• Hey Guest. Check out your NeoGAF Wrapped 2025 results here!

The Math Help Thread

Status
Not open for further replies.
GiantEnemyGoomba said:
I'm gonna be taking integral calculus this upcoming semester. I did really well in my differential calc class (we ended with the fundamental theorem of calculus and u-substitution), but that was last fall semester, so I was wondering what I should brush up on before I start school again since it's been more than half a year. I'm guessing pretty much everything, but idk if stuff like related rates or optimization comes up much in integral calc.
Click to expand...

I would brush up on the topics covered in differential calc. but I wouldn't worry about brushing up on the optimization problems. Also, familiarize yourself with Sigma notation because the hardest part of integral calculus is power series.

Integral calc. flows like this: Learn integration, mostly u-sub and integration by parts; apply integration, which involves finding volumes of functions rotated around axises; learn power series, this will suck; and then a short introduction to DEQ.
 
Would anyone be willing to look at my Nelder-Mead code and help troubleshoot? It seems to just run infinitely and I am not sure why.
 
Fou-Lu said:
Would anyone be willing to look at my Nelder-Mead code and help troubleshoot? It seems to just run infinitely and I am not sure why.
Click to expand...

I'm unfamiliar with the NM algorithm. Can you tell us if the algorithm and your code are easy to follow when laid side-by-side, and what language your code is written in?
 
It appears to be an optimization algorithm.

Two thoughts without looking at anything or knowing NM?
1) Is it possible that you have a loop in your code that is not iterating properly. Use debug commands to output to verify that everything you think is changing between iterations actually is.

2) Are you sure that the algorithm is guaranteed to converge after a finite number of iterations for the input you have? If not, consider modifying your code to die after, say, 500 or 1000 iterations. In case the solutions traverse local maxima--i.e. the answer at iteration 1000 might be worse than the answer at iteration 999--modify your code to store the best result so far (for reference on this, look at the pocket algorithm in machine learning / neural networks)
 
Suppose I had the integral from 0 to infinity of f(b)e^(-sb)db. Moreover, one is told that b = a - k for some constant k.

How is this integral equivalent to the Laplace transform of f(a)?.

Because the above integral should be Laplace transform of f(b). And then you substitute b for a - k, so shouldn't the integral be equivalent to Laplace transform of f(a-k)?

This is regarding the Second Shifting Theorem of Laplace transforms.
 
How can I resolve 3^(1/log2(3)) without a calculator?

EDIT
problem solved
I didn't remember that a^loga(b)=a per definition
 
Need some help with set theory. We went over Cantor's theorem in class and we're asked how to define the size of some set A greater than equal to the size of some set B. I know I can use pairing to determine if the size is the same or not, such that if there is an element x in B that doesn't map to an element y in A then B is larger(provided everything else was mapped 1-1). I'm not sure how I would define one set being greater than equal to though. Any hints? Tried googling to get help but all of the examples are strictly equal to or greater than.
 
Not sure if this would be a good place to ask for some Physics help.

"A hoodlum throws a stone vertically downward with an initial speed v0 from the roof of a building, a height h above the ground.

(a) How long does it take the stone to reach the ground?

(b) What is the speed of the stone at impact? Give your answers in terms of the given variables and g."

I'm completely lost on this question and this course so far in general. The only reason I've gotten to all but 4 problems done on my Wileyplus homework would be thanks to finding stuff on google to help me solve the problems. All I know is acceleration is g in this free fall case and that's 9.8.
 
Nirran said:
Not sure if this would be a good place to ask for some Physics help.

"A hoodlum throws a stone vertically downward with an initial speed v0 from the roof of a building, a height h above the ground.

(a) How long does it take the stone to reach the ground?

(b) What is the speed of the stone at impact? Give your answers in terms of the given variables and g."

I'm completely lost on this question and this course so far in general. The only reason I've gotten to all but 4 problems done on my Wileyplus homework would be thanks to finding stuff on google to help me solve the problems. All I know is acceleration is g in this free fall case and that's 9.8.
Click to expand...

Use the kinematic equations.

There are 4 kinematic equations. Identify which of the 4 to use based on the variables you know and the variables you need to find.

Vf^2 = Vi^2 + 2 * accel * distance
Vf = Vi + accel * t

These are the two equations you can use to solve (a).

