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We were talking about graphing last time so we will do more on graphing. As I said,
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Professor Sutherland and I realized that a lot people are uncomfortable with graphing. Which is understandable,
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because schools use graphing calculators for graphing, and a graph is just some picture that comes out from your calculator.
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I kind of want you to think about graphing a little more. So remember I talked about the absolute value of sine,
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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.
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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,
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what happens when x is very big? When x is very big, 1 over x is going to be close to zero.
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So when x is a million, 1 over x is 1 over a million. Close to zero.
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So way out here, we really almost taking the sine of zero and the sin(0)=0. So, far out is just
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how the value of zero and this is kind switch the sine. It sort approaches the axis like this.
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The end behavior will approach zero. Now what happens as you get close to x? close to
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the x axis. So now when x is small, when x is one this is the sine one 1, 1 radian.
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When x is 1/2 the sine of 2, when x is 1/3 the sine of 3, when x is 1 over 1000
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the sine of one thousand. So to get the sine of a bigger and bigger number, what happens to the sine?
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It just goes up and down. The sine of 1000 is not necesarely bigger than sine of 999. It depends on the radians.
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Since we keep switching up and down as we get different numbers and as
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x gets closer and closer to zero and 1/x is getting bigger and bigger so this is squeezing on the side.
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It starts doing this. What happens is it wiggles like crazy as it goes into zero. It is the same on the other side
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and then it goes like that. Pretty looking graph. Maybe it it was the same in both directions.
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The sin(1/x) is very strange looking and that is because 1 over x behaves kind in a
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funny way. So, why does it do this? When x is very large, 1 over x approaches zero so
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you are looking at basically sine of zero ok? So when x is 10 you are looking
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at sin(1/10), when x is 1000 you are looking at sin(1/1000), so for x big it is close
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to sine of zero, but it is going to go up and down so sometimes you get negative values
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and sometimes you get positive values, approaches zero. When x gets very close to 0,
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1 over x gets very large. So, the sine will start alternating back and forth as you
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come up with other numbers. Sine of, let me do it in the calculator sin(100), sin(101),
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equals... I dunno. So you cannot use a calculator but I can use a calculator so
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you know, I am in front of the room. Kind of weird I have half on my audience today.
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So, sin(100)= -.5 and sin(101)= +.45, sin(102), sin(103), sin(104) is negative again.
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So every a few numbers it switches from positive to negative. So when x gets closer to zero,
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remember this number gets bigger and bigger. You take the sine of bigger
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and bigger numbers which sometimes is positive and sometimes is negative. That
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is why is going up and down. Remember you are getting close to zero these values keep
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switching the signs. That is why is like that. So that is a fun graph. We love to
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use fun graphs, we can ask you questions sin(1/x). Why? Because other stuff is easy
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and I have to ask you the hardest ones too. Ok everyone got the idea? I am not asking you
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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.
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Ok, that was sine(1/x). What about 1/sin(x)?
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Also known as cosecant x. Let's figure it out!
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So that is our sine graph again. So what is 1 over that look like?
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The first thing is remember that the sine of pi is zero.
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Sin(2pi)=0. What happens when you do 1 over zero? Cannot get over zero right?
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It is undefined. So we will have a vertical asymptote at all places when sine is zero.
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And that one in the middle, OK?
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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
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we get that value. So now what happens between zero an pi/2, you get 1 over these numbers.
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These are numbers between zero and 1. So when you flip them, they get bigger than one.
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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
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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
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bigger than 1 and it is going to approach the vertical asymptote. The other side, you do
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the same thing. So it kind looks like a parabola but it is not. On this side, bigger values going
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down to bigger once. So this graphs is the same as this one. So if the sine graphs looks
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like that then you broke it and it is this way instead. So you take this "u" and you turn it up side down
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and then you take this and you turn it up side down and you get this. So that is what the
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cosecant 1/sin(x) looks like. So 1 over graph is another entertaining type of graph.
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because the sine has places where it is is zero.
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Alright. Let's see if you guys can figure out what the 1 over cosine graph looks like.
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We got y=cos x. Remember it kind look like that. pi/2, 3pi/2, 2pi, pi. ok!
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That is cosine x. Why don't you take a minute and see if you can figure out what secant graph looks like.
