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[cpp_is_king]

I'm not sure how you jumped from the equation for the triangle to the evaluated integral. What happened to x_a , y_a ? I treated them as constants during integration. With the linear equations that define the regions, they will not change once fixed (if that makes sense). I don't know if I'm hitting on what you are talking about but here is what I had laid out to integrate along the y-axis:

thats just it, when you're integrating, they aren't fixed. we used x_a and y_a initially to get a formula for the area of the region given some fixed point A. But in the original problem, the point A is chosen randomly, so EVERY choice of A yields a different triangle, and hence a different contribution to the final probability.. thats exactly what the integration does, varies A over the entire region.

So the inside of the double integral is the formula for area, by choosing the same integration variable as is used in the area formula, we are adding the contributions of every possible choice of A.

We could have left the formula in terms of x_a and y_a, but then we would have needed to use dx_a and dy_a.


btw, assuming you integrate in the correct order, and repeat this strategy for the top right (integrating the formula for the area of the quadrilateral), and add the 2 together, the resulting answer is correct.

Ive just never encountered a double integral where order of integration mattered, so im not sure what to make of it

 

[Troll]

10. Consider the ruled surface M given by x(u,v)=(vcosu, vsinu, uv), v>0.

c.Check that the v-curves are lines of curvature


*First of all, I am not really clear on what "v-curves" is referring to. I think it is the first and second partial derivatives of x, but I am not 100%. So I was hoping someone could help me clear that up. And second of all, I am not sure how to check whether they are lines of curvature.

Also, I feel really stupid asking this, but I suck at identifying curves. What kind of geometric surface is this?

 

[cpp_is_king]

10. Consider the ruled surface M given by x(u,v)=(vcosu, vsinu, uv), v>0.

c.Check that the v-curves are lines of curvature


*First of all, I am not really clear on what "v-curves" is referring to. I think it is the first and second partial derivatives of x, but I am not 100%. So I was hoping someone could help me clear that up. And second of all, I am not sure how to check whether they are lines of curvature.

Also, I feel really stupid asking this, but I suck at identifying curves. What kind of geometric surface is this?

First of all, this is a 4-dimensional surface, so it's kind of hard to just describe it. But I can try.

Not sure I'll be able to help with the rest of the problem, but imagine that this is a 2 dimensional equation f(u) = (v cos u, v sin u). Notice that I'm holding v constant here. In that case, it's easy to see you're dealing with a circle of radius v. But you've got this additional coordinate, so suppose v is still constant and you've got f(u) = (v cos u, v sin u, uv). The circle part is periodic with period 2pi in u, but the z coordinate is not. It will always grow linearly with u. So while the xy part (the circle) is infinitely tracing over itself as you increase u, the z part is going into (or coming out of) the sheet of paper.

What shape is this? A cylinder! Well, almost. You can fit it in a cylinder, that's for sure. But it's actually more like a corkscrew, because you're going into (or out of) the page AS you are tracing the circle.

So for a fixed v, you've got this corkscrew. But alas, we don't have a fixed v, it varies. So to go to the 4th dimension, you can imagine that you've got all corkscrews of all possible radii. Each time you change v, you get a corkscrew of a different radius and height, but they're always centered at the origin.

 

[Partial Gamification]

thats just it, when you're integrating, they aren't fixed. we used x_a and y_a initially to get a formula for the area of the region given some fixed point A. But in the original problem, the point A is chosen randomly, so EVERY choice of A yields a different triangle, and hence a different contribution to the final probability.. thats exactly what the integration does, varies A over the entire region.

So the inside of the double integral is the formula for area, by choosing the same integration variable as is used in the area formula, we are adding the contributions of every possible choice of A.

We could have left the formula in terms of x_a and y_a, but then we would have needed to use dx_a and dy_a.


btw, assuming you integrate in the correct order, and repeat this strategy for the top right (integrating the formula for the area of the quadrilateral), and add the 2 together, the resulting answer is correct.

Ive just never encountered a double integral where order of integration mattered, so im not sure what to make of it

I see how you are looking at this, just catching on that you setup the double integral for the area. I looked for some exception to Fubini's Theorem but I don't think there is one, order of integration should not matter. I googled it and found a write-up on a generic counter-example to the theorem, I didn't catch any sort of "why" explaining how order matters. The wiki page stated that with the absolute value of the integrand, of a double integral, evaluated and equalling infinity then the two ways to integrate may have different finite values (but I don't think that is applying). Now that I understand what the ship you are talking about I'll keep looking.

 

[Desmond]

EDIT: Nevermind

 

[TimesEunuch]

10. Consider the ruled surface M given by x(u,v)=(vcosu, vsinu, uv), v>0.

c.Check that the v-curves are lines of curvature


*First of all, I am not really clear on what "v-curves" is referring to. I think it is the first and second partial derivatives of x, but I am not 100%. So I was hoping someone could help me clear that up. And second of all, I am not sure how to check whether they are lines of curvature.

Also, I feel really stupid asking this, but I suck at identifying curves. What kind of geometric surface is this?

I plotted this surface in Mathematica for you. If you have access to Mathematica, I used the command
ParametricPlot3D[{v Cos, v Sin, u v}, {u, -2 \[Pi], 2 \[Pi]}, {v, 0, 1}, BoxRatios -> {1, 1, 2}, AxesLabel -> {x, y, z}]

This plot is for for u from -2pi to +2pi, and v from 0 to 1. The ruled lines on the surface are the constant u and constant v curves (the ones moving radially outward correspond to constant u). It's a 2-dimensional surface btw.

