103), sin(104) is negative again.
So every a few numbers it switches from positive to negative. So when x gets closer to zero,
remember this number gets bigger and bigger. You take the sine of bigger
and bigger numbers which sometimes is positive and sometimes is negative. That
is why is going up and down. Remember you are getting close to zero these values keep
switching the signs. That is why is like that. So that is a fun graph. We love to
use fun graphs, we can ask you questions sin(1/x). Why? Because other stuff is easy
and I have to ask you the hardest ones too. Ok everyone got the idea? I am not asking you
to graph sin(1/x), I am asking you to understand what sin(1/x) looks like. If we ask you to graph that on the exam that would be very mean.
Ok, that was sine(1/x). What about 1/sin(x)?
Also known as cosecant x. Let's figure it out!
So that is our sine graph again. So what is 1 over that look like?
The first thing is remember that the sine of pi is zero.
Sin(2pi)=0. What happens when you do 1 over zero? Cannot get over zero right?
It is undefined. So we will have a vertical asymptote at all places when sine is zero.
And that one in the middle, OK?
How do I write the middle? At pi/2, the sine of pi/2 is 1. So here is pi/2. 1 over 1 is also one so
we get that value. So now what happens between zero an pi/2, you get 1 over these numbers.
These are numbers between zero and 1. So when you flip them, they get bigger than one.
In other words say you have pi/6. So 1 over half is 2. You guys see that pi/6 sine is 1/2 so
1 over 1/2 is 2 and one over radical 3 which is about 0.6 you do one over that. You get a number
bigger than 1 and it is going to approach the vertical asymptote. The other side, you do
the same thing. So it kind looks like a parabola but it is not. On this side, bigger values going
down to bigger once. So this graphs is the same as this one. So if the sine graphs looks
like that then you broke it and it is this way instead. So you take this "u" and you turn it up side down
and then you take this and you turn it up side down and you get this. So that is what the
cosecant 1/sin(x) looks like. So 1 over graph is another entertaining type of graph.
because the sine has places where it is is zero.
Alright. Let's see if you guys can figure out what the 1 over cosine graph looks like.
We got y=cos x. Remember it kind look like that. pi/2, 3pi/2, 2pi, pi. ok!
That is cosine x. Why don't you take a minute and see if you can figure out what secant graph looks like.
So cosine and sine are the same graph. They are just shifted by
pi/2 (90 degrees). So 1 over sine and 1 over cosine also be shifted. So first you want
to find where it is zero. So it is zero at -pi/2, -3pi/3, and +pi/2 and +3pi/2.
So on. So you will have vertical asymptote there. ok!
Is equal to one at zero so 1/1 is going to be one. It is going to be -1 at pi. So 1 over
-1 is -1. And you get "u" shape again. This one approaching infinity and
down to negative infinity. Again these going toward infinity is these as they get close
to zero and if you do 1 over a small number you get a big number. This is something that
you have to really grasp at the beginning of calculus. 1 over a very small number is very big number,
1 over a very big number is a very small number. Ok? So as x gets very large, approaches infinity
1 over x approaches to zero. When x gets very close to zero, 1 over x goes to infinity
You got this? You can draw it nice and pretty, maybe some color, stars on it.
This is y=cos x, this is y= 1/cos(x) also known as sec(x). So that is the cosecant and
the secant graphs. We don't do much with them, you should just understand where they come from and what they look like.
So far so good? I'm gonna cover this up in a second.
Let's go back to this 1/x concept just for another minute.
This whole ties in something called the limit that you will see at the beginning of calculus.
The graph of y= 1/x looks like this. So notice what happens to 1/x. This is (1,1),
That is a hyperbola by the way. As x gets very large this graph appraoches the
x-axis as it gets close to zero.
Let's think about what happens when x gets large.
As x gets larger and larger this is 1/1, 1/10, 1/100,
1/1000 and so on. This number is getting very small. 1/1000 is not a very big number.
Its 0.001. So when x is a trillion, this is one trillionTH.
So when x is very large the y value is very close to zero.
It is 1 over that number. And, when x gets very small, so
what do me mean by small?
when x is 1/10 y is 10, when x is 1/100 y is 100, when x is 1/1000 y is 1000 and so on.
So now when x gets very small, y is very large. That is why blows up here.