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So cosine and sine are the same graph. They are just shifted by
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pi/2 (90 degrees). So 1 over sine and 1 over cosine also be shifted. So first you want
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to find where it is zero. So it is zero at -pi/2, -3pi/3, and +pi/2 and +3pi/2.
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So on. So you will have vertical asymptote there. ok!
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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
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-1 is -1. And you get "u" shape again. This one approaching infinity and
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down to negative infinity. Again these going toward infinity is these as they get close
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to zero and if you do 1 over a small number you get a big number. This is something that
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you have to really grasp at the beginning of calculus. 1 over a very small number is very big number,
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1 over a very big number is a very small number. Ok? So as x gets very large, approaches infinity
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1 over x approaches to zero. When x gets very close to zero, 1 over x goes to infinity
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You got this? You can draw it nice and pretty, maybe some color, stars on it.
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This is y=cos x, this is y= 1/cos(x) also known as sec(x). So that is the cosecant and
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the secant graphs. We don't do much with them, you should just understand where they come from and what they look like.
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So far so good? I'm gonna cover this up in a second.
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Let's go back to this 1/x concept just for another minute.
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This whole ties in something called the limit that you will see at the beginning of calculus.
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The graph of y= 1/x looks like this. So notice what happens to 1/x. This is (1,1),
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That is a hyperbola by the way. As x gets very large this graph appraoches the
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x-axis as it gets close to zero.
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Let's think about what happens when x gets large.
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As x gets larger and larger this is 1/1, 1/10, 1/100,
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1/1000 and so on. This number is getting very small. 1/1000 is not a very big number.
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Its 0.001. So when x is a trillion, this is one trillionTH.
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So when x is very large the y value is very close to zero.
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It is 1 over that number. And, when x gets very small, so
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what do me mean by small?
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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.
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So now when x gets very small, y is very large. That is why blows up here.
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So you want to get that principle down.
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Imagine the sin(x) graph we had. The sin(x) graph kind of mirrors this graph. Imagine the second. make sense?
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The sin x graph looks kind of mirroring this graph.
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Imagine the second. does that make sense?
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This is a crucial thing to get in your head
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when you have to do these kinds of graphs.
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You will see a bunch of these in a calculus class, 1/x, 1/x+1, 1/x-1, things like that.
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You want to get an idea what the basic shape of that graph looks like.
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How about
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y=x+sinx?
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Remember what sin x does?
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Sine x just goes up to 1 and down to -1 and back and forth.
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Doesn't really do much anything.
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x in the mean time gets bigger and bigger.
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y=x
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So when x is 5, y is 5 right?
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So, what
happens is you take x and you add some values
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to it, it will go up and down from 1 to -1.
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Sometimes it can be add to it, you never add more than 1
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Sometimes you can subtract from it, subtract more than one.
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So if you think about it,
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this is the graph y = x.
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So what's going to happen is, sort of imagine y=x line.
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You are going to add to it. So you going to come up above it and you going to subtract a bit,
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are going to add a bit.
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you subtract. You can sort of follow the line y=x.
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Remember the dotted line
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has nothing to do with the graph. So the graph just keeps going higher and higher.
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Sometimes it goes under y=x
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and sometimes goes above y=x.
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Just depends on whether you are adding or subtracting to it.
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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.
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So if I had -x+sin x, it will be down here
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It kind of snakes along the line y=x.
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How are we doing so far?
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This is actually adding two different graphs together.
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Let's see if you can figure out what sin(x) + cos(x) looks like?
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Alright this will definitely be on the final.
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Yeah, I'm gonna put this on the final, no doubt. Sure.
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Lets just take some values.
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Alright, what happens at zero?
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Well sin(x) is zero, cos(x) is 1 and 1.
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At pi/6, this is 1/2.
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This is radical 3 over 2.
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So when you add
1/2 and radical 3 over 2 you get about 1.4, doesn't really matter.
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Well, let's make that not 1.4 let's make that 1.3.
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Because if you add these two you
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you get about 1.4.
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Then if you add, at pi/3 this is radical 3 over 2, this is 1/2.
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If you are adding the same thing you add before (1.3)
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and at pi/2 this is one and this is zero.
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So when you do sine and cosine shape, we just go up a bit and down a bit
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You can certainly figure out the first part of the graph
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It would be here at 1, it will go up and go down at pi/2.