To solve (b), use:
distance = vi*t + 1/2 * accel * t^2

Make sure you properly direction and origins in a consistent manner. Is up positive? If so, then acceleration should be negative (in this instance accel = -9.8 m/s^2). Is the height of the roof 0? Then distance would also be a negative number (e.g. if the ground is 20 m below the roof, then distance = -20 m).
 
Nirran said:
Not sure if this would be a good place to ask for some Physics help.

"A hoodlum throws a stone vertically downward with an initial speed v0 from the roof of a building, a height h above the ground.

(a) How long does it take the stone to reach the ground?

(b) What is the speed of the stone at impact? Give your answers in terms of the given variables and g."

I'm completely lost on this question and this course so far in general. The only reason I've gotten to all but 4 problems done on my Wileyplus homework would be thanks to finding stuff on google to help me solve the problems. All I know is acceleration is g in this free fall case and that's 9.8.
Click to expand...

I'm not sure if you're in high school or university, but here we go. If you're in university, do the integration. If you're in high school, you should have these formulas as given:
Given: acceleration = g (9.8 m/s^2)
Integrate with respect to time: Velocity = gt + c (c in this case = v_0)
Integrate against with respect to time: Distance = 1/2 gt^2 + ct + d (d in this case = distance at time 0, which is 0 in this case)

Let's say, for example, the building is 9.8 m high. Let's say the guy just lets the rock drop, so you have v_0 = 0.
At time 0, the stone is at d=0 (the top of the building). The speed is 0. Acceleration is constant, it's g.
At time 1, the stone is at 1/2(g) = 4.8m.
When does it get to 9.8m?
9.8 = 1/2 gt^2 + 0(t)
We can drop the last term
9.8/(1/2) = gt^2
t^2 = 19.6/g = 2
t = sqrt(2) seconds

How fast is it going?
Velocity at time t = gt + 0 = gt = 9.8t = 13.85 m/s

Okay, now let's just let the building be h high:
h = 1/2 gt^2 + 0(t)
2h = gt^2 + 0(t)
t^2 = 2h/g
t = sqrt(2h/g)

How fast is it going?
Velocity at time t = gt + 0 = g*sqrt(2h/g) = 9.8sqrt(2h/9.8) = sqrt(9.8) * sqrt(2h)

Okay, now let's let v_0 be non-zero:
h = 1/2 gt^2 + v_0*t
h = 4.9 t^2 + v_0 t
-4.9t^2 + v_0t + h = 0

Quadratic formula to the equation; it will equal when t is either of the following values:
t = v_0 +- sqrt(v_0^2 + 19.6h)/9.8
Discard the negative solution because negative time isn't real
t = v_0 + sqrt(v_0^2 + 19.6h)/9.8

How fast is it going?
Velocity at time t = gt = 9.8t = 9.8(v_0 + sqrt(v_0^2 + 19.6h)/9.8

As Kieli notes, make sure your directional units are consistent. Here I make acceleration position and distance positive for simplicity's sake but it's more conventional to make acceleration -9.8m/s^2 and solve when distance is -h.
 
Stumpokapow said:
I'm not sure if you're in high school or university, but here we go. If you're in university, do the integration. If you're in high school, you should have these formulas as given:
Given: acceleration = g (9.8 m/s^2)
Integrate with respect to time: Velocity = gt + c (c in this case = v_0)
Integrate against with respect to time: Distance = 1/2 gt^2 + ct + d (d in this case = distance at time 0, which is 0 in this case)

Let's say, for example, the building is 9.8 m high. Let's say the guy just lets the rock drop, so you have v_0 = 0.
At time 0, the stone is at d=0 (the top of the building). The speed is 0. Acceleration is constant, it's g.
At time 1, the stone is at 1/2(g) = 4.8m.
When does it get to 9.8m?
9.8 = 1/2 gt^2 + 0(t)
We can drop the last term
9.8/(1/2) = gt^2
t^2 = 19.6/g = 2
t = sqrt(2) seconds

How fast is it going?
Velocity at time t = gt + 0 = gt = 9.8t = 13.85 m/s

Okay, now let's just let the building be h high:
h = 1/2 gt^2 + 0(t)
2h = gt^2 + 0(t)
t^2 = 2h/g
t = sqrt(2h/g)

