Author Archives: futurebird

Notebook Collection Part 2

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In which I continue to share my collection of blank notebooks and journals.

These large notebooks have unconventional papers: Rhodia "DOTS" paper, Whitelines graph paper, Bob's Your Uncle "pretty vacant" dot paper. Writersblock dots, and a custom notebook from etsy.com

These are some of my large notebooks, the theme here is "unconventional paper" -- blank paper and lined paper are very nice, but when you spend as much time as I do staring at notebooks a change of scenery is nice.

This is what whitelines paper looks like.

Whitelines paper is a love or hate thing. I know many people who love it and just as many who can't stand it. I'm on the love side. I find it much easier to see my drawings and diagrams without black lines cutting through them. I have rejected many notebooks for having lines that are far too dark. To my eye dark lines look cheap, like those cartoon character notebooks you see in dollar stores.

The Rhodia Dot Pad was my favorite paper until I discovered Writersblok dot paper. As I mentioned in my previous post Writersblok is very much underrated and they make the best dot paper around. I'll show you why:

Three kinds of dot paper: Bob's Your Uncle, Rhodia, Writersblok

Here you see the inside of the three notebooks with dotted paper. The Bob's Your Uncle notebook is very fun, but the dots are so big that they don't really work for graphing. (Bob's Your Uncle graph paper notebooks on the other hand are amazing, very relaxed light green grid, I'd share but I filled them all up so quickly that I have no blank pages!) Rhodia is better, but the dots still stand out on the very white paper. But, Writersblok has it perfect! Look at how fine those dots are, they almost don't even show up in the photo!

What about the green "custom" number? It's a mix of different types of graph paper, dot papers and blank pages.

The green "Custom" notebook.

Next up? Hardcover books:

Hardcover blank books.

I keep my eye out for re-purposed journals with scientific or mathematical themes.

The cover is from an old book of mathematical tables, pages from the original are mixed in.

The burgundy recycled journal has a Coptic binding. (I'm not a huge fan of Coptic binding, but this one is very nice!)

"Mathematics" you'll find the same title on the spine.

Open it to find...

The "mathematics" book has pure white blank pages.

I don't know what to write in the "mathematics" book it is very formal and commanding isn't it?

Boorum Pease makes the most amazing record books.

This RECORD book has a lovely red satin bookmark. These can be found at old house sales, or on Amazon for around $20.

Record book open.

I don't often used lined pages but I'll make an exception. The numbered pages and "Table of Contents" page are the kind of little touches that make me fall in love with a notebook. The red and black record book to the right also has numbered pages....

I think I'm in HEAVEN!

... BUT instead it has quad paper! My favorite. I first discovered this notebook at a Manhattan stationary store. It was very dusty and I literally fished it out from behind a cabinet. It was torn in a few places, but the store owner still wanted $90 for it! I was so in love I almost bought it.

So, when I discovered I could get one on Amazon for $30 ... well that seemed like a deal. I want to take this one camping with me this summer.

I still have a lot more to share, but that's all for today! If you have notebooks that you love let me know in the comments.

Notebook Collection

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There is nothing quite as exciting as a fresh blank notebook full of possibilities, well except, perhaps a full worn notebook filled with new mathematics! I can admit that I'm addicted to notebooks. Whenever I see one, with a new kind of color or material I buy it. As such, a shelf in our house is totally devoted to blank notebooks.

The Notebook Shelf

The Notebook Shelf

It was only a few months ago when I put them all in one place that it hit me how many I'd accumulated. These are all blank. I go through about one or two notebooks per week (of the Moleskine Cahier size) So, I have almost a whole bookcase full of full notebooks as well. I use Rhodia, Moleskine, Whitelines, Writer's block and other big brands quite often, In this post I'll focus on the more rare and unique stuff, the notebooks that make my hand tremble with excitement as I get ready to make that first mark on the unblemished page! The notebooks that I cherish, they are almost too good to write in, yet, I await the day that I mark them with such pent-up desire!

Okay, maybe I'm getting a little too excited here, on with the notebooks!

I like large notebooks best, they are best suited for lesson plans and proofs.

Although I like large notebooks best I do have a few small ones that I could not resist. Here we have: Quattero, a generic red notebook, an antique 1950s notebook, Writer's block "Dots" mini notebook and "Tidbit" Free Cut memo.