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Now what happens?
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Now cosine is going to be negative and sine still going to be
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positive when you start subtracting values.
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So If we had,
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sine of 2pi/3 is radical 3 over 2
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Cosine of 2pi/3 is negative 1/2
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so this is about .3
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3pi/4 this is radical 3 over 2 and negative radical 3 over 2, zero.
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Let me graph, it is zero somewhere around there.
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Think about now this is going to get smaller when you subtract more when we get down to
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here this is 0, 1, -1.
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So this sort of kind of behavior. I am going to leave up the graph now.
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This is going to behave a lot like what a sine and cosine graphs looks like.
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We don't really need points
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It is still looking at something like that.
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So whats another way we could have done this without plotting the points.
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I am going to draw them on the same set of axis.
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So that is sine. First graph. This is sin x
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and this is cosine x.
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So if we want to add them together, well here we adding zero and here we are adding 1
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So we are going to be at 1
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And now we are adding something that is coming down to 1 and something that
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is going toward 1.
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So if we are adding a little bit to the graph and right here
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sine will come down to radical 2 over 2 and cosine will go up to radical 2 over 2
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And that would be getting the biggest value we can get.
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And from over there
sine is getting bigger and cosine is getting smaller,
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cosine is heading down to zero.
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Now sine is getting smaller and cosine is now a negative value
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Right at this point.
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So we get to out most negative value.
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So, it is going to come down to there.
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So this is opposite radical 2, negative radical 2.
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This curve will go up again.
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So kind of like sine sort-of-graph except it has a different top and a different bottom.
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So how do I know the maximum is? the maximum is where i am adding together
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biggest possible combination of sine and cosine.
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where it adds up to radical two.
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And
the negative value, negative radical 2.
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How about non-trig graphs for a while. Those would be good right?
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These questions, this is a question a lot of people have trouble with on the minimum competence.
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Lets ask if a graph looks like this.
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I didn't graph anything in particular it is just a graph.
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For how many values, how many values of x where y=1?
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How do you figure that out?
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I am just waiting. (Audience: 5)
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How do we know that there are 5 values?
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well, you go to y=1 and you just draw an horizontal line.
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So where y equals 1 intersects the graph.
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Look what the graph is telling you.
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The graph is giving you information of the value of y.
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You got 1, 2, 3, 4, 5 intersections.
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Lets say where y=2, you have 4 intersections.
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This is where it is equal to 1/2, 3 intersections.
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So we gave a problem like this on web assign and we gave problems like this on
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the minimum competence.
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A lot of people have trouble with it.
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So if you want to know for what value, for how many values
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of x
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does f(x)=1?
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What you do is, you draw a horizontal line y=1 and see how many times intersect the graph.
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Make sense?
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What if I said for how many values
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does x =1?
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How many values of f(x) does x =1, the reverse of that.
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Then you draw a vertical line
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and in fact on functions you can only get either one or zero.
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Ok? Because you cannot more that one value otherwise it is not a function.
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So if they say for how many values of f(x) x=2, x=3.
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Always is going to come out 1 or is going to come out zero if the graph has a hole. a gap.
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Lets just practice a little more with logs and exponential kind of stuff.
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So remember what y=e^x looked like?
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y=e^x looks like something like that. Ok?
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It goes through point (0,1) Why does it goes through (0,1)?
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Well, e^0 is 1. In fact any number,
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other than zero, raised to zero will give you 1.
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So if I had to graph of 2^x it would be a little shallow
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here and a little flatter there ok?
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So this is e^x and this would be 2^x and 3^x goes the other way,
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3^x would be inside on this side.
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Why is that true?
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well just pick a number.
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When x=1 this is 2, this is 3. And "e" is at 2.7. ok?
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But all these graphs have the same shape.
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When x is equal to a large number they get very big and very fast.
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Because 2^10 for example is 1024 so when x is 10 it is way up there, 1024.
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So it blows up very quickly.
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And when it is on this side, we have 1/1024 so gets very close to zero.
00:27:57.960 --> 00:28:02.760
Notice, when x is a positive number we get a big positive number, when
00:28:02.760 --> 00:28:05.420
x is a negative number you get a very small positive number.
00:28:05.420 --> 00:28:06.960
Cannot be zero.