So you want to get that principle down.
Imagine the sin(x) graph we had. The sin(x) graph kind of mirrors this graph. Imagine the second. make sense?
The sin x graph looks kind of mirroring this graph.
Imagine the second. does that make sense?
This is a crucial thing to get in your head
when you have to do these kinds of graphs.
You will see a bunch of these in a calculus class, 1/x, 1/x+1, 1/x-1, things like that.
You want to get an idea what the basic shape of that graph looks like.
How about
y=x+sinx?
Remember what sin x does?
Sine x just goes up to 1 and down to -1 and back and forth.
Doesn't really do much anything.
x in the mean time gets bigger and bigger.
y=x
So when x is 5, y is 5 right?
So, what happens is you take x and you add some values
to it, it will go up and down from 1 to -1.
Sometimes it can be add to it, you never add more than 1
Sometimes you can subtract from it, subtract more than one.
So if you think about it,
this is the graph y = x.
So what's going to happen is, sort of imagine y=x line.
You are going to add to it. So you going to come up above it and you going to subtract a bit,
are going to add a bit.
you subtract. You can sort of follow the line y=x.
Remember the dotted line
has nothing to do with the graph. So the graph just keeps going higher and higher.
Sometimes it goes under y=x
and sometimes goes above y=x.
Just depends on whether you are adding or subtracting to it.
If you are adding or subtracting sin x. Make sense? That is kind of a fun graph! Sure, you can put this in the calculator and see what the graph look like.
So if I had -x+sin x, it will be down here
It kind of snakes along the line y=x.
How are we doing so far?
This is actually adding two different graphs together.
Let's see if you can figure out what sin(x) + cos(x) looks like?
Alright this will definitely be on the final.
Yeah, I'm gonna put this on the final, no doubt. Sure.
Lets just take some values.
Alright, what happens at zero?
Well sin(x) is zero, cos(x) is 1 and 1.
At pi/6, this is 1/2.
This is radical 3 over 2.
So when you add 1/2 and radical 3 over 2 you get about 1.4, doesn't really matter.
Well, let's make that not 1.4 let's make that 1.3.
Because if you add these two you
you get about 1.4.
Then if you add, at pi/3 this is radical 3 over 2, this is 1/2.
If you are adding the same thing you add before (1.3)
and at pi/2 this is one and this is zero.
So when you do sine and cosine shape, we just go up a bit and down a bit
You can certainly figure out the first part of the graph
It would be here at 1, it will go up and go down at pi/2.
Now what happens?
Now cosine is going to be negative and sine still going to be
positive when you start subtracting values.
So If we had,
sine of 2pi/3 is radical 3 over 2
Cosine of 2pi/3 is negative 1/2
so this is about .3
3pi/4 this is radical 3 over 2 and negative radical 3 over 2, zero.
Let me graph, it is zero somewhere around there.
Think about now this is going to get smaller when you subtract more when we get down to
here this is 0, 1, -1.
So this sort of kind of behavior. I am going to leave up the graph now.
This is going to behave a lot like what a sine and cosine graphs looks like.
We don't really need points
It is still looking at something like that.
So whats another way we could have done this without plotting the points.
I am going to draw them on the same set of axis.
So that is sine. First graph. This is sin x
and this is cosine x.
So if we want to add them together, well here we adding zero and here we are adding 1
So we are going to be at 1
And now we are adding something that is coming down to 1 and something that
is going toward 1.
So if we are adding a little bit to the graph and right here
sine will come down to radical 2 over 2 and cosine will go up to radical 2 over 2
And that would be getting the biggest value we can get.
And from over there sine is getting bigger and cosine is getting smaller,
cosine is heading down to zero.
Now sine is getting smaller and cosine is now a negative value
Right at this point.
So we get to out most negative value.
So, it is going to come down to there.
So this is opposite radical 2, negative radical 2.
This curve will go up again.
So kind of like sine sort-of-graph except it has a different top and a different bottom.
So how do I know the maximum is? the maximum is where i am adding together
biggest possible combination of sine and cosine.
where it adds up to radical two.
And the negative value, negative radical 2.
How about non-trig graphs for a while. Those would be good right?
These questions, this is a question a lot of people have trouble with on the minimum competence.
Lets ask if a graph looks like this.
