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- #27

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I saw both Taylor and MVT in high school, but only one induction proof.@etotheipi

Moreover, Taylor and MVT are not really school level. Induction is.

- #28

fresh_42

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We tried, but it didn't work. What is intermediate? Taylor and MVT in high schools (see post #24), or are even induction and limits too difficult? Especially school systems vary significantly all over the world.

And on the other end: Problems #4 and #5 are not very sophisticated. I would have called them "intermediate", and even the algebraic questions are not so difficult. We even had questions which could have been solved by looking up some definitions on Wikipedia, and even this was too much to do. People get scared by technical terms, but our questions are already heavily limited to only easy concepts. I tried to find some more interesting questions in my books ... no chance! I would need to hold a lecture beforehand, regardless whether a question would had been easy or difficult, simply to set up the language.

We have a thread with solutions to many questions:

https://www.physicsforums.com/threads/solution-manuals-for-the-math-challenges.977057/

where I think can be found problems at any level.

- #29

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I agree. I found problem #5 could be solved easily in a few minutes by referring to https://en.wikipedia.org/wiki/Matrix_representation_of_conic_sections. I like this problem because it showed me a beautiful connection between linear algebra and geometry. Thank you for making the effort to post the monthly challenges, I learn a lot from themAnd on the other end: Problems #4 and #5 are not very sophisticated. I would have called them "intermediate", and even the algebraic questions are not so difficult. We even had questions which could have been solved by looking up some definitions on Wikipedia, and even this was too much to do.

- #30

etotheipi

What is a Euclidean normal form, and what do we mean by ##(x,y)^{\tau}##?

- #31

member 587159

What is a Euclidean normal form, and what do we mean by ##(x,y)^{\tau}##?

I believe it means the vector ##(x,y)## transposed, so written as column vector ##\begin{pmatrix}x \\ y\end{pmatrix}##.

- #32

fresh_42

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Yes, it is only a habit in this case to write vectors or coordinates as column. The Euclidean normal form is, when the axis of the conic sections are the coordinate axis, and its center the origin of the Cartesian coordinate system: rotations, translations; basically such that all interesting quantities can directly be read.What is a Euclidean normal form, and what do we mean by ##(x,y)^{\tau}##?

- #33

etotheipi

- #34

fresh_42

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Well, you could have shown how you did it. It's a bit difficult to learn anything from your solution. And the foci are ##(0,\pm 1/\sqrt{2})## if I wasn't mistaken.

- #35

Office_Shredder

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- #36

fresh_42

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That isn't what the problem asks for.

- #37

etotheipi

Yes, sorry, I forgot to square root. Anyway, I'll give a more complete method here for anyone that wants it. First we need to find the centre of $$f(x,y) = 34x^2 + 24xy +41y^2 + 20x + 110y + 50=0$$Set ##f_y = 0## and ##f_x = 0## to get$$\begin{align*}68x + 24y + 20 &= 0 \\Well, you could have shown how you did it. It's a bit difficult to learn anything from your solution. And the foci are ##(0,\pm 1/\sqrt{2})## if I wasn't mistaken.