How fast is it going?
Velocity at time t = gt + 0 = g*sqrt(2h/g) = 9.8sqrt(2h/9.8) = sqrt(9.8) * sqrt(2h)

Okay, now let's let v_0 be non-zero:
h = 1/2 gt^2 + v_0*t
h = 4.9 t^2 + v_0 t
-4.9t^2 + v_0t + h = 0

Quadratic formula to the equation; it will equal when t is either of the following values:
t = v_0 +- sqrt(v_0^2 + 19.6h)/9.8
Discard the negative solution because negative time isn't real
t = v_0 + sqrt(v_0^2 + 19.6h)/9.8

How fast is it going?
Velocity at time t = gt = 9.8t = 9.8(v_0 + sqrt(v_0^2 + 19.6h)/9.8

As Kieli notes, make sure your directional units are consistent. Here I make acceleration position and distance positive for simplicity's sake but it's more conventional to make acceleration -9.8m/s^2 and solve when distance is -h.
Click to expand...

This was the best help so far still a bit stuck. It's a university class and she is wanting us to just give the formula with no numbers to show we know what we're doing.

For the first problem she wants it in quadratic formula. I have

1.) t=(v_o + sqrt(v_o^2 -4gh))/2g for the first but it's wrong. If I'm just given to insert the variables I feel I missed something.

2.) I got v= sqrt(v_o^2 + 2gh) since it is using v^2 =v_o^2 +2a(delta x so h?).

This is about the last problem I've been stuck on so far.
 
Anyone know how to map the natural number set N to the cartesian product NxN? I need to show that they are the same size, as well as NxNxN ect. I know that I can map the elements of N to NxN by 0 -> 0,0 1 -> 0,1 2->1,0 3-> 1,1 ect but I can't think of how I would come up with a function for it, let alone NxNxN ect.
 
Exuro said:
Anyone know how to map the natural number set N to the cartesian product NxN? I need to show that they are the same size, as well as NxNxN ect. I know that I can map the elements of N to NxN by 0 -> 0,0 1 -> 0,1 2->1,0 3-> 1,1 ect but I can't think of how I would come up with a function for it, let alone NxNxN ect.
Click to expand...

It is the same idea for 3D. One way to easily visualize the problem is trying to draw a continuous line (i.e. don't lift the pen) that passes through all the points and does not intersect/cross itself. If we can draw such a line, then we have just come up with the map (since a line is a 1D object).

Let's go back to the 2D example. See how you can draw a continuous line as you follow the steps below. One way to traverse so that we meet all the natural number pairs (x, y) is,

1. Start from (0, 0). This is like the base case in an induction.
2. Find all pairs (x, y) such that x + y = 1.
3. Find all pairs (x, y) such that x + y = 2.
4. And so on...
 
Exuro said:
Anyone know how to map the natural number set N to the cartesian product NxN? I need to show that they are the same size, as well as NxNxN ect. I know that I can map the elements of N to NxN by 0 -> 0,0 1 -> 0,1 2->1,0 3-> 1,1 ect but I can't think of how I would come up with a function for it, let alone NxNxN ect.
Click to expand...

So you basically have to show that the cartesian product NxN is countable, right? Try looking at the Pairing Function article on Wikipedia. The picture in particular is a nice illustration of what kgtrep just explained.
 
Thanks guys. That gives me some help. One issue I have though is how am I supposed to derive the function pi(k1, k2) := 1/2(k1 + k2)(k1 + k2 + 1) + k2? I understand following the NxN grid and assigning pairs to N, increasing by their pair sum, will show that they have the same size intuitively but how to I derive the function from that to prove it myself?

Also the page says it can be inductively generalized as N*n -> N which I also need to prove, but I don't quite see how. How would the function change as we add another dimension to the product?
 
Exuro said:
Thanks guys. That gives me some help. One issue I have though is how am I supposed to derive the function pi(k1, k2) := 1/2(k1 + k2)(k1 + k2 + 1) + k2? I understand following the NxN grid and assigning pairs to N, increasing by their pair sum, will show that they have the same size intuitively but how to I derive the function from that to prove it myself?

Also the page says it can be inductively generalized as N*n -> N which I also need to prove, but I don't quite see how. How would the function change as we add another dimension to the product?
Click to expand...


My guess is, the formula was derived from using summation identities or recognizing a pattern and using induction. The difficulty lies in showing bijectivity, so I think proof-by-drawing is better and more convincing answer.