 

[Mudkips]

Regarding the square / probability problem from the other day, even though I solved it, something is still bothering me. This was the approach I used for the bottom left half of the square;

The area of the triangle in this picture is x_a y_a / [2(1-x_a)]. Hence, this is the function you need to integrate over the bottom left portion of the triangle.

There are two ways to go about this: You can integrate x from 0 to 1-y (inner), and y from 0 to 1 (outer). Or you can integrate y from 0 to 1-x (inner), and x from 0 to 1 (outer).

If you do the former, the first integral evaluates to -y(x + ln(x))/2, and you will need to evaluate this from x=0 to x=1-y . However, this is obviously nto possible, as ln(0) is undefined.

On the other hand, if you do the latter, the first integral evaluates to xy^2/[2(1-x)] evaluated from y=0 to y=1-x. You can see that the 1-x on the numerator will cancel out and you will be left to integrate x(1-x)/2 from 0 to 1.

So, here we have an example of where order of integration matters, and I can't figure out what I'm missing here. The same situation arises when you try to integrate over the top-right portion of the square, but it's harder to figure out how to make it work there.

Anyone want to take a stab at what's going on here?

I figured this was your issue when you were hinting at it a while ago.
Doesn't the problem state that the point has to be within the square (so it can't be an the edge)?

Regardless, even if the problem didn't state that, since the points at the edges result in undefined shit, what you're really looking for is the limit of the result as your start and end points approach 0,0 0,1 1,0 and 1,1. Alternatively, think of it this way: What percentage of the time will that first point lie on the edge? It would be the area of the edge / the area of the square. But an edge is one dimensional!

Even if the integration when including the edges gives you horse shit results in the infinite (limit of ln(x) as approaches 0 is negative infinity), and even if you have an infinite number of such points on the edge causing your shit to make no sense, don't worry.

The fact that the edge is one dimensional and the area of the inside is 2 dimensional means the number of times (infinity) of the point being in the square outclass the number of times (infinity) of it being on the edge. You've got a bigger infinity (by an entire dimension) when dealing with the inside of the square, so you can safely discount all results from the edge of the square, even if they are infinite.

Regardless, this is incorrect:

On the other hand, if you do the latter, the first integral evaluates to xy^2/[2(1-x)] evaluated from y=0 to y=1-x. You can see that the 1-x on the numerator will cancel out and you will be left to integrate x(1-x)/2 from 0 to 1.

When x=1 you're still fucked. They don't cancel out unless you restrict x to never be 1 when you do that cancellation.

f(x) = x/x
f(x) != 1
f(x ; x != 0) = 1
f(0) = UNDEFINED

 

[cpp_is_king]

I figured this was your issue when you were hinting at it a while ago.
Doesn't the problem state that the point has to be within the square (so it can't be an the edge)?

Regardless, even if the problem didn't state that, since the points at the edges result in undefined shit, what you're really looking for is the limit of the result as your start and end points approach 0,0 0,1 1,0 and 1,1. Alternatively, think of it this way: What percentage of the time will that first point lie on the edge? It would be the area of the edge / the area of the square. But an edge is one dimensional!

Even if the integration when including the edges gives you horse shit results in the infinite (limit of ln(x) as approaches 0 is negative infinity), and even if you have an infinite number of such points on the edge causing your shit to make no sense, don't worry.

The fact that the edge is one dimensional and the area of the inside is 2 dimensional means the number of times (infinity) of the point being in the square outclass the number of times (infinity) of it being on the edge. You've got a bigger infinity (by an entire dimension) when dealing with the inside of the square, so you can safely discount all results from the edge of the square, even if they are infinite.

Regardless, this is incorrect:


When x=1 you're still fucked. They don't cancel out unless you restrict x to never be 1 when you do that cancellation.

f(x) = x/x
f(x) != 1
f(x ; x != 0) = 1
f(0) = UNDEFINED

Hmm.. I wonder then, if it's possible to actually integrate them in the opposite order but just use the letter a for a limit instead of the number 0 or 1, and after both limits have been completed, use l'hopitals rule to evaluate the limit.

 

[cpp_is_king]

Just solved probably the hardest problem I think I've ever done, and in the process I discovered a quite remarkable mathematical identity. I'll post it after a few days, but in case anyone wants to try this problem on their own, I don't want to spoil it yet. I'm happy this is over with now though, I think I spent probably 15-20 hours on this problem. Ugh.

Anyway, evaluate the sum:

Sum[n=1, Infinity] ln(1 + 1/n) * ln(1 + 1/(2n)) * ln(1 + 1/(2n+1))

 

[coldvein]

back with some basic math stuff for you guys!!! can anyone explain this:

A company has a manufacturing plant that is producing quality jackets. They find that in order to produce 150 jackets in a month, it will cost $7900. Also, to produce 350 jackets in a month, it will cost $13500. Find an equation in the form y=mx+b where x is the number of jackets produced in a month and y is the monthly cost to do so.

to me?

edit: another question ...
Find an equation of the line that goes through the points (-8,34) and (4,-26). Write your answer in the form y=mx+b.
i know how to find the slope but how do i find the y intercept?

 

[Vaporak]

back with some basic math stuff for you guys!!! can anyone explain this:

A company has a manufacturing plant that is producing quality jackets. They find that in order to produce 150 jackets in a month, it will cost $7900. Also, to produce 350 jackets in a month, it will cost $13500. Find an equation in the form where is the number of jackets produced in a month and is the monthly cost to do so.

to me?