82y + 24x + 110&=0

\end{align*}$$this is solved by ##\mathbf{x}_c = (x_c,y_c)^{\tau} = (\frac{1}{5}, \frac{-7}{5})^{\tau}##, i.e. this is the centre of the ellipse. Now we perform a coordinate transformation ##\mathbf{x}' = \mathbf{x} -\mathbf{x}_c## so that the centre is at our new origin ##\mathcal{O}'##, i.e.$$34(x' + \frac{1}{5}) + 24(x' + \frac{1}{5})(y' - \frac{7}{5}) + 41(y' - \frac{7}{5})^2 + 20(x' + \frac{1}{5}) + 110(y' - \frac{7}{5}) + 50 = 0$$ $$34x'^2 + 24x'y' + 41y'^2 = 25$$Now we perform a further coordinate transformation, this time a rotation (preserving the origin, ##\mathcal{O}'' = \mathcal{O}'##), so that the new axis intersect the ellipse at the vertices. This will be ##\mathbf{x}'' = R\mathbf{x}'##, where ##R = \begin{pmatrix} c & s \\ -s & c\end{pmatrix}##, and ##c=\cos{\theta}##, ##s = \sin{\theta}## for some angle ##\theta## to be determined. Running through the algebra with ##x' = cx'' - sy''## and ##y' = sx'' + cy''## yields$$34(cx'' -sy'')^2 + 24(cx'' - sy'')(sx'' + cy'') + 41(sx'' + cy'') = 25$$ $$x''^2 [ 34 c^2 + 41s^2 + 24cs] + y''^2 [41c^2 + 34s^2 - 24cs] + x''y''[24c^2 - 24s^2 + 14cs]$$We require that the coefficient of ##x''y''## is brought to be zero, i.e. that ##s = 4/5## and ##c=3/5##. Now it's just a case of substituting those back into the equation,$$50x''^2 + 25y''^2 = 25 \iff 2x''^2 + y''^2 = 1$$and for brevity I'll re-label ##x'' \equiv u## and ##y'' \equiv v##. And that completes the transformation!

- #38

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How did you learn this stuff when you're a first year uni student? I'm at the top of my first year math course for chemistry, but I haven't even seen half of this stuff before.Yes, sorry, I forgot to square root. Anyway, I'll give a more complete method here for anyone that wants it. First we need to find the centre of $$f(x,y) = 34x^2 + 24xy +41y^2 + 20x + 110y + 50=0$$Set ##f_y = 0## and ##f_x = 0## to get$$\begin{align*}68x + 24y + 20 &= 0 \\

82y + 24x + 110&=0

\end{align*}$$this is solved by ##\mathbf{x}_c = (x_c,y_c)^{\tau} = (\frac{1}{5}, \frac{-7}{5})^{\tau}##, i.e. this is the centre of the ellipse. Now we perform a coordinate transformation ##\mathbf{x}' = \mathbf{x} -\mathbf{x}_c## so that the centre is at our new origin ##\mathcal{O}'##, i.e.$$34(x' + \frac{1}{5}) + 24(x' + \frac{1}{5})(y' - \frac{7}{5}) + 41(y' - \frac{7}{5})^2 + 20(x' + \frac{1}{5}) + 110(y' - \frac{7}{5}) + 50 = 0$$ $$34x'^2 + 24x'y' + 41y'^2 = 25$$Now we perform a further coordinate transformation, this time a rotation (preserving the origin, ##\mathcal{O}'' = \mathcal{O}'##), so that the new axis intersect the ellipse at the vertices. This will be ##\mathbf{x}'' = R\mathbf{x}'##, where ##R = \begin{pmatrix} c & s \\ -s & c\end{pmatrix}##, and ##c=\cos{\theta}##, ##s = \sin{\theta}## for some angle ##\theta## to be determined. Running through the algebra with ##x' = cx'' - sy''## and ##y' = sx'' + cy''## yields$$34(cx'' -sy'')^2 + 24(cx'' - sy'')(sx'' + cy'') + 41(sx'' + cy'') = 25$$ $$x''^2 [ 34 c^2 + 41s^2 + 24cs] + y''^2 [41c^2 + 34s^2 - 24cs] + x''y''[24c^2 - 24s^2 + 14cs]$$We require that the coefficient of ##x''y''## is brought to be zero, i.e. that ##s = 4/5## and ##c=3/5##. Now it's just a case of substituting those back into the equation,$$50x''^2 + 25y''^2 = 25 \iff 2x''^2 + y''^2 = 1$$and for brevity I'll re-label ##x'' \equiv u## and ##y'' \equiv v##. And that completes the transformation!

- #39

etotheipi

How did you learn this stuff when you're a first year uni student? I'm at the top of my first year math course for chemistry, but I haven't even seen half of this stuff before.

Coordinate transformations I learned from Douglas Gregory, Classical Mechanics! It's not so bad, you just need to be careful about when you're leaving the vector alone but changing its representation i.e. w.r.t. a new coordinate system (passive), or if you're actually transforming the vector (active), and not getting the signs mixed up!

- #40

fresh_42

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It is not too difficult:How did you learn this stuff when you're a first year uni student? I'm at the top of my first year math course for chemistry, but I haven't even seen half of this stuff before.

