Stony Brook MAT 123 Fall 2015
Lecture 26: Graphs and solving equations
November 23, 2015

Start   A possible misinterpretation. I was graphing sin (1/x). Remember what happens with this graph. It sort goes up and down like this
0:42But as x gets bigger, still going up and down
1:20but it is continuing, it is stretching out. How it still goes up and down between 1 and -1, I am flatting out to zero, because I wasn't concetrating. Ok. That is what it looks like. Lets do it. Working your way back. Anyway you should just make sure that you have the graph correct.
1:46How many intersections are there between y=e^x and y=x? Well the graph is very helpful.
2:06If you wanted to solve, you said you want to solve the equation e^x=x look like.
e^x looks like this and y=x looks like this. There are no intersections. Right?
Why there are no intersections? Because e^x will always stay above y=x.
2:30If you try to graph them, we find that they they don't overlap.
However if we had lets say y=e^x, by the way that is (0,1). How do we know that they don't overlap? well when x=1 this is (1,e) which is about 2.7. This is (1,1). The y value is the same as the x value. This is (2,2),
3:10this is (2, e^2), e^2 is about 8. So, e^x is growing much faster than y=x so they are not going to overlap.
There is no intersections. But if I take let's say y=3x. It might something like that
3:46So what happens? When x=1, y=1 on the graph y= x. And when x=1, y=e which is less than 1 so something like that. So they cross each other twice. Solving for those values it is not simple,
4:16but if they ask you, you can see that there are two values. Could it be a third place, where the y is equal to some ax and y=e^x. We will make them cross three times. How many times
4:34does a curve cross a line? Intersection of a graph. Do you find a way to cross? The answer will be no.
I see how it could be target if it touches, it crosses once. Right? I see how it can cut through and gets twice. Do you see where they make three cuts? I don't. So, you can look
5:02at the graph if you can figure out things like how many possible solutions there are.
Because, the problem is y=e^x and y=3x. You can say, just by looking at the picture, I know that there are two solutions. Finding them might not be easy but that is okay because we have mathematical tools to find them. We can learn in the first semester of calculus.
So why am I doing this? Why switching and thinking about graphs in terms of solving equations. We had a graph and we know that they intersect. Okay? Then you know there
5:34is a solution to the equation wheres one graph is equal the other graph. If they don't intersect then there is no solution. If they intersect a bunch of times like there then we have a bunch of solutions.
Alright! Now we are going to solve some equations. Let's solve the equation.
6:25From the graphs there must be solutions to that! So how can we find them analytically?
6:36So this by the way is the same thing saying where sin2x is the same as sin x. Right?
Be careful that you don't lose certain solutions. So you write like this and you say, how do I solve this? Any ideas?
Ok. sin 2x is one of the double angle formula from before.
The problem is that this is 2x and that is a single x.
So why don't I take this and use the double angle formula. Now you can factor out sine x. Now I know that in order for this to be zero,
7:33either sin x is zero or 2 cosx -1 is equal to zero. So sine is zero at a lots of values. So let's restrict this.
Just values between 0 and 2pi not counting 2pi because 2 pi is the same thing as zero.
Ok, so where is sine equal to zero? Well sine
8:06is zero at zero, sine pi is zero, sine of 2pi but we are not counting that.ok? And what about 2 cosx -1?
When is that equal to zero?
So, 2cos x = 1, cos x=1/2. And where is cos x equal to 1/2? At 60 degrees also known as pi/3 and (5pi)/3.
8:45So that is the example of trigonometric equations.
So it is just like algebra equation but instead of using x we use sin x. What is another type of equation that I can use? How about
9:23Any idea how to solve that? Figure there is a quick quadratic equation.
9:40Remember we can factor, one way you can factor is doing substitution. Let's say u =sin x. You can think of this as 2u^2-5u+3=0. This now will factor very easily.
10:13You get 2u +1 =0, u = -1/2 , u=3 and then you have to remember to substitute back.
So this would be: sin x= -1/2 or sin x=3. Remember u=sin x. If you substitute sometimes
10:36is easier to do the quadratics. So sin x =-1/2. Sine is negative at x= third quadrant. 7pi/6,
11:0011pi/6 and sin x is never equals to 3. Why is sin x never equal to 3? Because sin x always between -1 and 1. You can through that answer out. Ok? Let's have you guys try one!
11:33Why don't you solve that?
Ok, Let's see if we can solve this one.So this again, it is just a quadratic equation, It looks like a quadratic equation. I think this it does. So one way to make this simple is to change the variables so make up a letter for cos x. What about y.
So you could re-write this. y^2 -3y+2 =0. Then you factor it (y-2)(y-1)=0. So y=2 or
12:26y=1. So y is equal to 2 or 1, which means cos x=2 or cos x=1. Why do we through out
12:39cos x =2, because cosine is always between 1 and -1. Cos =1, well again if we did this on an internal 0 and 2 pi, this would be x=0. But otherwise this has an infinite number of solutions: 2pi, 4pi, 6pi,-2pi,-4pi. Goes on forever. That is why you need an interval to solve the equation.