00:28:06.960 --> 00:28:10.940
What value, lets see 3^x.
00:28:13.400 --> 00:28:15.400
What would I raise x to, to get zero?
00:28:17.120 --> 00:28:18.520
You cannot raise to anything.
00:28:19.520 --> 00:28:24.740
If I do 3^0 is 1, 3 to anything grater than zero is a big number.
00:28:25.260 --> 00:28:29.240
3 to anything less that zero is a small number but positive
00:28:30.180 --> 00:28:33.720
If I do a fraction, if I do a negative, it does not matter I cannot get zero
00:28:33.720 --> 00:28:38.540
This graph will always going to have the x-axis be the horizontal asymptote.
00:28:41.000 --> 00:28:48.020
So of course I can now give an asymptote [unintelligible] For example y=e^x +1.
00:28:54.140 --> 00:28:56.140
It'd look like that, OK?
00:29:03.800 --> 00:29:05.800
the whole graph will shift up one.
00:29:06.800 --> 00:29:10.240
And if I have e^x-1 the graph will shift down 1.
00:29:14.880 --> 00:29:15.600
Simple?
00:29:16.860 --> 00:29:22.600
So we have y=e^x+k, k is going to be the our horizontal asymptote
00:29:26.960 --> 00:29:28.040
you sure?
00:29:29.000 --> 00:29:29.960
How about
00:29:31.280 --> 00:29:37.060
now if you know what e^x looks like, let's remember again what natural log of x looks like
00:29:41.300 --> 00:29:44.920
So e^x looks like this, natural log of x
00:29:47.480 --> 00:29:49.480
looks like that.
00:29:50.240 --> 00:29:53.380
ok? e^x has an horizontal asymptote
00:29:53.380 --> 00:29:56.160
and the natural log of x has a vertical asymptote.
00:29:56.500 --> 00:29:58.500
e^x goes through (0,1).
00:29:59.600 --> 00:30:05.080
Log of x goes through (1,0). Ok?
00:30:05.980 --> 00:30:11.340
And log of x looks
like maybe is flatting out but it does not.
00:30:11.900 --> 00:30:14.560
Its just going up slower and slower.
00:30:14.560 --> 00:30:15.780
Its going up.
00:30:15.780 --> 00:30:19.460
Because remember in exponential graph
00:30:19.460 --> 00:30:28.080
it goes up very quickly because say this was 2^x, 2^10 is 1024 you get a very big number.
00:30:28.680 --> 00:30:34.320
So log of 1024 is only 10, so now you got way out and going up 10.
00:30:34.720 --> 00:30:39.980
So the natural log graph is very flat but, it's never actually flat it's just very slowly
00:30:42.220 --> 00:30:45.260
So as log of x looks like so we can shift it
00:30:45.260 --> 00:30:49.660
and have the vertical asymptote so we can shift left and right.
00:30:49.660 --> 00:30:55.080
So if you have y= ln(x-1),
00:30:55.080 --> 00:31:00.460
that looks the same as before except there is (2,0).
00:31:02.420 --> 00:31:04.420
And now the vertical asymptote is at 1
00:31:07.740 --> 00:31:15.080
Log is very useful. Lots of times graphing something, like exponential, could be very
00:31:15.080 --> 00:31:21.600
messy. You cannot stay on the graph. So what you do is graph the log. Instead of
00:31:21.600 --> 00:31:33.680
graphing x vs y you graph, you make y = the log of x. The log of y. So, thats take the exponential
00:31:33.690 --> 00:31:40.619
graph and turns it into a log. So the idea is, the main value of logs of
00:31:40.619 --> 00:31:47.819
these kinds of graphs is just it is easy to graph. As you can see there is a lot of arithmetic
00:31:47.820 --> 00:31:56.040
relationship then you can get relationship of a line and that tell you that it is exponential So when you graph
00:31:56.049 --> 00:32:03.759
axis for example, you cannot really graph 10^5, 10^6, 10^7, 10^8 all on the same asymptote,
00:32:03.759 --> 00:32:09.820
all on the same axis because you are getting 10 thousand, one hundred thousand, million,
00:32:09.820 --> 00:32:16.900
very big numbers but its easy to graph 5,6,7,8, So you just graph the log of what the number
00:32:16.900 --> 00:32:20.829
is, it is much easier to fit all that in.