Given the red one, asked for the blue one. You have to rotate the curve, stretch it, and shift the center into the origin. You can either handle this with linear algebra (hint in post #29, or wait for the long version in my solution manual), or simply perform coordinate transformations (post #44) until it fits. We want to have an ellipse at the end: ##\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1##.

Imagine the ellipse is a crystal instead, and the red one is what you see under the microscope, whereas you need the blue version to identify it by the pictures in the book.

- #41

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In physics i risk to say there is change of coordinates in all place, mainly in mechanic, since it is important to all the laws be the same in all coordinates, i think the college starts to talk about it very early when the course is physics. For example, in my first semester of physics i saw it, probably the same with ethoteipiHow did you learn this stuff when you're a first year uni student? I'm at the top of my first year math course for chemistry, but I haven't even seen half of this stuff before.

- #42

fresh_42

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I would even go as far as to say: physics is all about coordinates. Without coordinates, how do you measure? And it is the reason why general relativity is hard to understand: all of a sudden, coordinates become dependent on the observer! Mathematically you just want to know whether it is an ellipse, a parabola or an hyperbola.In physics i risk to say there is change of coordinates in all place, mainly in mechanic, since it is important to all the laws be the same in all coordinates, i think the college starts to talk about it very early when the course is physics. For example, in my first semester i saw it.

- #43

etotheipi

Problem 5, I can do the first two parts today, too tired to attempt the harder final parts,$$\begin{align*}

\text{det}\begin{bmatrix}5 - \lambda &0&1&6\\3&3-\lambda&5&2\\0&0&3-\lambda&0\\6&0&3&-\lambda\end{bmatrix} &= \text{det}\begin{bmatrix}0&0&3-\lambda&0\\6&0&3&-\lambda\\5 - \lambda &0&1&6\\3&3-\lambda&5&2\end{bmatrix} \\ \\

&= (3-\lambda) \text{det} \begin{bmatrix} 6 & 0 & -\lambda \\ 3-\lambda & 0 & 6 \\ 3 & 3-\lambda & 2 \end{bmatrix} \\ \\

&= (3-\lambda)\left( 36(\lambda -3) -\lambda(\lambda-3)^2 \right) \\

&= (\lambda - 3)^2 (\lambda + 4)(\lambda -9) = 0

\end{align*}$$so ##\lambda_1 = 3##, ##\lambda_2 = -4##, ##\lambda_3 = 9## are the eigenvalues of ##A##. Now for the eigenvectors, ##\vec{u}, \vec{v}## and ##\vec{w}##,$$\begin{align*}\begin{bmatrix} -4 & 0 & 1 & 6 \\ 3 & -6 & 5 &2 \\ 0 & 0 & -6 & 0 \\ 6 & 0 & 3 & 9\end{bmatrix} \begin{bmatrix} u_1 \\ u_2 \\ u_3 \\ u_4 \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \iff u_1 = \frac{3}{2} u_4, u_2 = \frac{13}{12}u_4, u_3 = 0 \\ \\\begin{bmatrix} 9 & 0 & 1 & 6 \\ 3 & 7 & 5 &2 \\ 0 & 0 & 7 & 0 \\ 6 & 0 & 3 & 4\end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ v_3 \\ v_4 \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \iff v_1 = -\frac{2}{3}v_4, v_2 =0, v_3 = 0 \\ \\ \begin{bmatrix} 2 & 0 & 1 & 6 \\ 3 & 0 & 5 &2 \\ 0 & 0 & 0 & 0 \\ 6 & 0 & 3 & -3\end{bmatrix} \begin{bmatrix} w_1 \\ w_2 \\ w_3 \\ w_4 \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \iff w_1 = w_3 = w_4 = 0\end{align*}$$

So ##\vec{u} = \begin{bmatrix} 18 \\ 13 \\ 0 \\ 12 \end{bmatrix}##, ##\vec{v} = \begin{bmatrix} -2 \\ 0 \\ 0 \\ 3 \end{bmatrix}##, ##\vec{w} = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}##

