Fancy Maths Stuff!

[Xei]

I just discovered quite a neat little thing which has made me happy. I thought it'd be cool to set it as a puzzle; you only need elementary mathematics. Also it'll be interesting to see if people have ever been taught this (I've never seen it). When you get it, it's very satisfying to watch how all the algebra comes together. Puzzle: find the expression in terms of x for the gradient of the general quadratic curve, y = ax^2 + bx + c, for every arbitrarily chosen value of x, without the use of any calculus. Spoiler for Big clue: Draw a tangent line to a quadratic curve an arbitrary value of x, and think about how it's different from all the other lines which touch the curve at this value of x (but don't have the right gradient).

[Phion]

Without using derivatives, and since the graph is the same for y = ax^2, I would set the equation for the slope at (x_0,y_0) equal to f(x) and solve for m. I'm pretty sure this problem can be solved with power series and vector analysis as well.

[Xei]

I don't know what you mean. How do you know what the slope is at (x0, y0) without calculus (patently it's 0 but that's not thorough, and I don't see how it helps find the slope anywhere else). Also I don't see how power series are involved. Also, you need calculus to find power series. Edit: Just in case this was causing your confusion, x0 is just standard notation for any point on the x axis, chosen anywhere. It doesn't mean x = 0.

[Invader]

I've never heard of expressing the gradient of a function before. Is that a linear algebra thing?

[Xei]

Really? We were introduced to this in school when I was 15. The term is differentiation. Linear algebra is about linear equations, so finding gradients isn't really a part of it. Calculus is the relevant subject; differentiation is the first thing you learn in calculus and is totally central to the entire subject, and because of the importance of calculus, it turns out pretty much to be the most important thing in any mathematics beyond basic algebra. You can find gradients of straight lines easily, but with curves like quadratics, the gradient changes for every infinitesimal change in x so it's a lot harder. This is why calculus was invented; to deal with infinitesimal things. The differential (gradient function) of x^n turns out to be nx^(n-1). Therefore the gradient of ax^2 + bx + c at x, using this rule, is a*2x^1 + b*1x^0 + c*0x^-1 = 2ax + b. This is why I was surprised you can find the gradient of a quadratic, a smooth curve, without using calculus. I don't think you can use it for anything else though. For more information, check out the MIT lectures: http://www.youtube.com/watch?v=7K1sB...eature=related

[Invader]

Huh, I've seriously never heard of differentiation referred to as finding the gradient of a function before. Maybe it's just because I'm schooled in the U.S.? I've gone through calc 1 and 2 (there are 3 levels that are taught for calculus here, the third of which I'm taking in the coming year, and I'm not sure how that differs from European-taught math). I wasn't aware that the derivative of a curve could be found without the use of calculus either.. I'll give it some thought.

[Xei]

Weird... all it means is 'the gradient of the tangent to the curve at x'; does that make sense to you?

[Invader]

Otherwise known as the slope of the curve at x, in which case it does make sense to me.

[Xei]

Ah okay. Yeah, things like 'slope' and 'rise over run' I've only ever heard from Americans I think. Also you all seem to write straight 'x's. I do that too but my mathematician pals think it's weird.

[Invader]

You mean instead of x0 and xl? We usually only use those if we're talking about integration between two points, or in physics to denote the initial and final state of something that occurs over time. If x is just one arbitrary value, it typically just stays x. It's strange though, I used to think that the English used in math was standard all over the world.

[Xei]

Nah I mean handwriting-wise. We do 'em curly. But yeah, there are various idiosyncrasies I've noticed. You're much happier to abuse differential notation for instance (e.g. dy/dx = 1 <=> dy = dx).

[Phion]

I don't know what you mean. How do you know what the slope is at (x0, y0) without calculus (patently it's 0 but that's not thorough, and I don't see how it helps find the slope anywhere else). Also I don't see how power series are involved. Also, you need calculus to find power series. Edit: Just in case this was causing your confusion, x0 is just standard notation for any point on the x axis, chosen anywhere. It doesn't mean x = 0. I would set the formula for slope equal to the quadratic, y-y_1=m(x-x_1), (equation for slope-intercept) would then be, m(x-x_1)+y_1 = ax^2+bx+c, setting the discriminant equal to zero and factoring to obtain a real solution. Solving for m at an arbitrary value of x would give you the tangent at some point. What you're after though is the rate of change in that slope as x approaches the normal, otherwise known as finding a gradient. Admittedly, this technique would not work in most cases without utilizing limits, as the error in calculation would be fairly substantial. So, really, what the question is reduced to is how to find a tangent without calculus. Also, power series has been around since Babylonian times, much before calculus. You're probably thinking about Taylor or Maclaurin series.

[Phion]

I used to think that the English used in math was standard all over the world. Same.