You were in some lower level algebra class right? If you were given two points, would you know how to construct a line between them? That's what they're asking you to do in this question. You have two variables, # of jackets and cost, and two ordered pairs, (150, 7900), (350, 13500). From here it's all mechanical, recognizing what the question is asking is the important part.

 

[cpp_is_king]

back with some basic math stuff for you guys!!! can anyone explain this:

A company has a manufacturing plant that is producing quality jackets. They find that in order to produce 150 jackets in a month, it will cost $7900. Also, to produce 350 jackets in a month, it will cost $13500. Find an equation in the form y=mx+b where x is the number of jackets produced in a month and y is the monthly cost to do so.

to me?

edit: another question ...
Find an equation of the line that goes through the points (-8,34) and (4,-26). Write your answer in the form y=mx+b.
i know how to find the slope but how do i find the y intercept?

replace m with the slope you calculated.

since you know the point (-8, 34) is on the line, what does that tell you?

 

[coldvein]

You were in some lower level algebra class right? If you were given two points, would you know how to construct a line between them? That's what they're asking you to do in this question. You have two variables, # of jackets and cost, and two ordered pairs, (150, 7900), (350, 13500). From here it's all mechanical, recognizing what the question is asking is the important part.

okay, so i was having trouble picturing those numbers as two ordered pairs. as far as having two ordered pairs .. yes, i know how to construct a line between the two. i simply take out a ruler and draw the line!! and where that line intersects the y axis is the y intercept. but that isn't an "exact" way of finding the y intercept...right? i'm doing this stuff online, the system wants one answer, i can't just say "the line intersects y vaguely at 3050" or something.. is there something "mechanical" as you say that i can do to find the exact y intercept?

(im not allowed to use a graphing calculator in this class btw)

 

[Vaporak]

okay, so i was having trouble picturing those numbers as two ordered pairs. as far as having two ordered pairs .. yes, i know how to construct a line between the two. i simply take out a ruler and draw the line!! and where that line intersects the y axis is the y intercept. but that isn't an "exact" way of finding the y intercept...right? i'm doing this stuff online, the system wants one answer, i can't just say "the line intersects y vaguely at 3050" or something.. is there something "mechanical" as you say that i can do to find the exact y intercept?

I see you said you know how to find the slope of the line. cpp_is_king gave the same kind of hint that I'm going to give. If you know m, the slope, and you know a point, certain values of x and y, then what does your knowledge of algebra let you know about the equation y = mx +b?

hint/description of the general solution

Spoiler

if an equation of the form y = mx + b has only one unknown variable you can use the algebra you already know to solve for that variable and find it's value. b is the y intercept because y intercept means what the value of y is when x=0, so since m*0 = 0, y=b at the y intercept, so the question of find the y intercept is about finding b. From your equation, you can see that y - mx = b by simple subtracting mx from both sides. And finally, you said you already know the values of y, m, and x, so you can directly compute the value of b.

 

[coldvein]

replace m with the slope you calculated.

since you know the point (-8, 34) is on the line, what does that tell you?

(first of all, thank you gentlemen for your time. im tired and my brain is feeling kinda frazzled right now and maybe i shouldn't be trying to do my math homework, i feel like i'm overlooking something very simple here.)

okay, SO i've got .. (-8,34) and (4,-26) .. i'm getting the slope to be -5.

so:

y = -5x + b.

do i just insert -8 for x and 34 for y and solve for b? b = -6?

 

[Vaporak]

(first of all, thank you gentlemen for your time. im tired and my brain is feeling kinda frazzled right now and maybe i shouldn't be trying to do my math homework, i feel like i'm overlooking something very simple here.)

okay, SO i've got .. (-8,34) and (4,-26) .. i'm getting the slope to be -5.

so:

y = -5x + b.

do i just insert -8 for x and 34 for y and solve for b? b = -6?

Yup it's that easy. So the equation that connects the two lines (-8, 34) and (4, -26) is y = (-5)x - 6

 

[cpp_is_king]

do i just insert -8 for x and 34 for y and solve for b? b = -6?

and if you want to check your work, you can use the other number and make sure the answers are the same

 

[Memento_Mori15]

I'm back to ask another question
So I'm prepping for my next exam on Monday and I'm reviewing Lagrange multipliers.
Main question:
For this problem, f(x,y,z)=xy+3xz+2yz with the constraint of: 5x+9y+z=10. I got x=-5/3, y=20/9, z= -5/3. Plugging these values back into the equation, I got -25/9. Now I assumed it was going to be a minimum but when I checked the back of the book, it says no max and minimum. Why?

This question doesn't have to be answered since I'm assuming that with more practice, you get better at solving these quickly but is there a faster method to tackle this problem or Lagrange multipliers?
This was my method:

Lambda is denoted by L:
I noticed that Fz gives you 3x-2y-L. From that, L= 2x-2y.
From Fx: y+3z-5L. L=(y+3z)/5
Set the lambdas equal to each other.
3x+2y= (y+3z)/5
5(3x+2y)= y+3z
15x+10y=y+3z
9y=-12z
y=-4/3z
Fy: x+2z-9L
9L=x+2z
L=x+2z/9
Fy: x+2z-9L 9L=x+2z L=x+2z/9
Let Lambda equal to 3x+2y.
2x+2y=(x+2z) /9
9(3x+2y)=x+2z
27x+18y=x+2z
26x+18y=2z
Isolate z to get z =13x+9y
Plug that into y=-4/3z. I get x=z.
Knowing all this, I plugged it into the constraint to get 5z-12z+z=10 Thus z=-5/3. Since x=z, x=-5/3. y= 20/9

It took me like 30 minutes to play around with it and get the values for each variable. I definitely can't afford to spend that much time on one problem on the exam. These types of problems usually take me a long time to figure out. Definitely when you have two variables multiplying each other like x*y+...