Quattero notebooks come in all sizes, they are a little pricy, but the paper is so smooth and thick it's worth it, the only thing I don't like about them is that the backs of the papers are blank, you only get the grid on one side, considering that the paper is bleed-proof this seems like a missed opportunity.

Here's what they look like inside: Writerblock "dots", Quattro, and the little red one.

Writersblok is one of the most underrated notebooks around. They are a less expensive version of the "Moleskine Cahier" but, even if they cost more than Moleskine I'd still take them over the Moleskine any day! Why? DOTS. Dots are the ideal method of ruling the blank page, grid paper is too distracting, lined paper looks awful if you stop to make drawings and graphs as often as I do. So dotted paper is one of the main things I look for in a notebook. The dots in the Writersblok notebooks are the best, they are whisper light, so that you'll almost forget they are there. Yet when you look at your pages you'll see that the lines are straighter, graphs are more accurate and the text is evenly spaced. I've been seeing more and more kinds of dot paper, I really hope it catches on!

Antique notebooks fascinate me. Especially those that have survived totally blank and untouched like this one for decades. I love that this little memo book has an even smaller notebook (not an address book!) tucked inside. If I ever think of the perfect use for it I will use it, but for now it remains untouched, it is one of my most cherished notebooks. (*sniff*)

Notebook within a notebook...

Notebook within a notebook...

Ever since I started loving blank books and shopping for them I have starting HATING address books, planners and photo albums. Why? Often I'll see a notebook on the shelf, the perfect size! The perfect color! I reach for it and.... it's a PHOTO ALBUM! ugh. So annoying. I end up wondering who are all of the people who like photo albums and address books so much, have they not heard of Flikr and gmail? I suppose they must be wondering what's with all of the blank books... who needs paper? I have an iPad.

(I will continue sharing my collection in the next post.)

More Hints

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Here is the problem:

Prove that the volume of a cone, (defined as the set of points on line segments joining a vertex, v, with a set of points in the same plane called the base) with height H, and base area B is \frac{BH}{3}.

What makes this tricky is at first it seems, with so many possible shapes for the base, there is no good way to express all of these 'different' areas with the same A(x). But, since the slices are similar we need only concern ourselves with the ratio of any side, radius, diagonal or width of a given slice to that same side, diagonal radius, or width on the base. With this we can find the area, independent of any specific formula (such as s^2 = A or \pi r^2 =A)

To do this use the fact that the ratio of areas of similar plane figures is equal to the ratio of any one of their dimensions squared. And if we are looking for the area slice at x that ratio will be the same as the ratio of x to H, the total height.

So, this means that \frac{A(x)}{B} = \frac {x^2}{H ^2}. So, now you have a formula for A(x).

Of yarn and math.

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A collection of hand-spun yarn and my spindle. I did not spin all of this yarn, some is from etsy.com

Yarn has been on my mind. It's on my mind because I love to knit, but also because of this post I read some weeks ago on the mathematics stack exchange. The author presents a paper about an "optimal yarn ball" that looks like this:

via. http://arxiv.org/abs/1005.4609

Very cool, but it would be impossible to do that with real yarn! (Though, I doubt that was their goal here.) When wrapping yarn around a ball you cannot make sharp turns or the yarn will fall off. If we think of the yarn as a curve on the surface of the sphere, we would say it must have curvature less than some small constant m. (meaning the best fit tangental circle must have radius greater than \frac 1m)

The yarn should be wrapped evenly: it should not overlap itself too often. (This would lead to the yarn ball taking on a non-spherical shape over time. No good!) The distribution of the self-intersections should be of nearly consistent density over the surface of the sphere. The yarn will divide the surface of the sphere in to regions. If A_i, A_j are the areas of any two of these regions then for some small constant F, |A_i- A_j| < F, \forall i, j.

As the curve is extended further and further (wrapped around more times) the density of intersections should increase and A_i \rightarrow 0 for all of the regions.

I know one answer is a randomly generated curve that deviates from a straight path. Overtime it produces a perfect yarn ball. That's how I make my yarn balls, randomness produces even inward tension which forms a sphere.

But, I want to know if there is a non-random answer to this question.