\text{det}\begin{bmatrix}5 - \lambda &0&1&6\\3&3-\lambda&5&2\\0&0&3-\lambda&0\\6&0&3&-\lambda\end{bmatrix} &= \text{det}\begin{bmatrix}0&0&3-\lambda&0\\6&0&3&-\lambda\\5 - \lambda &0&1&6\\3&3-\lambda&5&2\end{bmatrix} \\ \\

&= (3-\lambda) \text{det} \begin{bmatrix} 6 & 0 & -\lambda \\ 3-\lambda & 0 & 6 \\ 3 & 3-\lambda & 2 \end{bmatrix} \\ \\

&= (3-\lambda)\left( 36(\lambda -3) -\lambda(\lambda-3)^2 \right) \\

&= (\lambda - 3)^2 (\lambda + 4)(\lambda -9) = 0

\end{align*}$$so ##\lambda_1 = 3##, ##\lambda_2 = -4##, ##\lambda_3 = 9## are the eigenvalues of ##A##. Now for the eigenvectors, ##\vec{u}, \vec{v}## and ##\vec{w}##,$$\begin{align*}\begin{bmatrix} -4 & 0 & 1 & 6 \\ 3 & -6 & 5 &2 \\ 0 & 0 & -6 & 0 \\ 6 & 0 & 3 & 9\end{bmatrix} \begin{bmatrix} u_1 \\ u_2 \\ u_3 \\ u_4 \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \iff u_1 = \frac{3}{2} u_4, u_2 = \frac{13}{12}u_4, u_3 = 0 \\ \\\begin{bmatrix} 9 & 0 & 1 & 6 \\ 3 & 7 & 5 &2 \\ 0 & 0 & 7 & 0 \\ 6 & 0 & 3 & 4\end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ v_3 \\ v_4 \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \iff v_1 = -\frac{2}{3}v_4, v_2 =0, v_3 = 0 \\ \\ \begin{bmatrix} 2 & 0 & 1 & 6 \\ 3 & 0 & 5 &2 \\ 0 & 0 & 0 & 0 \\ 6 & 0 & 3 & -3\end{bmatrix} \begin{bmatrix} w_1 \\ w_2 \\ w_3 \\ w_4 \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \iff w_1 = w_3 = w_4 = 0\end{align*}$$

So ##\vec{u} = \begin{bmatrix} 18 \\ 13 \\ 0 \\ 12 \end{bmatrix}##, ##\vec{v} = \begin{bmatrix} -2 \\ 0 \\ 0 \\ 3 \end{bmatrix}##, ##\vec{w} = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}##

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- #44

fresh_42

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Sleep over it. It is problem 4. And what is the difference between 3 and -4? And what are 9,12,13,18?Problem 5, I can do the first two parts today, too tired to attempt the harder final parts,$$\begin{align*}

\text{det}\begin{bmatrix}5 - \lambda &0&1&6\\3&3-\lambda&5&2\\0&0&3-\lambda&0\\6&0&3&-\lambda\end{bmatrix} &= \text{det}\begin{bmatrix}0&0&3-\lambda&0\\6&0&3&-\lambda\\5 - \lambda &0&1&6\\3&3-\lambda&5&2\end{bmatrix} \\ \\

&= (3-\lambda) \text{det} \begin{bmatrix} 6 & 0 & -\lambda \\ 3-\lambda & 0 & 6 \\ 3 & 3-\lambda & 2 \end{bmatrix} \\ \\