 

[coldvein]

my dudes. i made some mathematical breakthrus tonight.

 

[Partial Gamification]

It took me like 30 minutes to play around with it and get the values for each variable. I definitely can't afford to spend that much time on one problem on the exam. These types of problems usually take me a long time to figure out. Definitely when you have two variables multiplying each other like x*y+...

My advice is to, first and foremost, get good at both extracting the needed information and neatly listing out your knowns. You seem to have a handle on the process and so I would suggest technology (if available). If you don't have the legal use of a graphing calc, even older models (typically the programmable ones) have linear equation solvers available, I feel the linear algebra makes it easier. Four variables, four equations. I find the row-reduction tedious and adding steps increases the chance for human error, so I'll leave it to you and anyone else that might have some helpful tactics and suggestions. Note that if I did this by hand, I would choose my rows differently so that they would (hopefully) require fewer steps for producing the reduced matrix. For instance, swapping row-one and row-two would give me the "1" in the right place... but then again, I suppose RREF is unneccessary blah blah blah, good luck!

 

[Memento_Mori15]

My advice is to, first and foremost, get good at both extracting the needed information and neatly listing out your knowns. You seem to have a handle on the process and so I would suggest technology (if available). If you don't have the legal use of a graphing calc, even older models (typically the programmable ones) have linear equation solvers available, I feel the linear algebra makes it easier. Four variables, four equations. I find the row-reduction tedious and adding steps increases the chance for human error, so I'll leave it to you and anyone else that might have some helpful tactics and suggestions. Note that if I did this by hand, I would choose my rows differently so that they would (hopefully) require fewer steps for producing the reduced matrix. For instance, swapping row-one and row-two would give me the "1" in the right place... but then again, I suppose RREF is unneccessary blah blah blah, good luck!
http://www.abload.de/img/lagrangemultzxs2t.png[IMG][/QUOTE]
Hmm, that looks very familiar but I haven't taken a linear algebra course yet. I think that's going to be introduce in the differential equation course next semester. However, my professor decided not to put Lagrange multipliers on this upcoming exam and maybe save it for the final(which I bet it will be on there). So for now, I'll save that image and examine it more in depth after my exam. Thank you for the help again!

 

[cpp_is_king]

So here's that identity I discovered the other day. It's quite remarkable IMO. Since then I've been trying to generalize it to higher orders, but unsuccessfully.

Let a,b,c > 0 such that abc = 1. Then:

ln(a)ln(b)ln(c) = (1/3)(ln(a)^3 + ln(b)^3 + ln(c)^3)

I'll leave the proof as an exercise (it's not that difficult), and it leads to a really elegant solution of the above sum.

 

[pompidu]

hey guys. I first want to say thank you all for the help you gave me a while ago. My Discrete Math class has taken a turn for the absolute worse. Proofs and induction/strong induction are lost on not just me but the whole class. Test grade average for the class has fallen from 65% to 50% since from the begging till now. Our textbook is the biggest pile of shit, most unhelpful. The hw that is assigned out of it has no examples anywhere in the book, and the teacher just lifts examples straight out of there.

The internet seems to not help either. Can't find and help on induction with summations, subsets, unions, intersects etc... If someone can point me to a resource that would be most helpful. Theres is only few hours a week that our teacher has available and most are when I am working. The math tutor center has been of some help but they get lost just as easy. And half the time what they are showing you is incorrect. I just don't know what to do anymore. This is my sophomore year and this class is a prerequisite for the rest of my classes for this major. Doesn't help the fact that I can't withdraw and I can't fail without my financial aid disappearing forever. End rant.

 

[Partial Gamification]

MIT has this on induction with summation. Think of induction as like the slices of bread that "sandwich" the proof, or object to be proved. Usually, it is evaluating what you got at the first trivial value, usually variable equals one; and then, evaluating at the n+1 value where n is the finite ceiling of the object (set or equation). I'll look for some more and tack what I find to this post.


edit: lecture study guide in related resources might be more than what you are looking for. ..and, if you haven't had calc... let me keep looking.



and cpp, I expanded out the original expression with the natural logs... holy crap no wonder it took 20hrs!

 

[cpp_is_king]

hey guys. I first want to say thank you all for the help you gave me a while ago. My Discrete Math class has taken a turn for the absolute worse. Proofs and induction/strong induction are lost on not just me but the whole class. Test grade average for the class has fallen from 65% to 50% since from the begging till now. Our textbook is the biggest pile of shit, most unhelpful. The hw that is assigned out of it has no examples anywhere in the book, and the teacher just lifts examples straight out of there.

The internet seems to not help either. Can't find and help on induction with summations, subsets, unions, intersects etc... If someone can point me to a resource that would be most helpful. Theres is only few hours a week that our teacher has available and most are when I am working. The math tutor center has been of some help but they get lost just as easy. And half the time what they are showing you is incorrect. I just don't know what to do anymore. This is my sophomore year and this class is a prerequisite for the rest of my classes for this major. Doesn't help the fact that I can't withdraw and I can't fail without my financial aid disappearing forever. End rant.

throw up somme questions

 

[pompidu]

Thanks guys! Such a frustrating class. Here's a problem that im not sure what to do.