Once we draw a line that passes through each point once, we "straighten it out" to get the bijective map (notice, injectivity and surjectivity are satisfied by definition). It's like untangling Christmas lights.

So for 3D, you would look for all pairs (x, y, z) such that

1. x + y + z = 0
2. x + y + z = 1
3. x + y + z = 2
4. And so on.

SKjPuL8.png


This is not the only way to connect all the points.
 
Alright. I see how that works. I'm not sure how my professor wants us to show it. He went over NxN counting to show its countable by using the pairing scheme above so hopefully that should be fine using a picture as proof for it and the NxNxN ect cases.

Thanks.
 
MisterLuffy said:
I'm doing stats homework, what does typical value, highly concentrated, and representative value mean?
Click to expand...

"typical value" (central tendency) refers to the average value of a distribution. Given the level of course I'd expect to ask this question, you can say that you obtain the typical value using a measure of central tendency, like mean, trimmed mean, median, or mode, or if you have a multivariate distribution then a joint mean. The term "representative value" is also used to refer to this; presumably your text draws some distinction between the two, perhaps that representative value involves some sort of subject-matter analysis for choosing which measure of central tendency is most appropriate given the context we're talking about.

I'm not familiar with the expression "highly concentrated" related to statistics, but presumably this would refer to a distribution that has a relatively low "spread". We define distributions by the value of their central tendency and the shape and size of their spread. You measure spread using interquartile range, standard deviation/standard error, range, minimum, maximum, and other such functions.

Just to be clear, I think the idea of this thread is to get help understanding stuff, not to have people do questions for you. Especially since your questions are just definition help, you'd probably be better served by just reading your notes or textbook.
 
Exuro said:
Thanks guys. That gives me some help. One issue I have though is how am I supposed to derive the function pi(k1, k2) := 1/2(k1 + k2)(k1 + k2 + 1) + k2? I understand following the NxN grid and assigning pairs to N, increasing by their pair sum, will show that they have the same size intuitively but how to I derive the function from that to prove it myself?

Also the page says it can be inductively generalized as N*n -> N which I also need to prove, but I don't quite see how. How would the function change as we add another dimension to the product?
Click to expand...

Once you know that N^2 and N are of the same cardinality, you can generalize to N^k for any finite k, using induction. The idea is that N^k can be thought of as N x N^(k-1). The N^(k-1) is in bijection with N by the induction hypothesis, and you can use this fact to trivially build a bijection between N^k = N x N^(k-1) and N x N. We know that there is a bijection between N^2 and N, so by composing the two bijections we get one from N^k to N, establishing the induction step.
 
AgentChris said:
Anyone know how to attack this problem?
Click to expand...

This is the image you posted, right? Its not showing here, so I had to look in the source code

http://i4.minus.com/jKxTXQKKHlwbf.JPG

First of all, do you know how summation work?

∑X = sum of all elements of X

Also written in the following form:

k
∑(Xi)
i=0

meaning the sum of all X with index from 0 to k (int his case, X0 + X1 + X2 + ... + Xk)

Well, to solve the first one, ∑X, you have to add up all elements of X

This means 3 + 3 + 1 + 2 + 4 = 13

The next one is ∑X^2. Please note that this means ∑(X^2), and not (∑X)^2

3^2 + 3^2 + 1^2 + 2^2 + 4^2 = 9 + 9 + 1 + 4 + 16 = 39

The third one is (∑X)^2, but we already calculated ∑X, ∑X = 13, so (∑X)^2 = 13^2 = 169

Now, the fourth one is ∑(Y+3)

∑(Y+3) is the sum of (Y+3) for all Y, this means (1 + 3) + (2 + 3) + (1 + 3) + (3 + 3) + (5 + 3) = 4 + 5 + 4 + 6 + 8 = 27

Can you do the rest by yourself? Try to do the second half and post your answers :D
 
Ahnez said:
Can you do the rest by yourself? Try to do the second half and post your answers :D
Click to expand...

You are awesome.

5.
(∑Y)^2 = 144
1 + 2 + 1 + 3 + 5 = 12
12^2 = 144

6.
∑X∑Y = 25
∑X = 13
∑Y = 12
13 + 12 = 25

7.
∑XY = 25
(3 + 1) + (3 + 2) + (1 + 1) + (2 + 3) + (4 + 5) = 25

8.
N for the X Scores = 3, 3, 1, 2, 5 = 5
There are 5 total numbers.

Thank you!!
 