First Thoughts:
In trying to solving this myself I thought it might be a good idea to project the sphere on to the plane, as a Riemann surface. But, since so many of the requirements focus on uniformity, this makes the problem a little strage. Observe: The area of the regions will need to *increase* as we move away from the unit circle toward infinity. If the curve ever passes through the north pole, then the plane version would shoot off to infinity. What I found even more weird is what would happen if the curve simply passed very near, but not through, the north pole. Then, on the plane, the curve would go very far from the origin and then loop back. By making the curve near enough to the north pole the loops can be as large as we please.

Since wrapping increases the density uniformly, if we observe the wrapping pattern projected on to the plane it would be like loopy knot that, over time, keeps casting bigger and bigger outlier loops, it would always grow and be un-bounded. (even without any intersection with the north pole!)
The requirement about curvature, would mean that the knot could be more curvy inside of the unit circle, and then grow less curvy as we move away from the origin.

Next to find such a curve... Maybe if I projected the following polar plot back on to the Riemann sphere?

The graph of the form r= \theta^2 \sin (k \theta) produced good-looking results for 0 < k <1... but all of the graphs are far too dense near the origin. Also, wrapping would start in a lop-sided manner, filling in the bottom of the sphere first then moving up... and never reaching the north pole! But, maybe there is some way to tweak this?

(This graph shows the kind of non-random pattern I have in mind for a solution. In a practical sense, a solution to this problem could be used to wind spherical balls of ropes, yarns, or cords in a factory setting. Now, I will think about how I would instruct a robot to wind a yarn ball... if the robot is unable to do things randomly. What would you tell the robot to do?)

This post grew out of a question I asked at the Stack Exchange.

Hints

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Take home

#2 Hint:

#3 Hint:

You can use this trig identity
\sin x \cos x = \frac12 \sin 2x

#5 Hint:

Consider f(t) = g(t) - h(t) , where g and h are the position functions of the two runners.

#6 Hint:

\frac{d}{dx} |x| = sgn x = \frac{x}{|x|}

#7 Hint:

Induction would be much much harder than L'Hopital.

#8 (b)Hint:

Use \int_a^b f(t)dt =-\int_b^a f(t)dt

#8 (d)Hint:

Divide the square root of x in to each number in the numerator to make it easier to see what the anti derivative should be.

#9 Hint:

Draw a diagram look at the triangles.

Answers to Routine Practice Problems.

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  1. \lim_{x \rightarrow 0} \frac{\sin(x)}{x}
    This shows how powerful Hospital's rule really is. Recall, that we did a rather long geometric proof of this limit earlier in the semester. So, from that we know it's 1. This is the indeterminate form \frac{0}{0}, consider
    \lim_{x \rightarrow 0} \frac{\cos(x)}{1} and this is clearly 1. 

     

  2. \lim_{x \rightarrow \infty} \left( \sqrt{x} - \sqrt{x+1} \right) Use the conjugate.
    = \lim_{x \rightarrow \infty} \frac{ \left( \sqrt{x} - \sqrt{x+1} \right)\left( \sqrt{x} + \sqrt{x+1} \right)}{\sqrt{x} + \sqrt{x+1} } = \lim_{x \rightarrow \infty} \frac{ x - x+1}{\sqrt{x} + \sqrt{x+1} } = \lim_{x \rightarrow \infty} \frac{1}{\sqrt{x} + \sqrt{x+1} } = 0  

     

  3. \lim_{x \rightarrow \infty}\frac{x \ln(x)}{\sqrt{x}} We have \frac{\infty}{\infty} So, consider:
    \lim_{x \rightarrow \infty}\frac{\frac{x}{x}+ \ln x}{\frac{1}{2\sqrt{x}}} = \lim_{x \rightarrow \infty}\frac{2\sqrt{x} + 2\sqrt{x}\ln x}{1} = \infty 

     

  4. Find an equation of the tangent to the curve y=x^2-x that is parallel to y=3x+3.
    y'=2x-1, we want the derivative to be 3, the slope of y=3x+3. So, solve 3=2x-1, and you get x=2. Now find the tangent line through the point (2, 2). y=3x-4. 