Using Mathematical Induction:

Thanks Partial Gamification, gonna check it out.

 

[Partial Gamification]

Thanks Partial Gamification, gonna check it out.

Np, sematically, I'll go through that example: "The intersection of all the A-type sets is a subset of the intersection of all B-type sets." You are given A-type sets are a subset of B-type sets. What if j=1? What of j= n+1? Do you still have A-type sets as a subset of B-type sets?

post a few more or pick some from here: http://www.math.uiuc.edu/~hildebr/347/induction1.pdf

 

[pompidu]

gonna try to figure this one out and post it and maybe you can show me what is right and wrong.

Edit: Finished the problem but imgur has trouble uploading the picture so hopefully I can post it soon. It is most likely wrong

 

[cpp_is_king]

Thanks guys! Such a frustrating class. Here's a problem that im not sure what to do.

Using Mathematical Induction:

Thanks Partial Gamification, gonna check it out.

Is using induction a requirement for this problem? it seems easier to just apply the definitions of intersection and subset.
1. Let x be in the intersection of A's
2. Then x is in every A_i.
3. Then x is in every B_i.
4. Then x is in intersection of all B's.
5. Then A_int is a subset of B_int

 

[Partial Gamification]

gonna try to figure this one out and post it and maybe you can show me what is right and wrong.

Edit: Finished the problem but imgur has trouble uploading the picture so hopefully I can post it soon. It is most likely wrong

First, your intersections look like "n"s, but if I posted my handwriting...

I like the use of the variable "x" as an arbitrary element.

Here is what I threw together, your base step could have one more line that makes the trival case explicit. For the inductive step, it is the intersection of all the A's and B's, not just the (k+1) step. edit: image removed (redundant- quoted below)

edit: Looking at the last line, the indexes should be on the Union and not the sequence.

 

[pompidu]

Is using induction a requirement for this problem? it seems easier to just apply the definitions of intersection and subset.
1. Let x be in the intersection of A's
2. Then x is in every A_i.
3. Then x is in every B_i.
4. Then x is in intersection of all B's.
5. Then A_int is a subset of B_int

Yeah we do have to use induction.

First, your intersections look like "n"s, but if I posted my handwriting...

I like the use of the variable "x" as an arbitrary element.

Here is what I threw together, your base step could have one more line that makes the trival case explicit. For the inductive step, it is the intersection of all the A's and B's, not just the (k+1) step.

edit: Looking at the last line, the indexes should be on the Union and not the sequence.

Ok this looks alot better, thank you! This stuff is so confusing and our teacher just does a terrible job of explaining what to do. THis is actually a big help.

 

[oxrock]

I know my math is lowly compared to that of others posted but please bare with me almighty math gaf, I need help yet again! For 2 problems this time

first problem
Instructions: Write equation in it's equivalent exponential form, then solve for x.
log base 64 x= 2/3


My solution: 64^2/3 = x x= 16 I think this is right, I'm kind of confused about logs honestly so not sure how I'm doing honestly.

Second problem
instructions: evaluate expression without using a calculator
log(ln e)


That's all they gave me, I have no clue what the hell I'm supposed to do with this.
I know it's log base 10 ( log base e, e). That's about as far as I know how to go with this. Is it possible for someone to talk me through what it is I'm supposed to be doing here? I'm totally clueless.

edit: While waiting I'm continuing to work on this and perhaps I'm on to something? log base 10 = 1 right? so it's 1( e^x = e) Well x = 1 but who cares because I just need to know ln e = e correct? So the equation simplifies to 1(e) = e? (2.71828...) Does this look right at all?

 

[coldvein]

I know my math is lowly compared to that of others posted but please bare with me almighty math gaf, I need help yet again! For 2 problems this time

first problem
Instructions: Write equation in it's equivalent exponential form, then solve for x.
log base 64 x= 2/3


My solution: 64^2/3 = x x= 16 I think this is right, I'm kind of confused about logs honestly so not sure how I'm doing honestly.

Second problem
instructions: evaluate expression without using a calculator
log(ln e)


That's all they gave me, I have no clue what the hell I'm supposed to do with this.
I know it's log base 10 ( log base e, e). That's about as far as I know how to go with this. Is it possible for someone to talk me through what it is I'm supposed to be doing here? I'm totally clueless.

your math isn't as lowly as mine, no worries. i have no idea what that problem you posted is even talking about, don't feel so bad.

 

[big ander]

I know my math is lowly compared to that of others posted but please bare with me almighty math gaf, I need help yet again! For 2 problems this time

first problem
Instructions: Write equation in it's equivalent exponential form, then solve for x.
log base 64 x= 2/3


My solution: 64^2/3 = x x= 16 I think this is right, I'm kind of confused about logs honestly so not sure how I'm doing honestly.

Second problem
instructions: evaluate expression without using a calculator
log(ln e)


That's all they gave me, I have no clue what the hell I'm supposed to do with this.
I know it's log base 10 ( log base e, e). That's about as far as I know how to go with this. Is it possible for someone to talk me through what it is I'm supposed to be doing here? I'm totally clueless.