AgentChris said:
You are awesome.

5.
(∑Y)^2 = 144
1 + 2 + 1 + 3 + 5 = 12
12^2 = 144

6.
∑X∑Y = 25
∑X = 13
∑Y = 12
13 + 12 = 25

7.
∑XY = 25
(3 + 1) + (3 + 2) + (1 + 1) + (2 + 3) + (4 + 5) = 25

8.
N for the X Scores = 3, 3, 1, 2, 5 = 5
There are 5 total numbers.

Thank you!!
Click to expand...

Just two things,

6. ∑X∑Y is actually (∑X)*(∑Y), so its actually 12 * 13 = 156

The same for 7,

∑XY = ∑(X * Y), so (3*1) + (3*2) + (1*1) + (2*3) + (4*5) = 3 + 6 + 1 + 6 + 20 = 36

You actually solved ∑(X+Y)

edit: Also, you're welcome :D
 
Could anyone give me some pointers for this problem I have in my integral calculus class:

j47T98H.png


I've only learned up to u-substitution, so it shouldn't require integration by parts or anything more complex than that, but I can't seem to work this one out by using what I know.
 
GiantEnemyGoomba said:
Could anyone give me some pointers for this problem I have in my integral calculus class:

j47T98H.png


I've only learned up to u-substitution, so it shouldn't require integration by parts or anything more complex than that, but I can't seem to work this one out by using what I know.
Click to expand...

u-substition is enough. Try to evaluate what e^(ln x) means.
 
Could anyone help me find the partial derivatives of ln(x+ln(y))?

I'm supposed to go up to 2nd order but I'm lost. My teacher makes her own problems and all the ones in the book are way more simple.
 
RELAYER said:
Could anyone help me find the partial derivatives of ln(x+ln(y))?

I'm supposed to go up to 2nd order but I'm lost. My teacher makes her own problems and all the ones in the book are way more simple.
Click to expand...

In a partial derivative, treat the variable you are NOT operating on as a constant. First order partial derivatives means taking the derivative with respect to each variable individually. Second order partial derivatives means taking the derivative with each possible sequence of two variables (xx, xy, yx, yy). The cross partials (xy and yx) are equivalent*

First order derivative dx: chain rule:
1/(x+ln(y)) * 1 = 1/(x+ln(y))

Second order derivative dx dx: quotient rule
d/dx f(x)/g(x) = g(x)f'(x) - f(x)g'(x) / g(x)^2
f(x) = 1
f'(x) = 0
g(x) = x+ln(y)
g'(x) = 1
0 - 1 / (x+ln(y))^2 = -1/(x+ln(y))^2

Second order cross dx dy: quotient rule
f(x) = 1
f'(x) = 0
g(x) = x+ln(y)
g'(x) = 0 + 1/y = 1/y
(0 - 1/y) / (x+ln(y))^2
-1/(y(x+ln(y))^2)

Now do first order partial dy with chain rule; you don't need the other cross partial because they're equivalent; then do second order dy dy. You've got this, it's all elementary differentiation. Partial derivatives is just about using them in the right order.

* Provided both functions have continuous second derivatives.
 
Im back again for this semester with another simple question. This time it's finding linear models.

I'm given a graph with information on meat production by metric tons and years 2000-2005. I know that the data I'm inputting to my list on my TI-84 is not wrong when looking at it point for point. The issue is that the book is either rounding it up or down(?) because it claims my equation numbers are off +/- 1.0 or so.

This is the graph I'm given:

rZL70SN.jpg
Click to expand...

This is what my data input for the list.

J22U17V.jpg
Click to expand...

I follow all the right steps to use the LineReg (ax+b) L1, L2 command and I get an output of:
a = 2.494285714
b = 131.7142857

This would make my equation "M = 2.49t + 131.71". That is wrong according my textbook as their equation comes out to be " M = 2.3t + 132.1".

This is really boggling my mind as I don't know why I'm of by so much on both numbers. am I not rounding up/down correctly? is there something in the data that I'm doing wrong? should my LineReg not include both L1 and L2?
 
Daria said:
*snip*
Click to expand...

