Algebra (all content)
- Interpreting equations graphically
- Interpreting equations graphically (example 2)
- Interpret equations graphically
- Solving equations graphically (1 of 2)
- Solving equations graphically (2 of 2)
- Solving equations graphically
- Solve equations graphically
Solving equations graphically (1 of 2)
Sal solves the equation e^x=1/[x(x-1)(x-2)] by considering the graphs of y=e^x and y=1/[x(x-1)(x-2)]. Created by Sal Khan.
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- I don't get it... how is 7.846 and 7.633 within 0.01 ?(14 votes)
- Those are the values of E(x) and R(x). You want x within 0.01 of the actual value of x, not E(x) within 0.01 of R(x).(24 votes)
- At2:05Sal uses his calculator and chooses "e" and raises that to a power of 2.1. What is "e" and where did that irrational come from?(13 votes)
- e is a constant, just like PI is a constant. e is something like: 2.71828.... e is the base of natural logarithm. I believe that e stands for Euler's number.(18 votes)
- I have never learned this stuff. are there videos before this that will help me do this?(12 votes)
- This video gives an introduction to euler's number (e).
This might help too (from wikipedia): The number e is an important mathematical constant that is the base of the natural logarithm. It is approximately equal to 2.71828, and is the limit of (1 + 1/n)n as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series.
- At the point Sal was looking for, the y values change enormously for tiny changes in x. Sal guesses a starting x of 2.1 by looking at the x axis. So would there be a way to use an estimate on the graph for the y axis, in order to go back and get a better starting point for the x value?(4 votes)
- If you used an inverse function for either R(x) or E(x) , then you could plugin an estimate of maybe 7.8 into that function, to come up with a starting value of x. e^x conveniently has an inverse function, which is ln(x).
ln(7.8) comes up as 2.05412373369554605284 on my calculator, which gives you a very good starting point, yes. :)(8 votes)
- What is the value of e at4:07?(3 votes)
- e is an irrational number like π (meaning it cannot be written as the ratio of 2 integers and thus in its decimal form it will go on forever without any pattern). The first few digits of e are:
e ≈ 2.718(3 votes)
- What is "e"? I've run into it, and I know it's aprox. 2.71812, but what is it's specific "job" in mathematics?(2 votes)
- The number represented by the variable 'e' is called Euler's Number. In math, it represents the base for something called the natural logarithm, which is a process that counteracts something being raised to a power, much like addition counteracts subtraction.(4 votes)
- Please, how exactly is E(x)=R(x) within 0.01 ?(3 votes)
- When it's much closer than either 2.05 and 2.07.(1 vote)
- I seems that the correct Algebra way to solve this would be to subtract e^x from 1/(x(x-1)(x-2) and set it equal to zero. i.e. 1/(x*(x-1)*x-2))-e^x=0. This is actually kind of what I did in Excel. I just made two rows of formula and subtracted the results. If I knew how to solve for x with that natural log (e) that is exactly what I would do.(2 votes)
- the solution suggested by jeffery works well
log natural in excel is ln()
charting in excel is as good as a calculator(2 votes)
- What does "within 0.01" mean exactly?(2 votes)
- That probably means the answer which you obtain in the end should be limited to the hundredths digit or 0.01...and thats what he did, he got the answer till the hundredths digit and did not solve it any further even though it is possible.(3 votes)
- Isn't this technically trial and error/Guess and Check?(2 votes)
- To some extent, yes. When you use graphing to find the solution to a system of equations, your answer will only be as good as the accuracy of your graphing. If you do the graph by hand, it is very likely that lines won't be straight, your scale might be off a little, etc. All these types of small deviations open up the potential for a close, but inaccurate solution. Thus, it is always a good ideal to check whatever solution you find to verify that it really is a solution.(1 vote)
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.