First problem you're done.
Second problem: you have written that the full expression out and for the inner part you have
log base e of e
So the question you want to ask yourself is: to what power would I need to raise the base to get e? Well, here the base is e. Raising e to the first power, or doing e^1, gives you e. Thus,
log base e of e = ln(e) = 1
By the way, this should also be something you hold as an identity just for quick solving. when you see "log base b of b" the answer is 1.
so now you're at:
log base 10 of 1
Again you could just come to this case and say "to what power do I need to raise 10 to get 1?" And, of course, realize that 10^0 = 1. But again, this is better to just keep in your head as an identity. for any base b, log base b of 1 is 0. so the answer:
0

 

[Partial Gamification]

First problem you're done.

Hold on there, plug in that value. 64*16 = 2/3 ???

There is nothing “almighty” here. Mathematics is a cumulative practice that branches into more areas than I could only pretend to understand. It might be cliché, and I have said it before, but “we stand on the shoulders of giants,” you included. Don't be afraid to ask and there is no need for groveling. People help because they like to share the knowledge. The algebra you are doing took Centuries to develop, and the notation pretty much took Euler's body of works to standardize it.

 

[oxrock]

ooooh, so the question becomes log base 10, 1. so 10^x = 1 so x obviously equals 0. I gotcha. Thanks guys! =)

 

[big ander]

Hold on there, plug in that value. 64*16 = 2/3 ???
http://www.abload.de/img/exponetoupma.png

There is nothing “almighty” here. Mathematics is a cumulative practice that branches into more areas than I could only pretend to understand. It might be cliché, and I have said it before, but “we stand on the shoulders of giants,” you included. Don't be afraid to ask and there is no need for groveling. People help because they like to share the knowledge. The algebra you are doing took Centuries to develop, and the notation pretty much took Euler's body of works to standardize it.

I'm fairly certain you misread his first problem. I do have it correct, though I did skip the step of converting to an equivalent exponential form.
I believe you're reading it as
log base e of (64x) = ln(64x)
Though, that being equal to 2/3 would still make the rest of your solution shaky to me, because ln(2/3) =/= 2/3. And I'm not sure why you're checking it with "64*16 =? 2/3". [EDIT: are you reading it as "log(64x = 2/3)" or "log(64x) = log(2/3)"?] Please tell me if I'm making some grave, basic error here, but I'm fairly sure you read over this problem too quickly.

Anyway, looking at the notation used through out his entire post, I'm almost positive op meant log base 64 of x, or

which undeniably results in the solution of x = 16.
Still, in my original post I didn't type out the exponential form because the OP already had the more important second half of it and it seems they already get it, but here we go:
64^(log base 64 of x) = 64^(2/3)
x = 64^(2/3) = 16

 

[Partial Gamification]

I'm fairly certain you misread his first problem.

I am too now, good eyes!

[edit:] and here's hoping for a html enabled math-help thread that takes mathjax (latex) and won't compromise the site!

 

[wbsmcs]

Just took a math contest at my school and one of the question I couldn't get was:

Determine a pair of positive integers (x,y) with y>100 that satisfy the equation: x^2 - 2y^2 = 1 ... How would you do this without randomly punching numbers into your calculator?

 

[Leezard]

Just took a math contest at my school and one of the question I couldn't get was:

Determine a pair of positive integers (x,y) with y>100 that satisfy the equation: x^2 - 2y^2 = 1 ... How would you do this without randomly punching numbers into your calculator?

There are algorithms for solving that. Look this up: http://en.wikipedia.org/wiki/Diophantine_equation
edit: oh right, i was thinking of a linear Diophantine equation.

 

[cpp_is_king]

Just took a math contest at my school and one of the question I couldn't get was:

Determine a pair of positive integers (x,y) with y>100 that satisfy the equation: x^2 - 2y^2 = 1 ... How would you do this without randomly punching numbers into your calculator?

Often times trial and error is the best way. For example, when you're looking for roots of a high degree polynomial, the first thing you do is plug in -1, 0, and 1 since they're easy to compute and you can use the sign of the result to construct intervals for where the roots must lie before trying more.


Generally, if trial and error is the solution, the numbers will be very small. The neat thing about diophantine equations is that once you have one solution, you can often construct additional solutions. But this problem only asks for 1 solution, so trying a few simple values is the way to go as your first step in exploring the problem.

x=1:
1 - 2y^2 = 1
y = 0
This one violates the restriction that x,y > 0, so we discard it.

x=2:
4 - 2y^2 = 1
-2y^2 = -3
y^2 = 3/2
This one violates the restriction that x,y be integers so we discard it.

x=3:
9 - 2y^2 = 1
-2y^2 = -8
y^2 = 4
y = 2

Oh hey, we found one!

So anyway, yea. The answer for this one is (3,2).

BTW, this equation is called Pell's Equation, and it's quite famous. There's no way you would have been expected to find all solutions for this equation, because it's quite difficult.


There are many other strategies for solving Diophantine Equations, but they don't apply to this problem so I won't bother mentioning them.

Edit: just saw it said y > 100. Makes it much harder, but google Pell's Equation to see how to construct additional solutions from an existing solution

 

[Troll]

Find the principal curvatures, the principal directions, and asymptotic directions(when they exist) for each surface. Identify the lines of curvature and asymptotic curves when possible.
(a)the cylinder: x(u,v)=(acosu, asinu, v)

I have the principal curvatures as -1/a and 0 and the principal directions as x=x_1{1,0} where x_1 is not zero and x=x_2{0,1} where x_2 is not zero. Now I have that the asymptotic direction exists as (-1/a)(0)=0 so that shows that it exists. Now the next part is what I just have no clue how to do. The asymptotic direction is when II_p(V,V)=0, and II_p(V,V)=K(kappa), where K=0 in this case. Now I'm not sure what to do with that....do I just state what K is in each case? I also have no clue what the lines of curvature are. My book says that asymptotic curves are when the tangent vector at each point is an asymptotic direction...

