I actually didn't follow on where you go lol...
What I just did is to just integrate 2 - sec^2 x dx from -pi/4 to pi/4
integral 2- sec^2 x dx
= 2x - tan x from -pi/4 to pi/4
= 4x - 2tan x from 0 to pi/4 (even graph)
= 4(pi/4) - 2 tan (pi/4)
= pi - 2
-
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The Math Help Thread
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I actually didn't follow on where you go lol...
What I just did is to just integrate 2 - sec^2 x dx from -pi/4 to pi/4
integral 2- sec^2 x dx
= 2x - tan x from -pi/4 to pi/4
= 4x - 2tan x from 0 to pi/4 (even graph)
= 4(pi/4) - 2 tan (pi/4)
= pi - 2
That is the correct answer. Thanks alot
Still a little confused on how you got this though
4x - 2tan x from 0 to pi/4 (even graph)
If a graph is even, that means that I can slice the area in y axis and they will be equal, so instead of integrating from -pi/4 to pi/4 which is a pain at times, im just gonna integrate the one half and double the answer I get. Its way easier to do since you use 0 as a bound, making your calculations waaay easier and faster.
MathGAF I need your help, I'm totally lost with this problem:
Let U⊆ℂ be an open set and K⊆ℂ a compact (closed and bounded) set, so that K⊆U. Show that there exists a real number ε>0 so that K + B[0,ε]⊆ U. (B[0,ε] is a closed circular region with radius ε and centre z=0.)
There is a hint about using sequence convergence (maybe Bolzano–Weierstrass?), but that hasn't really gotten me anywhere.
Let U⊆ℂ be an open set and K⊆ℂ a compact (closed and bounded) set, so that K⊆U. Show that there exists a real number ε>0 so that K + B[0,ε]⊆ U. (B[0,ε] is a closed circular region with radius ε and centre z=0.)
There is a hint about using sequence convergence (maybe Bolzano–Weierstrass?), but that hasn't really gotten me anywhere.
My guess, warning this is all really rough and I'm kind of tired.
-Assume for every ε>0 K + B[0,ε] isn't contained in U.
-Use this to get a sequence x_n so each x_n is in K + B[0,1/n] and the compliment of U.
-{x_n} is contained in K + B[0,1], a compact set, so by sequential compactness or B-W or whatever it has a convergent subsequence, with a limit x*.
-{x_n} is contained in the compliment of U, which is closed, so x* is not in U.
-(bit I haven't really thought much about) x* is in K. Otherwise for large enough n it wouldn't be in K + B[0,1/n] or something. The convergence should give us this or something anyway.
-but K is contained in U, so x* is in U, so we have a contradiction. So there must be some ε>0 so that K + B[0,ε]⊆ U.
Anticitizen One
Banned
Hey guys where can I find out how to multiply/divide using Pi.
For example I'm doing this problem where I have to find amplitude, period, and phase shift.
y=-5cos(4x-pi/2)
I know the amplitude = 5
The period is 2pi/4 which = 1/2pi
but how do I do the phase shift? The formula is c/b so it would be -pi/2/4 but how do I calculate that?
Also another example we did in class:
y=-3sin(2X+pi/2)
The phase shift is -pi/2/2 and somehow the professor got -pi/4 as the answer. Can anyone explain?
For example I'm doing this problem where I have to find amplitude, period, and phase shift.
y=-5cos(4x-pi/2)
I know the amplitude = 5
The period is 2pi/4 which = 1/2pi
but how do I do the phase shift? The formula is c/b so it would be -pi/2/4 but how do I calculate that?
Also another example we did in class:
y=-3sin(2X+pi/2)
The phase shift is -pi/2/2 and somehow the professor got -pi/4 as the answer. Can anyone explain?
cpp_is_king
Member
pi is irrelevant in this problem. replace pi with 1. What's 1/2 divided by 2?
Anticitizen One
Banned
Oh ok I see. So in what circumstances is pi relevant? Only in degrees/radian conversions?
cpp_is_king
Member
I'm not totally sure what you mean. When I said it wasn't relevant, I mean that going from -pi/2/2 -> -pi/4 is no different than going from -1/2/2 to -1/4. It's just the way fractions work.