13:10How do we do on this? Easy? Good. The answer is all values for cos x=1 so the answer is x= 0, 2pi, 4pi, etc. ok? Or we have to give you the interval, so we usually say the interval is between 0 and 2pi. Ok? Let's do another one.
13:37You guys love trigonometry!
14:04ok? Let's see who can solve that? So I would take sin^2 and replace it with 1-(cos^2)x.
This is one of the trig identity thingies. So if you see this equation you should look for this.
Ok, add cosine squared on the other side.
14:30Subtract the one.
Now it looks suspiciously like that, does'nt it! In fact, it is identical, except for cos(theta) in there. See, its a disguised equation.
Well I look and I say: when I have mixed sine and cosine,
15:04it is very hard to solve unless you can factor and separate one from the other. So you should be trying to put everything to one variable. Same thing when x and y are mixed together you have to find a way to isolate the x, isolate the y So here I look and say there is no way I can pull a cosine square out [unintelligible]
15:30So one last example of that before we try a different type. Now that you all have seen you can do that trick lets see something similar. try that one!
16:07So Nikki, whats our first step? [unintelligible] Beautiful, so you can take cos^2 x and replace it with 1- sin^2 x.Then you distribute it. [unintelligible] Now I put everything on the other side.
16:50I get 2 sin^2 x + sin x -1 = 0. So far so good? Alright! So you can either factor that with
17:03ways of quadratics or if it bothers you can change sin x to variable like a, or y, or u, or anything. And we write it so it is easier for you to see. So if you say something like a=sin x, that would be 2A^2 +A-1=0 and that factors very nicely. (2A-1) (A+1).
17:41You don't have to do that step, but you can! So this means that either A=1/2 or A=-1 and then you substitute back. So you get sin x =1/2 or sin x= -1. So x is, where is sin x=1/2 Nikki?
18:07Sin x =1/2 where? pi/6 and 5pi/6. Exactly! That is what you have right? That is what I thought! And sin x = -1 where? at 3pi/2. So you can solve these equations, trigonometric
18:35equations, you solve like a regular algebra equation and then instead of getting x is a number you get sin x equals a number or cos x and then you can solve from there.
We can make this very nasty but we don't need to. What is something that we can do other than sine and cosine?
19:17What about something like that? So that's is 2^(2x)+2^x -12=0.
That would give you a good quadratic right? So I could do a variable u= 2^x. Well,
19:342^ (2x) is just u^2 +u-12=0. Does that factor? u=-4 or u=3. so far so good?So that means that 2^x = -4
20:12or 2^x=3 Alright, 2^x can never be equal to -4, it can never be a negative or even zero, only positive. So, through that out. Where is 2^x =3? When x= log base 2 of 3.
20:35That would be the log form. You don't have to give me an actual decimal. So these kinds of problems are going to have logs solutions. So much fun!
21:40Let's see if you can solve that!
So, this is 2^x then I look at this and say can I make this something like 2^x? Sure! because 4 is 2 squared.
22:02So this is 3 times 2^2x ( I wrote 2x sorry that should be 2^x).
ok, now I can pick a letter for 2^x and make this: 3y^2 +4y+8=0. I think that is no solutions.
Then we are all done. Does that factor?
22:36I copied this from the book. Let's see!
It has no real solution so you kind of stop there.[unintelligible] Alright, this looks like the [unintelligible] right?
23:00How about y= e^x?
This becomes y^2-6y+5=0. Cannot do that. That factors into (y-1)(y-5)=0 so y= 1 or y=5. Which means e^x =1 or e^x=5.
23:41e^x=1, that is x=zero. and e^x=5 that is x=the natural log of 5.Let's try something slightly more entertaining.
24:04That is a fun one! So what do you do is you multiply by e^x, so this becomes e^x squared or e^2x +3 =4e^x. You can leave it like that or you can make it (e^x)^2.
24:43Really it does not matter. So then we need substitution? Lets use some letters. What about B? this becomes: B^2 +4B+3=0. And that factors into (B-3)(B-1)=0 where B=3 or B=1. But, e^x =3 and e^x =1.
25:25e^x =3, then x=ln 3 and e^x =1, x=ln1. Also know as zero.
26:06That would work. I think! Let's see if it does!
Nick has a clue! [unintelligible] This is 4 times 4^x. Alright this is 4 squared to the x. You can make that
26:404^(2x) + 4* 4^x -12=0. How about w=4^x. w^2+4w-12=0.
So this becomes (w+6)(w-2)=0. Ok!
27:11that means that w=-6 or w=2. 4^x =-6 (no solution) or 4^x=2. What do I have to do to 4 to get 2?
27:35I need the square root! so x=1/2. Yes, you can write it as log. You should know that, that is 1/2.