Hopefully someone can help me make some sense out of this problem.

 

[rjc571]

Just took a math contest at my school and one of the question I couldn't get was:

Determine a pair of positive integers (x,y) with y>100 that satisfy the equation: x^2 - 2y^2 = 1 ... How would you do this without randomly punching numbers into your calculator?

Write y = x - k (k is some positive integer) (since equation can only be satisfied if x > y)
Substitute to get x^2 - 4kx + 2k^2 + 1 = 0
Quadratic formula -> x = 2k + sqrt(2k^2 - 1)
So 2k^2 - 1 is a perfect square, so write 2k^2 - 1 = m^2, m = positive integer
-> 2k^2 - m^2 = 1 (this can only be satisfied if m > k)
Now repeat the process a few more times:
Write m = k + j, j positive integer
Substitute: k^2 - 2jk - j^2 - 1 = 0
Quadratic formula: k = j + sqrt(2j^2 + 1)
So 2j^2 + 1 is a perfect square, so 2j^2 + 1 = z^2, z = positive integer
z^2 - 2j^2 = 1 (can only be satisfied if z > j)
z = j + f, f positive integer
Substitute: j^2 - 2fj - f^2 + 1 = 0
Quadratic form: j = f + sqrt(2f^2 - 1)
Note: this square root term for f is the same as the square root term you had earlier for k. This is relevent because you're trying to find values for f, j and k which ensure that the terms inside the square roots are perfect squares. Since the square root terms involving k and f are the same, any value which works for k will also work for f.

Start with f = 1. Working your way backwards, you can find values for j, then k, then x and y. But if this doesn't give you a large enough value for y, you can take what you have for k and make that your new f. Then use this to find new values for j, k, x and y.
Repeat this until you get y > 100.

You should get:
f = 1
j = 2
k = 5
x = 17
y = 12 -> not big enough
So set f = 5 (the value you have for k):

f = 5
j = 12
k = 29
x = 99
y = 70 -> not big enough
so set f = 29

f = 29
j = 70
k = 169
x = 577
y = 408 -> answer

 

[FrigidEh]

Can I get help with this optimization question?

Let T be the triangle with vertices at a= (-1,0), b= (1,0), and c= (0,2).

a) a point p lies inside T along the y-axis. Find the smallest possible value for the sum
lp-al + lp-bl + lp-cl (absolute value bars)(where lp-al is the distance between p and a)
b) A smaller triangle S, with its lower vertex (0,0) and its upper edge parallel to the x-axis, is inscribed in T. Determine the minimum possible perimeter of S.

 

[coldvein]

coldvein baby math checking in. this is getting frustrating. we're using systems of equations to solve word problems or something .. i have to read a word problem and come up with a system of equations, then find a solution. this is bad.

In 1985, college graduates average $508 in weekly earnings. This amount has increased by approximately $25 in weekly earnings per year. By contrast, in 1985, people with less than four years of high school averaged $270 in weekly earnings. This amount has only increased by approximately $4 in weekly earnings per year. How many years after 1985 will college graduates be earning three times the amount per week that people with less than four years of high school earn? In which year will this occur? What will be the weekly earnings for each group at that time?


this shit is CRAZZZAAAYYY!!! okay, so i need to create a system of equations to figure this out. this is the hard part. once i have the system i think i can find the solution .. so .. two variables?

y = years after 1985
x = weekly earnings students
??????

so

y = 508 + 25x
y = 270 + 4x
??????

where does the "three times the amount per week" part come in? i am drawing a blank here.

???????

 

[big ander]

coldvein baby math checking in. this is getting frustrating. we're using systems of equations to solve word problems or something .. i have to read a word problem and come up with a system of equations, then find a solution. this is bad.

In 1985, college graduates average $508 in weekly earnings. This amount has increased by approximately $25 in weekly earnings per year. By contrast, in 1985, people with less than four years of high school averaged $270 in weekly earnings. This amount has only increased by approximately $4 in weekly earnings per year. How many years after 1985 will college graduates be earning three times the amount per week that people with less than four years of high school earn? In which year will this occur? What will be the weekly earnings for each group at that time?


this shit is CRAZZZAAAYYY!!! okay, so i need to create a system of equations to figure this out. this is the hard part. once i have the system i think i can find the solution .. so .. two variables?

y = years after 1985
x = weekly earnings students
??????

so

y = 508 + 25x
y = 270 + 4x
??????

where does the "three times the amount per week" part come in? i am drawing a blank here.