     

  5. Find an equation of the tangent to the curve y=\ln x that will pass through the origin.
    We are looking for a line, so y=mx+b. It goes through the origin so we already know b=0. y'=\frac{1}{x} so, if the point where this line is tangent is ( x_0, y_0) then y_0 = \frac{1}{x_0}x_0 +b means b = y_0 -1. If b=0, then y_0 =1. This happens when x_0 =e. So, the slope, m = \frac{1}{x_0} = \frac{1}{e}. Our equation is: y= e^{-1}x 

     

  6. A man walks away from a bird perched on a 40-foot-high flag pole. The man walks at a rate of 5 feet per second. How fast is the distance (as the crow flies) between the man and the bird increasing when he is 30 feet away from the pole? 

    Find the situation is a right triangle. Call the base b, the height (the pole) a, and the hypotenuse c. We can use a^2 + b^2 = c^2 to find c when b=40... c=50. First a, the height of the pole is constant. so: 1600 + b^2 = c^2 Now take the derivative of our equation w/ respect to time: 0 + 2b\frac{db}{dt} = 2c \frac{dc}{dt} since \frac{db}{dt} = 5 \frac{feet}{second} and we know c=50 and b=30. Hence, 2* 30 * 5 \frac{feet}{second} = 2*50 * \frac{dc}{dt}. We want to know \frac{dc}{dt}, solving we get \frac{dc}{dt}= 3 \frac{feet}{second}.

     

  7. For f(x) = 3x^4 - 12x^3+17: State any interval(s) of increase. State any relative minimum(s) and relative maximum(s). State the (x,y) location of any point(s) of inflection. 

    Take the derivative: f'(x) = 12x^3 - 36x^2 = 12x^2(x -3) our critical points are 0 and 3. Test values in each interval:

    f'(-1) = -12 - 36 = neg So, (-\infty, 0) is an interval of decrease.
    f'(2) = 12 \dot 8 - 36 \dot 4 = neg So, (0, 3) is also an interval of decrease
    f'(4) = 12 \dot 64 - 36 \dot 16 = pos So, (0, 3) is an interval of increase.

    We only have a relative min. and it's at x=3. Next for the inflection point we look at the 2nd derivative:

    f''(x) = 36x^2 - 72x = 36x(x -2) Possible inflection points at 0 and 2.

    f''(-1) = 36 + 72 = pos concave up.
    f''(1) = 36 -72 = neg concave down.
    f''(10) = 3600 - 720 = pos concave up.

    The concavity changes at both points so there are two inflection points: at 0 and 2. What could such a function look like?

     

  8. Find \frac{dy}{dx} for \sin^3(x+y) =1.

    Using the chain rule: 3 \sin^2 (x+y) \cos(x+y) (1 + \frac{dy}{dx}) =0 then solve for \frac{dy}{dx}. You get -1.
    This may seem like too simple of a solution but, if sin^3 (x+y) = 1 then take the cube root of both sides and:
    sin (x+y) = 1

    The sine is 1 at \frac{\pi}{2}, \frac{\pi}{2} + 2\pi, \frac{\pi}{2} + 4\pi, ... , \frac{\pi}{2} + n 2\pi ... so:

    • x+y = \frac{\pi}{2}

    • x+y = \frac{\pi}{2} + 2\pi

    • x+y = \frac{\pi}{2} + 4\pi

    • \vdots

    Then subtract x in each equation:

    • y = -x + \frac{\pi}{2}

    • y = -x + \frac{\pi}{2} + 2\pi

    • y = -x + \frac{\pi}{2} + 4\pi

    • \vdots

    These are linear equations. If we graph all of them at once then that is the set of (x,y) pairs that solve our equation:

    (imagine more evenly spaced parallel lines that fill the whole graph...)

    The slope of any tangent to these lines in -1 everywhere. So, our solution makes sense.

     

  9. Find \frac{dy}{dx} for \sqrt{xy} = 2 + x^2 y^2. You will get \frac{4xy^2 \sqrt{xy} -y}{x- 4yx^2 \sqrt{xy} }= \frac{y(4xy \sqrt{xy} -1)}{x(1- 4yx \sqrt{xy}) } = \frac{-y}{x} 

     

  10. Find an antiderivative of f(x) = 3x^2 - 2x -1.
    F(x)= x^3 - x^2 - x + 5 (The constant can be anything including 0.) 

