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# Solving equations graphically (1 of 2)

Video transcript

Graphs of e of x
equals e to the x and r of x is equal to 1 over x
times x minus 1 times x minus 2 are shown below. Estimate the solution to e
to the x-- so that's e of x-- being equal to essentially
r of x within 0.01. So we want to
figure out for what value does e of x equal r of x? And they want us to estimate it. We can either just try
to get as close as we can from this graph. They want us to be within 0.01. And we can also
use a calculator, kind of try numbers out
to hopefully zero in on this point right
over here, where e of x is equal to r of x. So what I want to do is, let
me draw a little table here. Let's try out some x values. And then for each
of these x values, let's see where we land on e of
x and where we land on r of x, and then we can decide whether
we are too high or too low. And I encourage you to pause
this video before I actually go ahead and do this, and
try to do this on your own. But I do suggest using
some form of calculator or, well, probably a calculator. I'm assuming you've
given a go at it, and now I will attempt it. Now just eyeballing
it-- and eyeballing it is helpful, because that'll
give us kind of our first order approximation of at what x value
are these two functions equal. If I just look at this
graph the way it's drawn, it looks like this is
pretty close to 2.1. It looks like when x is
2.1, both of those functions look pretty close
to-- I don't know. This looks like about 7.7 or
7.8 or something like that. But let's figure out
what they're doing. So let's see, when
x is equal to 2.1-- get my calculator out-- when
x is equal to 2.1, well, e of x is just e to the x power. So e to the 2.1 power
is equal to 8.166. Let me write that down, 8.166. And what is r of x? r of x is 1 divided by x,
so that's going to be 2.1. Times x minus 1. Well, that's going to be
1.1, so that's times 1.1. Times x minus 2. Well, that's just going
to be 0.1, times 0.1. And that is equal to
4-- did I do that right? No, that can't be. 2.1 over-- 2.1
times 1.1 times 0.1. 1 over all of that. 4.32? Let's see, 2.1 r
of x is 4 point-- I guess that's possible. Actually, that looks
right, because r of x declines so sharply
right over here. So it's actually, at 2.1
where actually r of x is actually closer to right
over here, give or take. So it's equal to 4.32. So 2.1 e of x is actually a
much larger value than r of x. So e of x is clearly
too high. r of x is already dropped
a good bit by then. If I were to go all
the way down to 2, at 2 it looks like actually r
of x kind of spikes up. It just goes to infinity
as we approach 2. So we're not going to go
all the way down to 2, but why don't we lower
this a little bit. Why don't we try 2.05? So 2.05, what is e of x? e of x is e to the x, right? So e to the 2.05 power gets
us 7.76-- I'll round it, 8-- 7.768. Approximately 7--
actually, all these are approximate, so
I'll just write 7.768. And what is r of x? I'll just keep rounding
to the thousands. Here, well, we didn't have
to round too much just because that was so far
off, but I'll put it there. Actually, it was
329, so I could-- let me write it this way-- 3290. So let me throw
that 9 here, just so everything-- we evaluate
the function to thousandths. So let's evaluate r of
x, when we're at 2.05, it's going to be 1 divided by
x, which is now 2.05, times x minus 1, which is 1.05, times
x minus 2, which is 0.05. And that gets us to 9.29-- I'll
round to 2-- 9.292, so 9.292. So now we're on this
side, where r of x is roughly right over
here and it's more than e of x, which is at 7.7,
which is right around here. So now our x value is too low. So maybe let's see if we
can go a little bit higher. And let's try to go roughly
halfway between these two, but I don't want to get too
precise, because you have to get to the nearest hundredth. So let's go to 2.07. So e to the 2.07 is equal to
7.925 if I round it, 7.925. I want to do all this in green
just to be color consistent. And now let's evaluate r
of x at that same value. So 1 divided by x, which is
2.07, times that minus 1, which is 1.07, times that
minus 2, which is 0.07, which gives us 6.44, I
guess we could say 6.450. So at 2.05 that was too
low, 2.07 is too high. Now, r of x has
dropped below e of x. So we know the right answer is
in between these two numbers, and so if we select
2.06 that's definitely going to be within 0.01
of the right answer. So I would go with
2.06 is definitely going to be within the 0.01 of
the correct solution to this. But just for fun, let's
actually just try it out. So e to the 2.06 is 7.84--
I guess we could round to 6. And if we were to
evaluate r of x, it's 1 divided by 2.06
times that minus 1, which is 1.06, times 0.06. It gets us to 7.632. So we're also
getting pretty close, but our precision
that they gave, they don't say that they have
to be within each other of that, they say, let's
estimate the solution. So there's some actual
precise solution to this right over here, some x
value, where these are actually equal to each other. That's the x value, which gives
us this point of intersection. We just have to get within
0.01 of that x value, and 2.06 definitely
does the trick.