???????

what you actually have there is y=weekly earnings and x=years after 1985.
Let's actually break it up into y_1=weekly earnings of college graduates and y_2=weekly earnings of HS dropouts. so:
y_1 = 508 +25x
y_2 = 270 + 4x
The number of years since 1985 is the same for both types of people, so we just need x.
Here's where the three times the amount per week thing comes in: you're told to look at where weekly earnings of graduates equals three times weekly earnings of dropouts. In notation:
y_1 = 3 * y_2
Substitute and simplify.
508 +25x = 3 * (270 + 4x)
508 + 25x = 810 + 12x
508 + 25x - 508 - 12x = 810 + 12x - 508 - 12x
13x = 302
13x/13 = 302/13
x ~= 23.231
Thus, 23.231 years after 1985 college graduates will be earning 3 times as much weekly as HS dropouts.
Here's the part I myself don't know for sure: if we're starting from, say, January 1 1985, then the weekly pay becomes equal in 2008. If it we actually began later that might not be true, but I think assuming that is acceptable.
Solving for the weekly earnings at that time isn't bad from here. Let's calculate y_2 = weekly earnings of HS dropouts first.
y_2 ~= 270 + 4(23.231) = 362.924 = approximate weekly earnings for high school dropouts.
Now y_1.
y_1 = 3 * y_2 ~= 1088.772 = approximate weekly earnings for college graduates.

 

[Memento_Mori15]

Okay, I have yet another question but this time about Surface Integrals. I'm kinda confused about this concept but I'm getting the hw problems except for this one:
x² +y² +z²=4, 1≤z≤2; f(x,y,z)=z^2 (x² +y² +z²)^-1
My approach:
I immediately thought spherical and thus:
2cos(&#920sin(&#966, 2sin(&#920sin(&#966, 2cos(&#966, 0<&#920;<2pi, ?<&#966;<?. So I need to find the limits for &#966; before I move on.

Sorry for the horrible paint image.

Now I figured the hypotenuse will be 4 and the length will be 2, so I get pi/6. In phi, it'll be pi/3. Awesome, that's my lower limit. Thus pi/3<&#966;<?. Now the hypotenuse will be 4 still and the opposite length will be 1...I'm not going to get an exact result. What do I do now? Am I approaching this problem correctly?

 

[cpp_is_king]

Okay, I have yet another question but this time about Surface Integrals. I'm kinda confused about this concept but I'm getting the hw problems except for this one:
x² +y² +z²=4, 1&#8804;z&#8804;2; f(x,y,z)=z^2 (x² +y² +z²)^-1
My approach:
I immediately thought spherical and thus:
2cos(&#920sin(&#966, 2sin(&#920sin(&#966, 2cos(&#966, 0<&#920;<2pi, ?<&#966;<?. So I need to find the limits for &#966; before I move on.

Sorry for the horrible paint image.

Now I figured the hypotenuse will be 4 and the length will be 2, so I get pi/6. In phi, it'll be pi/3. Awesome, that's my lower limit. Thus pi/3<&#966;<?. Now the hypotenuse will be 4 still and the opposite length will be 1...I'm not going to get an exact result. What do I do now? Am I approaching this problem correctly?

this is purely algebraic, no need to visualize or use a geometric analysis.

if z = 2 cos(&#966 , then your limits are simply

z=1 -> &#966; = acos(1/2) = pi/3
z=2 -> &#966; = acos(1) = 0

when you replace x, y, and z with the spherical equivalent, the entire integral simplifies to just

cos(&#966^2

with 0<&#920;<2pi and 0 < &#966; < pi/3

 

[Troll]

Calculate the Christoffel symbols for the following parametrized surfaces. Then check in each case that the Codazzi equations and the first Gauss equation hold.
d. a surface revolution: x(u,v)=(f(u)cosv, f(u)sinv, g(u)), with f'(u)^2 + g'(u)^2 =1

Solution: So I calculated the Christoffel symbols as gamma_uv_v=f'(u)/f(u), gamma_vv_u=-f'(u)f(u) and the rest are zero. These are correct as they are in the back of the book. Now I checked the first Codazzi equation and it worked out, but the second Codazzi equation and the first Gauss equation are not working out, so I know I did something wrong.

For l, m, n, E, F, G and their u and v derivatives I got:
l=f'(u)g'(u)-g'(u)f''(u), so l_v=0.
m=0, so m_u=m_v=0.
n=f(u)g'(u), so n_u=f'(u)g'(u)+g''(u)f(u).
E=1, so E_u=E_v=0.
F=0, so F_u=F_v=0.
G=f(u)^2, so G_v=0, G_u=2f'(u)f(u).

Codazzi Equation 2:
m_v - n_u = l(gamma_vv_u) + m(gamma_vv_v + gamma_uv_u) - n(gamma_uv_v)

and I get that:

-f'(u)g'(u)-g''(u)f(u) = f'(u)g'(u)[-f''(u)f(u) + f(u)f'(u) + 1]

For the first Gauss equation it is:
EK = (gamma_uu_v)_v + (gamma_uv_v)_u + (gamma_uu_u)(gamma_uv_v) + (gamma_uu_v)(gamma_vv_v) - (gamma_uv_v)(gamma_uu_v) - (gamma_uv_v)^2

where K=(ln-m^2)/(EG-F^2)

I get that K=[(-g'(u)f''(u) + f'(u)g'(u))(f(u)g'(u))] / [f(u)^2]

And so I get that:
[(-g'(u)f''(u) + f'(u)g'(u))(f(u)g'(u))] / [f(u)^2] = [f(u)f''(u)-[f'(u)^2]] / [f(u)^2]

I know that I'm doing something wrong, but I just can't seem to figure out where the mistake is being made.

 

[big ander]

Maybe one of y'all will find this easy, I'm sillily stumped.

I know it does converge (because after being fairly sure I checked on wolfram, converges to 0.514987) and I'm quite certain I just need to use comparison test. i.e. find a sequence b_n such that 0<the given sequence =< b_n and the sum of b_n from 1 to inf converges. I always have trouble picking sequences though. Am I supposed to apply the concept of telescoping series somehow maybe?