Pi is just a number, like any other number. So in that sense, it's "relevant" whenever you need it, just like 8 is relevant whenever your problem involves the number 8.
MisterLuffy
Member
I need help with this question for Probability and Statistics:
Sorry about the numbers, I couldn't align them. I wish could copy and paste the problem here or an image but the file is pdf.
The article "Determination of Most Representative Subdivision"(J. of Energy Engr., 1993: 43-55) gave data on various characteristics of subdivisions that could be used in deciding whether to provide electrical power using overhead lines or underground lines. Here are the values of the variable x = total length of streets within a subdivision:
1280 5320 4390 2100 1240 3060 4770
1050 360 3330 3380 340 1000 960
1320 530 3350 540 3870 1250 2400
960 1120 2120 450 2250 2320 2400
3150 5700 5220 500 1850 2460 5850
2700 2730 1670 100 5770 3150 1890
510 240 396 1419 2109
a) Construct a stem-and-leaf display using the thousands digit as the stem and the hundreds digit as the leaf.
b) Construct a histogram using class boundaries 0, 1000, 2000, 3000, 4000, 5000, and 6000? What proportion of subdivisions have total length less than 2000? Between 2000 and 4000?
Sorry about the numbers, I couldn't align them. I wish could copy and paste the problem here or an image but the file is pdf.
The article "Determination of Most Representative Subdivision"(J. of Energy Engr., 1993: 43-55) gave data on various characteristics of subdivisions that could be used in deciding whether to provide electrical power using overhead lines or underground lines. Here are the values of the variable x = total length of streets within a subdivision:
1280 5320 4390 2100 1240 3060 4770
1050 360 3330 3380 340 1000 960
1320 530 3350 540 3870 1250 2400
960 1120 2120 450 2250 2320 2400
3150 5700 5220 500 1850 2460 5850
2700 2730 1670 100 5770 3150 1890
510 240 396 1419 2109
a) Construct a stem-and-leaf display using the thousands digit as the stem and the hundreds digit as the leaf.
b) Construct a histogram using class boundaries 0, 1000, 2000, 3000, 4000, 5000, and 6000? What proportion of subdivisions have total length less than 2000? Between 2000 and 4000?
Thanks for your help. Proving it that way makes a lot of sense actually, I never would've thought using sequences would be useful for this kinda thing.
For a) you just need to look up the definition of a stem and leaf display, it's just a way to group numbers of the same 'order'.
For example the numbers 2100, 2109, 2730 would be written together as follows:
thousands (stem)|hundreds(leaf(s))
2|117
To make a histogram you just make 6 bins as defined by those "class boundaries" and for each x you check in which bin it falls. All you do then is keep check of how many x have fallen into each bin and the result of that gives you your histogram.
Also if you copy the data into matlab you can do it in a minute or so, using the histc and bar functions:
MisterLuffy
Member
For a) you just need to look up the definition of a stem and leaf display, it's just a way to group numbers of the same 'order'.
For example the numbers 2100, 2109, 2730 would be written together as follows:
thousands (stem)|hundreds(leaf(s))
2|117
To make a histogram you just make 6 bins as defined by those "class boundaries" and for each x you check in which bin it falls. All you do then is keep check of how many x have fallen into each bin and the result of that gives you your histogram.
Also if you copy the data into matlab you can do it in a minute or so, using the histc and bar functions:
Thanks.
Edit: Do I have to sign up to use matlab?
FUBAR McDangles
Member
This is for my economics homework, but it involves math so I'm turning to you guys. I've been trying to figure out this problem for a good 20 minutes now. I don't know if it's because I'm only a few weeks removed from summer break and am out of practice, or what, but I have absolutely no idea how I am supposed to do this.
Suppose that Rob and Big both raise animals and sell them. Because Rob and Big have different talents, they have varying abilities to raise these animals. In 1 day, Rob can produce either 10 bulldogs or 20 mini-horses. In 1 day, Big can produce either 9 bulldogs or 36 mini-horses. Assume that Rob and Big decide to specialize completely and trade with one another. Over the course of 1 month (30 days), Rob will produce 300 bulldogs and 0 mini-horses while Big will produce 0 bulldogs and 1080 mini-horses.