     

  11. Find an antiderivative of f(x) = \frac {2x \sin (x) - x^2 \cos (x)}{\sin^2 x}
    F(x) = \frac{x^2}{\sin x} + 9 (The constant can be anything including 0.) 

     

  12. Find the linearization of y=x^3 -2x +1 centered at 0.   
    This is the same as finding the tangent to the curve at (0, 1). The slope is given by y'=3x^2 -2 if x=0 then the slope is -2. Then find the equation y = -2 x +1 

     

Almost Convergence.

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The title of my last post "Almost Cauchy Sequence" made me wonder if "Almost Cauchy" was already defined in some formal sense. I could not find much on "Almost Cauchy" ... but I did find out about "Almost Convergence." (I think I have ran in to this before... but it is very neat... observe:)

A sequence S = S_n is almost convergent to L if for any \epsilon > 0 we can find an integer n such that the average of n or more consecutive terms in the sequence is within \epsilon of L. Formally, \forall \epsilon > 0 \quad \exists N \ni \quad \displaystyle \left| \frac{1}{n} \sum_{i=0}^{n-1} S_{k-i} -L \right| < \epsilon \quad \forall n > N, \; k \in \mathbb{N}. (I took this definition from here, and I wonder why they didn't mention that we need k \geq n or else we are looking at negative terms in the sequence... But, maybe there is an interpretation of something like S_{-3}? Or maybe the author thinks that would be obvious? I'm going to ask at the mathematics stack exchange!)

Almost Cauchy Sequence, unfinished business.

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I just remembered a question from beginning analysis that my professor mentioned in class:

"Find a sequence A that is not Cauchy, yet any consecutive terms A_n, A_{n+1} will become arbitrarily close."

It was one of those questions that he mentioned in class for us to do on our own, but I never quite got to it. I was thinking about convergence and complete metric spaces today (and about unfinished business, as I've made a lot of my own since I started teaching.) And the question came back to me.

We seek a sequence, A, where \forall \; \epsilon > 0 \quad \exists N \in \mathbb{N} \; \ni \; \forall n > N \quad |A_n - A_{n+1}| < \epsilon and \exists \epsilon > 0 \; \ni \quad \forall N \in \mathbb{N} \quad \exists m,n > N \quad |A_m - A_n| > \epsilon. (Writing the negation of what it means for a sequence to be Cauchy was a little more tricky than I thought...)

The natural log function is the most obvious answer. For large vales of n \ln (n) - \ln (n+1) is small. That is, \displaystyle \lim_{n \rightarrow \infty} \ln \frac{n}{n+1} = 0. This is true becuase after taking EXP of both sides we have: \displaystyle \lim_{n \rightarrow \infty} \frac{n}{n+1} = 1

Next, it seems obvious that the natural log can't be Cauchy, but just to be certian if we let \epsilon = \frac{1}{2} then, \forall N \in \mathbb{N} if we choose some n > N and let m= ne (so, n,m > N) then

| \ln m - \ln n | = |\ln \frac{m}{n}| = |\ln \frac{ne}{n}| = 1 > \frac{1}{2} = \epsilon.

NOT Cauchy at all!

Ok, I imagine my professor was thinking of the natural log... but, what other functions also have this property? \sqrt[n]{x}? So, I guess this is really asking when \displaystyle \lim_{x \rightarrow \infty} f(x) - f(x+1) = 0 while \displaystyle \lim_{x \rightarrow \infty} f(x) = \pm \infty. It is necesairy but not sufficent for f'(x) > 0 \; \forall x and f''(x) < 0 \; \forall x , both will have the x-axis as a horizontal asymptote.

Nice to have finally gotten around to doing this problem! I think it will be good practice to find more unfinished business in my notes.

Answers to homework.

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1. Find an equation of the tangent to the curve y = e^x that is parallel to x−4y = 1.

Y = \frac{1}{4}x + \frac{1-\ln (1/4)}{4}

(It is better to leave the natural log in the equation, don't find a decimal value... Though you may use such value for graphing.)

20110404-125754.jpg

2. Find an equation of the tangent to the curve y = e^x that will pass through the origin.

The the tangent to the curve at x_0 will have the form y = e^{x_0}(x+1-x_0) ... Plug in (0,0) and you find you are seeking the x-value where the natural log is zero. So the equation is y= ex.

20110404-014525.jpg