&= (3-\lambda)\left( 36(\lambda -3) -\lambda(\lambda-3)^2 \right) \\

&= (\lambda - 3)^2 (\lambda + 4)(\lambda -9) = 0

\end{align*}$$so ##\lambda_1 = 3##, ##\lambda_2 = -4##, ##\lambda_3 = 9## are the eigenvalues of ##A##. Now for the eigenvectors, ##\vec{u}, \vec{v}## and ##\vec{w}##,$$\begin{align*}\begin{bmatrix} -4 & 0 & 1 & 6 \\ 3 & -6 & 5 &2 \\ 0 & 0 & -6 & 0 \\ 6 & 0 & 3 & 9\end{bmatrix} \begin{bmatrix} u_1 \\ u_2 \\ u_3 \\ u_4 \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \iff v_1 = -\frac{2}{3} v_4, v_2 = \frac{13}{12}u_4, v_3 = 0 \\ \\\begin{bmatrix} 9 & 0 & 1 & 6 \\ 3 & 7 & 5 &2 \\ 0 & 0 & 7 & 0 \\ 6 & 0 & 3 & 4\end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ v_3 \\ v_4 \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \iff u_1 = \frac{3}{2}u_4, u_2 =0, u_3 = 0 \\ \\ \begin{bmatrix} 2 & 0 & 1 & 6 \\ 0 & 7 & 5 &2 \\ 0 & 0 & 0 & 0 \\ 6 & 0 & 3 & -3\end{bmatrix} \begin{bmatrix} w_1 \\ w_2 \\ w_3 \\ w_4 \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \iff w_1 = w_3 = w_4 = 0\end{align*}$$So ##\vec{u} = \begin{bmatrix} 18 \\ 13 \\ 0 \\ 12 \end{bmatrix}##, ##\vec{v} = \begin{bmatrix} -2 \\ 0 \\ 0 \\ 3 \end{bmatrix}##, ##\vec{w} = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}##

- #45

etotheipi

Sleep over it. It is problem 4. And what is the difference between 3 and -4? And what are 9,12,13,18?

Think I got the numbers mixed up, they should be ok now!

- #46

fresh_42

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Let's agree on a single set of representatives, not on many at the same time.Think I got the numbers mixed up, they should be ok now!

- #47

mathwonk

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historical hints: 3-weierstrass, inverse mapping theorem, 7-sylow, 9-nakayama.

Last edited:

- #48

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1. Finding the stationary points of ##f(x) \rightarrow f'(x)=0 ##

##4x^3 + 18x^2 + 22x + 6 = 0##

##(2x+3)(4x^2+12x+4)=0##

##x=-\frac{3}{2} , x=\frac{-3 \pm \sqrt{5}}{2}##

Checking the sign diagram of first derivative:

a. For ##x<\frac{-3 - \sqrt{5}}{2} \rightarrow f'(x) = (-)##

b. For ##\frac{-3 - \sqrt{5}}{2}<x<-\frac{3}{2} \rightarrow f'(x) = (+)##

c. For ##-\frac{3}{2} <x<\frac{-3 + \sqrt{5}}{2} \rightarrow f'(x) = (-)##

d. For ##x>\frac{-3 + \sqrt{5}}{2} \rightarrow f'(x) = (+)##

##x=\frac{-3 - \sqrt{5}}{2} \rightarrow y=-1## so ##(\frac{-3 - \sqrt{5}}{2} , -1)## is local minimum

##x=-\frac{3}{2} \rightarrow y=\frac{9}{16}## so ##(-\frac{3}{2} , \frac{9}{16})## is local maximum

##x=\frac{-3 + \sqrt{5}}{2} \rightarrow y=-1## so ##(\frac{-3 + \sqrt{5}}{2} , -1)## is local minimum

2. Checking the end behaviour of ##f(x)##

##\lim_{x \to \pm \infty} {f(x)=\infty}##

So the local minimum found above is also global minimum

3.

##f(x)=a## will have no solution if ##a## is below global minimum ##\rightarrow a < -1##

No ##a## will result in ##f(x)## has one solution

##f(x)=a## will have two solutions if ##a## is global minimum or ##a## is above global maximum ##\rightarrow a=-1## or ##a>\frac{9}{16}##

##f(x)=a## will have three solutions if ##a## is global maximum ##\rightarrow a=\frac{9}{16}##

##f(x)=a## will have four solutions if ##a## is between local minimum and local maximum ##\rightarrow -1<a<\frac{9}{16}##

I am not sure how to write good conclusion for final answer

- #49

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After expanding everything: ##z=\frac{7a^3+84a}{4b^2 +12}##

Since ##a=2b## :

##z=\frac{56b^3 +168b}{4b^2 +12}##

##=\frac{56b(b^2+3)}{4(b^2+3)}##

##=14b##

- #50

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[\spoiler]

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