After specialization, suppose that Rob and Big trade with each other and Rob sends Big 171 bulldogs. In order for both Rob and Big to benefit from trade, Big must send Rob more than: (round your answers to two decimal places)
more than _____ mini-horses
and fewer than ______ mini-horses
I just have no idea what to do. Can anybody help me out here? I'd really appreciate it, everyone.
Suppose that Rob and Big both raise animals and sell them. Because Rob and Big have different talents, they have varying abilities to raise these animals. In 1 day, Rob can produce either 10 bulldogs or 20 mini-horses. In 1 day, Big can produce either 9 bulldogs or 36 mini-horses. Assume that Rob and Big decide to specialize completely and trade with one another. Over the course of 1 month (30 days), Rob will produce 300 bulldogs and 0 mini-horses while Big will produce 0 bulldogs and 1080 mini-horses.
After specialization, suppose that Rob and Big trade with each other and Rob sends Big 171 bulldogs. In order for both Rob and Big to benefit from trade, Big must send Rob more than: (round your answers to two decimal places)
more than _____ mini-horses
and fewer than ______ mini-horses
I just have no idea what to do. Can anybody help me out here? I'd really appreciate it, everyone.
Does anyone have a site where I can learn algebra as I just don't understand it and I would love to understand itthanks in advance !
http://patrickjmt.com/
FreddieQuell
Member
So I know there's some way to algebraically represent this, but I couldn't figure it out so I did it in plain english, lol.
So, looking at it from the perspective of Big, he is getting 171 bulldogs. This is the same as him spending (171bulldogs/9bulldogsPerDay)=19 days producing bulldogs. This means he would spend the other 11 days producing mini-horses, giving him 36*11=396 minihorses. So, if Big doesn't end up with at least 396 horses, this trade wasn't a benefit for him. So he can give Rob up to 1080-396=684 horses.
From Rob's side of things, he ends up producing 300-171=129 bulldogs. This equates to 129/10=12.9 days producing bulldogs. Leaving him 17.1 days to produce mini-horses, which would give him 17.1*20=342 horses.So Big has to send him at least that many.
FUBAR McDangles
Member
So I know there's some way to algebraically represent this, but I couldn't figure it out so I did it in plain english, lol.
So, looking at it from the perspective of Big, he is getting 171 bulldogs. This is the same as him spending (171bulldogs/9bulldogsPerDay)=19 days producing bulldogs. This means he would spend the other 11 days producing mini-horses, giving him 36*11=396 minihorses. So, if Big doesn't end up with at least 396 horses, this trade wasn't a benefit for him. So he can give Rob up to 1080-396=684 horses.
From Rob's side of things, he ends up producing 300-171=129 bulldogs. This equates to 129/10=12.9 days producing bulldogs. Leaving him 17.1 days to produce mini-horses, which would give him 17.1*20=342 horses.So Big has to send him at least that many.
Ah, see, I just needed the answer written out like that. I really appreciate it. It makes sense now but I had absolutely no idea what to do when I looked at it earlier. Thanks a bunch.
And yeah, my econ stuff has quite a few pop culture nods. Vinny/Paulie from Jersey Shore producing T-Shirts and Ice-Cream, Beyonce and Kelly from Destiny's Child producing music, and then this. And we're only a week in haha.
Thanks once again man.
Hey everyone, I'm assuming this is the best place to ask. I'm taking a signals and systems class and I'm trying to understand some basic definitions for input/outputs systems. Linearity is one I feel I only understand half of.
So I have small problems like y[n] = cos(x[n]) and I have to prove a few terms including linearity. In this superposition would fail as Ay[n1] + By[n2] =/= cos(Ax[n1]+Bx[n2]), thus is isn't linear.
There's a problem that passes the superposition test so I assume it fails the homogeneity but I don't quite understand that law.
It's y[n] = 2x[2^n]
If anyone could explain homogeneity to me through this problem I would be grateful.(and if they know signals and systems can they explain why this is memoryless/causal? I figured because n=0 made it x[1] it would be considered a reliance on a future signal.)