The way I see it there is not a single correct model, they are asking you to find one equation for a model of this data and answer the rest of the questions using it.

Looking at the picture again, they even say it. "For Ex. 21 and 22, the answers may vary".
 
Ended up doing this a few ways and couldn't get there answer.

Then I looked above the question and saw "For Exercises 21 and 22, the answers may vary"
 
Leezard said:
The way I see it there is not a single correct model, they are asking you to find one equation for a model of this data and answer the rest of the questions using it.

Looking at the picture again, they even say it. "For Ex. 21 and 22, the answers may vary".
Click to expand...

Yes, it depends on what the linear model equation is minimizing (absolute error, sum of squared errors, etc.) and what the estimator used is. Judging from the fact that the person is just using a calculator and that the book is not typeset like a college level math text, I'm guessing this is not an OLS / regression class but rather some high school intro stats type class.

Using OLS with non-denormed data (i.e. inputting years as 2000, 2001, ...):
> x = c(2000,2001,2002,2003,2004,2005)
> y = c(132.1, 133.2, 137.7, 139.0, 141.2, 144.5)
> model = lm(y~x)
> summary(model)

Call:
lm(formula = y ~ x)

Coefficients:
(Intercept) -4856.8571
x 2.4943

Using OLS with denormed data (i.e. inputting years as 0, 1, ...)

> x2 = c(0,1,2,3,4,5)
> model2 = lm(y~x2)
> summary(model2)

Call:
lm(formula = y ~ x2)
Coefficients:
(Intercept) 131.7143
x2 2.4943
 
Leezard said:
The way I see it there is not a single correct model, they are asking you to find one equation for a model of this data and answer the rest of the questions using it.
Click to expand...

TUSR said:
Then I looked above the question and saw "For Exercises 21 and 22, the answers may vary"
Click to expand...

I should start paying more close attention to the whole page. I saw that but wasn't exactly sure what they were pertaining too when I kept turning back to the answer. Thanks guys.

Stumpokapow said:
Yes, it depends on what the linear model equation is minimizing (absolute error, sum of squared errors, etc.) and what the estimator used is. Judging from the fact that the person is just using a calculator and that the book is not typeset like a college level math text, I'm guessing this is not an OLS / regression class but rather some high school intro stats type class.
Click to expand...

Thank you Stump for the explanation! The class is a college course, but only Algebra II.
 
Aq29lRg.png
vhGXJKw.png

Having a little trouble with this one. The integral I came up with is is 2(16-x^2) evaluated from -4 to 4, which I thought for sure is correct. But when I solve it I get 512/3 which isn't an answer choice.
 
GiantEnemyGoomba said:
Aq29lRg.png
vhGXJKw.png

Having a little trouble with this one. The integral I came up with is is 2(16-x^2) evaluated from -4 to 4, which I thought for sure is correct. But when I solve it I get 512/3 which isn't an answer choice.
Click to expand...
My guess is you did an incorrect computation of the height of your equilateral triangles. Given that the sides of your triangles are 2y, consider just half of the volume of the shape so you get convenient right triangles. These triangles will then have a base of y and a hypotenuse of 2y.

Using the Pythagorean Theorem, you'll find the height of the right triangles to be √3y. So your differential element will be:

1/2bh dx= 1/2(y)(√3y) dx =1/2(√(16-x^2))(√(3*(16-x^2)) dx = √3/2*(16-x^2) dx

Multiply this by 2 and integrate from -4 to 4 and you should get the correct answer.
 
Biggest-Geek-Ever said:
My guess is you did an incorrect computation of the height of your equilateral triangles. Given that the sides of your triangles are 2y, consider just half of the volume of the shape so you get convenient right triangles. These triangles will then have a base of y and a hypotenuse of 2y.

Using the Pythagorean Theorem, you'll find the height of the right triangles to be √3y. So your differential element will be:

1/2bh dx= 1/2(y)(√3y) dx =1/2(√(16-x^2))(√(3*(16-x^2)) dx = √3/2*(16-x^2) dx

Multiply this by 2 and integrate from -4 to 4 and you should get the correct answer.
Click to expand...

Bleh, I feel dumb. I was calculating it as if the height of the triangle was the same as it's base. Appreciate the help.
 