[Xei]

Solving for m at an arbitrary value of x would give you the tangent at some point. What you're after though is the rate of change in that slope as x approaches the normal, otherwise known as finding a gradient. Admittedly, this technique would not work in most cases without utilizing limits, as the error in calculation would be fairly substantial. So, really, what the question is reduced to is how to find a tangent without calculus. Yes, considering tangents is the right way to do this problem. I would set the formula for slope equal to the quadratic, y-y_1=m(x-x_1), (equation for slope-intercept) would then be, m(x-x_1)+y_1 = ax^2+bx+c, setting the discriminant equal to zero and factoring to obtain a real solution. You're actually very close to an exact method, if you just expand on this a little. Also, power series has been around since Babylonian times, much before calculus. You're probably thinking about Taylor or Maclaurin series. Power series are infinite series, and you tend to find them by Taylor's method. Considering the Babylonians had only just invented 0, I find it hard to believe that they could conceptualise or even come close to requiring as advanced concepts as these. What do you understand by the term 'power series', Mr. Feynman?

[Alric]

Really? We were introduced to this in school when I was 15. The term is differentiation. Its actually possible to graduate collage with just algebra level math in the US. Unless you are studying math or science, people really neglect math. I always thought it was silly, that in high school you are only required to take a math class two out of the four years. Of course any one who is serious takes it all four years. I don't want to put down our math to much though, because we do have real math courses and stuff. They just make it really easy to slack off, and I suspect a lot of people do just that.

[Xei]

What's graduating college; 18 years old? If so it's the same in the UK. You can elect to drop maths completely when you're 16. Well, you can drop out of school completely when you're 16.

[Alric]

Well in the US you go to school until your 18, then collage is normally 4 years after that if you want a basic degree(which is of course optional). So technically you could be in school until you are 22 get a bachelors degree in say arts, and you never got passed algebra.

[Invader]

High school in the U.S. ends at about 18 years of age, yar. College can also be as short as two years if one is only going to receive "general education" credits or trade skills at a community college (the cheap alternative to universities). Some high schools, at least in California (or maybe just in the district I'm in) a minimum of three years of math classes are required. You're much happier to abuse differential notation for instance (e.g. dy/dx = 1 <=> dy = dx) Well, I've only ever seen it written dy/dx = 1. I'm going to assume that's the legit version. Also,

[Phion]

Yes, considering tangents is the right way to do this problem. You're actually very close to an exact method, if you just expand on this a little. Power series are infinite series, and you tend to find them by Taylor's method. Considering the Babylonians had only just invented 0, I find it hard to believe that they could conceptualise or even come close to requiring as advanced concepts as these. What do you understand by the term 'power series', Mr. Feynman? A power series is an infinite sum of terms in a sequence with a very specific multiplier of the form (x-a). The Babylonians where surprisingly advanced for there time and could even solve quadratics similar to what we're dealing with here. Plus, they did their calculations using a very obscure number base, and they could even take measurements geometric techniques. While I may be mistaken about the time frame of development, infinite series has been well thought about since at least the Roman times; see Zeno of Elea. Many of his proposed paradoxes survive to this day, it's all very interesting. Putting another couple minutes into this method couldn't hurt. Lets see. m(x-x_0)+y_0 = ax^2+bx+c [m(x-x_0)+y_0] = (ax^2+bx+c) (mx-mx_0+y_0) = (ax^2+bx+c) (mx-mx_0+y_0) - (ax^2+bx+c) = 0 (ax^2 + bx -mx) + (c-y_0-mx_0) = 0 ax^2 + x(b-m) + (c-y_0-mx_0) = 0 (b-m)^2 - 4(1)(c-y_0-mx_0) = 0 Solve for m, and plug that discriminant into the quadratic equation to obtain a real solution.

[Xei]

A power series is an infinite sum of terms in a sequence with a very specific multiplier of the form (x-a). The Babylonians where surprisingly advanced for there time and could even solve quadratics similar to what we're dealing with here. Plus, they did their calculations using a very obscure number base, and they could even take measurements geometric techniques. While I may be mistaken about the time frame of development, infinite series has been well thought about since at least the Roman times; see Zeno of Elea. Many of his proposed paradoxes survive to this day, it's all very interesting. The thing about Zeno though was they were only paradoxes because the idea of infinite series hadn't been properly developed yet. I'm still not sure how power series fit in with this. Solve for m, and plug that discriminant into the quadratic equation to obtain a real solution. Simply solving for m is sufficient. Working through the algebra is quite neat. Have a cookie. Invader: It pops up more when you integrate a separable differential equation or do integration by substituting the differential at the end of the integral. And nah, more like )(ei. That's how pretty much everyone writes maths xs, otherwise they look like multiplication. I use . for multiplication though so it's fine. Also do you do lines through your 7s? :V

[Phion]

High school in the U.S. ends at about 18 years of age, yar. College can also be as short as two years if one is only going to receive "general education" credits or trade skills at a community college (the cheap alternative to universities). Some high schools, at least in California (or maybe just in the district I'm in) a minimum of three years of math classes are required. Yea, I'll be finishing up two certificates in computer science this year (only 60 credits total), then after that I'll be going back to obtain a transfer degree in physics/engineering, which I should have gotten almost a year ago, to a university in the state of Illinois. I slacked off quite a bit in high school, but I've done a pretty good job at making up for it. Well, I've only ever seen it written dy/dx = 1. I'm going to assume that's the legit version. Me too, but I think he's talking about the actual operation of differentiating an equation, which involves multiplying the expression by dx.

[Phion]

Also do you do lines through your 7s? :V Habitually.

[Xei]

What about zs? :V

[Phion]

What about zs? :V Always.

[Xei]

Let's be friends.