I'm guessing that A2x[2^n] cannot possibly equate to 2x[2^(An)] Is this along the right track?
So I have small problems like y[n] = cos(x[n]) and I have to prove a few terms including linearity. In this superposition would fail as Ay[n1] + By[n2] =/= cos(Ax[n1]+Bx[n2]), thus is isn't linear.
There's a problem that passes the superposition test so I assume it fails the homogeneity but I don't quite understand that law.
It's y[n] = 2x[2^n]
If anyone could explain homogeneity to me through this problem I would be grateful.(and if they know signals and systems can they explain why this is memoryless/causal? I figured because n=0 made it x[1] it would be considered a reliance on a future signal.)
I'm guessing that A2x[2^n] cannot possibly equate to 2x[2^(An)] Is this along the right track?
Casanova
Member
I'm an engineering student(biomedical), so according to the "norms", I am supposed to be inherently good at math... but, I'm not. I am currently taking College Calculus, and while I understand the material, I don't necessarily find it natural. It's just that every now and then I will encounter what seems like an elementary/intermediate algebraic problem, and I find myself forgetting a lot of the "fundamentals" and having to go back and review stuff that I cannot seem to remember but I know I've seen in the past.
Is this normal for engineering majors (and college students in general)?
Is this normal for engineering majors (and college students in general)?
yes.
Not for Engineering majors, what year are you in?
I'd say it's kinda common, it depends on what type of engineering though. Some of them typically are a bit better than math and some a bit less good.
Anyone feel like helping me real quick? Just took a trig exam and i'm wanting to know if I got the one question right that I was unsure about.
Anyone want to check what the first trough and peak is of the graph y = -2sin(x -(pi/6)) ?
First trough and peak where x > 0
Thanks in advance!!
Anyone want to check what the first trough and peak is of the graph y = -2sin(x -(pi/6)) ?
First trough and peak where x > 0
Thanks in advance!!
FreddieQuell
Member
Woot! I hope so because that's what I got :3
FreddieQuell
Member
Yeah I went to wolfram first, but couldn't see where to locate the peak and trough. Care to enlighten me? Must I be a pro user or...am I dumb :3
FreddieQuell
Member
MisterLuffy
Member
How do I determine a trimmed percentage? There was a question on a problem stating "What are the corresponding trimming percentages?" How do I got about answer this question? Trimming percentage wasn't giving on this problem. Do I have to find it?
Think of it this way: if you have something of the form f(x-a) (with f a function and a some constant) then the function is 'delayed' as compared to f(x), so the graph gets shifted to the right.
There is no need to check both superposition and homogeneity in order to show linearity of a system, since superposition and linearity are equivalent and homogeneity is required for linearity. To check linearity you simply check if an input of the form x[n]=A*x1[n]+B*x2[n] (a linear combination of input signals x1[n] and x2[n]) gives an output of the form y[n]=A*y1[n]+B*y2[n], with y1[n] the output corresponding to the single input signal x1[n], etc.
For the system y[n]=2x[2^n]. Let x[n]=A*x1[n]+B*x2[n], then y[n]=2x[2^n]=2*(A*x1[2^n]+B*x2[2^n])=A*(2x1[2^n])+B*(2*x[2^n])=A*y1[n]+B*y2[n], so the system is linear. Homogeneity then follows from linearity, so it's automatically satisfied, you can set B=0 to check, since if x[n]=A*x1[n] then y[n]=A*y1[n].
The system is also not memoryless since the output depends on future values of the input signal for n=0,1,2,.. (as you said yourself). The same goes for causality since again y[n] depends on future values of the input signal, so the system cannot be causal.
Hi
I'm taking integral calculus and we just did the fundamental theorem, but I'm still having trouble wrapping my mind around thinking of how to get antiderivatives when it comes to relatively complicated functions.
For instance, how would you go about thinking of an antiderivative for something like this
sqrt(3/x)
Is it just 2/3(3/x)^3/2
?
I'm taking integral calculus and we just did the fundamental theorem, but I'm still having trouble wrapping my mind around thinking of how to get antiderivatives when it comes to relatively complicated functions.