This is more of an economics problem, but I just don't know what to do! Like, am I supposed to put in any random numbers? Any help is appreciated! Here are the questions:

Externalities–analysis and policy design: Suppose that in a competitive market, demand is given by the equation P = 600 – Q, and supply is given by the equation P = 160 + Q, where P is price and Q is quantity of some good or service. Production of each unit of output Q leads to a marginal external cost of $50, caused by pollutants emitted by the production of Q. If we add this marginal external cost to the market information, the equation for the social-cost supply
curve is given by P = 210 + Q.

a. Compute the unregulated market level of output and price as well as the socially
efficient level. By how much does the market output exceed the socially efficient
output, and by how much is the market price below the socially efficient price?

b. Compute the monetary value of the deadweight social loss from the market failure
that occurs if society lets firms to continue to produce negative externalities without
regulation

c. Suppose a tax per unit output (per-unit tax) is imposed on the production of Q with
the intent of making equal the market level of output and the socially efficient level.
How high should that per-unit tax be? What is the gain in net social benefits
(consumer plus producer surplus) that results from this per-unit tax?
 
We just finished talking about Bessel's Equation in my second DE class. I am having some difficulties figuring out how you would choose the substitutions to transform a second order DE into Bessel's Equation. For example we had the following as a bonus question that no one in the class could get:

x^2y'' + x(1-2xtanx)y' - (xtanx+k^2)y = 0

How would I go about figuring out how to transform that into Bessel's equation?:

x^2y'' + xy' + (x^2 - k^2)y = 0
 
Fou-Lu said:
We just finished talking about Bessel's Equation in my second DE class. I am having some difficulties figuring out how you would choose the substitutions to transform a second order DE into Bessel's Equation. For example we had the following as a bonus question that no one in the class could get:

x^2y'' + x(1-2xtanx)y' - (xtanx+k^2)y = 0

How would I go about figuring out how to transform that into Bessel's equation?:

x^2y'' + xy' + (x^2 - k^2)y = 0
Click to expand...
Multiply your DE by cos(x) to obtain:

(1) (x^2y'' + xy'-k^2y)cos(x) - (2x^2y' +xy)sin(x) = 0

It can tricky to figure out exactly what substitution you need. At this point, I would rearrange (1) with the goal of making it look like the desired Bessel equation:

x^2[cos(x)y''-2sin(x)y'] + x[cos(x)y' - sin(x)y] -k^2cos(x)y = 0

We're missing a term of the form x^2y to complete the (x^2 -k^2) term of the Bessel equation. So I would manipulate the equation algebraically to "force it" into that form:

x^2[cos(x)y''-2sin(x)y' - cos(x)y] + x[cos(x)y' +sin(x)y] +[x^2 -k^2]cos(x)y = 0

See if you can tell what substitution you need to make from here. I'll leave the rest of the work in spoilers if you want to view it after you try it on your own.

t = cos(x)y
t' = cos(x)y' - sin(x)y
t'' = cos(x)y'' - sin(x)y' -sin(x)y' - cos(x)y = cos(x)y'' - 2sin(x)y' - cos(x)y

Clearly, substituting this in to (1) leads to the desired Bessel equation in the form:

(2) x^2t'' + xt' + (x^2 - k^2)t = 0

Honestly, it might be clear to you once we obtain (1) that the desired substitution is cos(x)y. But hopefully the extra grunt work made it more apparent how to get to (2). :)
 
I'm doing linear programming, and while I can take duals just fine, I'm having a hell of a time figuring out what complementary slackness means or how to use it to check if a solution is optimal.
 
Hey guys do you have a recommendation on a resource I can use to brush up on statistics?

I am starting my Masters and based on some of the papers I have been reading my thesis will involve a fair amount of regression analysis, root mean square, that sort of stuff. It has been a while since I have done much stats so I thought it would be a good idea to work on it a bit. Also for those of you that use stats a lot, do you have any programs you recommend? I used SigmaPlot a bit a few years ago but dunno if that is the best option these days.
 
Status
Not open for further replies.

Similar threads

Plumbing GAF Help?
25
492
jason10mm replied
Unhinged Bios Official Thread
10
450
SJRB replied
2025 | The yearly positive thread
3
388
DKehoe replied
4K Blu Ray Collectors Help Me
13
549
JimmyRustler replied
NBA Thread 2026: There will be ballin'
2 3 4 5
216
3K
bender replied
Top Bottom