I didn't graph anything in particular it is just a graph.
For how many values, how many values of x where y=1?
How do you figure that out?
I am just waiting. (Audience: 5)
How do we know that there are 5 values?
well, you go to y=1 and you just draw an horizontal line.
So where y equals 1 intersects the graph.
Look what the graph is telling you.
The graph is giving you information of the value of y.
You got 1, 2, 3, 4, 5 intersections.
Lets say where y=2, you have 4 intersections.
This is where it is equal to 1/2, 3 intersections.
So we gave a problem like this on web assign and we gave problems like this on
the minimum competence.
A lot of people have trouble with it.
So if you want to know for what value, for how many values
of x
does f(x)=1?
What you do is, you draw a horizontal line y=1 and see how many times intersect the graph.
Make sense?
What if I said for how many values
does x =1?
How many values of f(x) does x =1, the reverse of that.
Then you draw a vertical line
and in fact on functions you can only get either one or zero.
Ok? Because you cannot more that one value otherwise it is not a function.
So if they say for how many values of f(x) x=2, x=3.
Always is going to come out 1 or is going to come out zero if the graph has a hole. a gap.
Lets just practice a little more with logs and exponential kind of stuff.
So remember what y=e^x looked like?
y=e^x looks like something like that. Ok?
It goes through point (0,1) Why does it goes through (0,1)?
Well, e^0 is 1. In fact any number,
other than zero, raised to zero will give you 1.
So if I had to graph of 2^x it would be a little shallow
here and a little flatter there ok?
So this is e^x and this would be 2^x and 3^x goes the other way,
3^x would be inside on this side.
Why is that true?
well just pick a number.
When x=1 this is 2, this is 3. And "e" is at 2.7. ok?
But all these graphs have the same shape.
When x is equal to a large number they get very big and very fast.
Because 2^10 for example is 1024 so when x is 10 it is way up there, 1024.
So it blows up very quickly.
And when it is on this side, we have 1/1024 so gets very close to zero.
Notice, when x is a positive number we get a big positive number, when
x is a negative number you get a very small positive number.
Cannot be zero.
What value, lets see 3^x.
What would I raise x to, to get zero?
You cannot raise to anything.
If I do 3^0 is 1, 3 to anything grater than zero is a big number.
3 to anything less that zero is a small number but positive
If I do a fraction, if I do a negative, it does not matter I cannot get zero
This graph will always going to have the x-axis be the horizontal asymptote.
So of course I can now give an asymptote [unintelligible] For example y=e^x +1.
It'd look like that, OK?
the whole graph will shift up one.
And if I have e^x-1 the graph will shift down 1.
Simple?
So we have y=e^x+k, k is going to be the our horizontal asymptote
you sure?
How about
now if you know what e^x looks like, let's remember again what natural log of x looks like
So e^x looks like this, natural log of x
looks like that.
ok? e^x has an horizontal asymptote
and the natural log of x has a vertical asymptote.
e^x goes through (0,1).
Log of x goes through (1,0). Ok?
And log of x looks like maybe is flatting out but it does not.
Its just going up slower and slower.
Its going up.
Because remember in exponential graph
it goes up very quickly because say this was 2^x, 2^10 is 1024 you get a very big number.
So log of 1024 is only 10, so now you got way out and going up 10.
So the natural log graph is very flat but, it's never actually flat it's just very slowly
So as log of x looks like so we can shift it
and have the vertical asymptote so we can shift left and right.
So if you have y= ln(x-1),
that looks the same as before except there is (2,0).
And now the vertical asymptote is at 1
Log is very useful. Lots of times graphing something, like exponential, could be very
messy. You cannot stay on the graph. So what you do is graph the log. Instead of
graphing x vs y you graph, you make y = the log of x. The log of y. So, thats take the exponential
graph and turns it into a log. So the idea is, the main value of logs of
these kinds of graphs is just it is easy to graph. As you can see there is a lot of arithmetic
relationship then you can get relationship of a line and that tell you that it is exponential So when you graph
axis for example, you cannot really graph 10^5, 10^6, 10^7, 10^8 all on the same asymptote,
all on the same axis because you are getting 10 thousand, one hundred thousand, million,
very big numbers but its easy to graph 5,6,7,8, So you just graph the log of what the number
is, it is much easier to fit all that in.