For instance, how would you go about thinking of an antiderivative for something like this
sqrt(3/x)
Is it just 2/3(3/x)^3/2
?
Absoludacrous
Member
You equate sqrt (3/x) to the sqrt (3) over the sqrt (x). You can bring sqrt (3) out of the integral, then you're just integrating 1/sqrt(x), which is x^(-1/2), which should be an easier integral.
I'm not quite sure what you're asking for. If it's for the answer to something like the exalple you made:
y = sqrt(3/x)
You can rewrite this as follows:
y = sqrt(3) * sqrt(1/x)
Which can then be rewritten as:
y = sqrt(3) * x^(-1/2)
Since the square root involves raising a function to (1/2) and an inverse function is just raising it to -1. Now I'll just pretend this is a derivative:
(dy/dx) = sqrt(3) * x^(-1/2)
dy = [sqrt(3) * x^(-1/2)]dx
Integral(dy) = sqrt(3) * 2 * x^(1/2) + C
Since by default, you add 1 to the previous power and divide by the new resulting power, you divide by (1/2), which is just multiplying by 2.
---
But if you're asking for how derivatives work in general, consider the purpose of derivatives. They allow one to determine the instantaneous rate of change at any given point on a function. Now, let's look at the graph of the example you gave:
Let's just say we'll be taking the integral from the value when x=0 to when it equals 1. You can take the area of an arbitrarily small rectangle at any given point (so small, it looks like a line), which just involves length * width. The width will be the value pumped out by the function at that point, while we'll just call the height dx.
This would be useless in of itself, since we don't know what dx means. But, when you take every single rectangle that can exist from 0 to 1, you get something that looks like the area under the particular curve. Basically, since the sum of all of those rectangles involving dx is the as the area, we just need to scoop them all up with our nets forged from the fires of Sohcahtoa.
As for why the integral of two functions is the difference between the two, take a look at this:
The black lines represent two functions, where the first is f(x) and the second is g(x). Each is going to have their own areas, represented by the blue and red sections, respectively. When you take the integral between two curves, you take the area of the lesser one and subtract it from the area of the greater one.
The remaining portion would then be the new area generated by the function.
Hope I helped you, whatever you meant by your question!.
This is not exactly a math problem...more of a physics one. I have been stuck on it for a couple of hours and no one in my study group has been able to come up with a final answer that we're confident in.
Given that the pulley at A is smooth, the tensions in AC and AD are equal. Find the force acting on AE when alpha is 32 degrees. The system is in equilibrium. CD is one string, BC is a separate string.
Thanks guys, any help is appreciated.
Given that the pulley at A is smooth, the tensions in AC and AD are equal. Find the force acting on AE when alpha is 32 degrees. The system is in equilibrium. CD is one string, BC is a separate string.
Thanks guys, any help is appreciated.
Show us your work! Did you get T_AC = 5313 N and T_BC = 546 N ?
Yes I looked at all the forces acting on point C first and got those two numbers. Then I would analyze point A with the forces acting on it being T_AC, T_AD, and F_AE. Now with T_AC = T_AD = 5313N, the only unknown force would be F_AE...but when I would solve for F_AE using the x&y components, I would get two different answers. 6430N when looking at the x-components, and 4070N when looking at y-components.
The arm is physically angled at 50 deg, but the force of the arm doesn't have to be in line with it. Imagine if the arm was just a block; F_AE wouldn't be any different.
So I should think of the force acting on F_AE as going vertically through point A?
And then for equilibrium and solving for F_AE I would do.... T_ACcos5 + T_ADcos30 = F_AE ?
F_AE acts at some unknown angle, but you can find its horizontal and vertical components separately. The final answer is probably supposed to be expressed as a magnitude and an angle.
Oh okay thank you finally got it. I didn't know that you weren't allowed to assume that the force acted at 50 degrees.
I'm doing a problem that requires completing the square and am stumped.
Yeah, completing the square....something I thought I knew how to do, but I can't get the same answer as my book for some reason.
It is 4x^2-4x+3
My book is giving an answer of (2x-1)^2 +2.
I keep messing with my answer and I can't get it to match this. I am surely overlooking something obvious here.
Yeah, completing the square....something I thought I knew how to do, but I can't get the same answer as my book for some reason.
It is 4x^2-4x+3
My book is giving an answer of (2x-1)^2 +2.
I keep messing with my answer and I can't get it to match this. I am surely overlooking something obvious here.
cpp_is_king
Member
Hmmm sorry I don't follow...what led you to take that first step? And how did the second step follow from there? I have a fairly formulaic way of completing the square, I divide to get rid of any constant on the lead coefficient, then move constants to other side of equation, then take half of the X^1 coefficient and square it, add to both sides, then rewrite...
This is to solve an integral that requires the integration of rational functions by partial fractions technique, btw.
Think of it like this: finding a perfect square form is like solving an equation, like so:
You have 4x^2 - 4x + 3. Let's call this "*"
Recall that two polynomials are equal if and only if they have the same degree and coefficients.
A perfect square is in the form of (ax+b)^2. Which is (a^2)*x^2 + 2ab*x + b^2.
If you look at * it almost has this form. If we want it to be a perfect square, then we have to find the values for a and b in our formula above.
We already have the value for a^2, which is 4. This means that a = 2 or a = -2 (since a^2 = 4).
Now, what happens with b? Well, the term 2ab*x has to match up with the -4x we have above. This means that 2a*b = -4, in other words, a*b = -2
If a = 2, then a*b = -2 implies that 2*b = -2. This means that b = -1.
From this we obtain the following perfect square (replacing in the general formula): (2x - 1)^2.
We also observe that (2x - 1)^2 = (-2x + 1)^2. Which means that no matter what sign a has, we obtain the same answer.
( This is a simple case of factoring -1: (2x - 1)^2 = ( (-1) * (-2x + 1))^2 = (-1)^2 * (-2x+1)^2 = 1 * (-2x+1) ^ 2 = (-2x+1) ^ 2 )
Where does this leave us? Well, (2x-1)^2 = 4x^2 - 4x + 1. In order to obtain the same polynomial we had at the beginning we only need to add two to our answer, and thus.
4x^2 -4x +3 = (2x-1)^2 + 2
I hope this helps.
You have 4x^2 - 4x + 3. Let's call this "*"
Recall that two polynomials are equal if and only if they have the same degree and coefficients.
A perfect square is in the form of (ax+b)^2. Which is (a^2)*x^2 + 2ab*x + b^2.
If you look at * it almost has this form. If we want it to be a perfect square, then we have to find the values for a and b in our formula above.
We already have the value for a^2, which is 4. This means that a = 2 or a = -2 (since a^2 = 4).
Now, what happens with b? Well, the term 2ab*x has to match up with the -4x we have above. This means that 2a*b = -4, in other words, a*b = -2
If a = 2, then a*b = -2 implies that 2*b = -2. This means that b = -1.
From this we obtain the following perfect square (replacing in the general formula): (2x - 1)^2.
We also observe that (2x - 1)^2 = (-2x + 1)^2. Which means that no matter what sign a has, we obtain the same answer.
( This is a simple case of factoring -1: (2x - 1)^2 = ( (-1) * (-2x + 1))^2 = (-1)^2 * (-2x+1)^2 = 1 * (-2x+1) ^ 2 = (-2x+1) ^ 2 )
Where does this leave us? Well, (2x-1)^2 = 4x^2 - 4x + 1. In order to obtain the same polynomial we had at the beginning we only need to add two to our answer, and thus.
4x^2 -4x +3 = (2x-1)^2 + 2
I hope this helps.
Thanks a bunch, I understand now.
I guess the only thing I need to decide is when to use each method. Doing it the way you showed, where you use what is already there, or the way I used to do it where you eliminate the coefficient first.
Doing it my old way gives an answer of (x-1/2)^2 + 5/4.
I'm thinking the way you showed was selected in the book because it was a more convenient form for solving the integral ( they used U substitution on the [2x-1]afterward ). Maybe like, as with most other stuff in integration, you have to use some trial and error.
Thanks for your help.
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