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The Math Help Thread
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Yep...
But the question goes like this: r(t) = (a, b) where r(t) is a vector function. It's derivative is r'(t) = <f(t), g(t), h(t)>...
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This high school student came to me with this question... and I could not figure it out. It seems like simple algebra but I don't know if it's solvable... part 4 of this question:
If he did the first question (makig the graph), the answer should be where both equations meet together. If he wants to do this algebraically, it just means setting both equations equal to each other...
5log_3 (-5x-10) + 4 = -.5 (3)^(-2x-5) +4
Im still solving it as I type this message... and Ill prefer the guy asking you to either go straight to grapher or solve it with me... lol
edit screw my equation.. use the transformation
log_b n = e
to
b^e = n
you can transform one equation andntry to isolate y, then you can set them up equally...
I've been at it for the past hour... when equating the y of both equations I always get some form of this equality:
(3^-24.3)^(3^(-2x)) = -5x-10
From which there is no way to isolate for x.... is the only way to solve this is guess and check? Do you just sketch the graph and start inputting values that are close to your intersection point on the graph? According to the student, this is a past exam question her teacher assigned to her.
(3^-24.3)^(3^(-2x)) = -5x-10
From which there is no way to isolate for x.... is the only way to solve this is guess and check? Do you just sketch the graph and start inputting values that are close to your intersection point on the graph? According to the student, this is a past exam question her teacher assigned to her.
Wait AAK, I gotta wash the kid lol...
but Im wondering how you get those ugly numbers... A rule of thumb in Math is if your problem gets too ugly, you're doing it wrong lol.
Don't do any calculator stuff yet till you fully manipulated the equations. This requires some exponent magic...
I just wrote the decimal so it'd be easier to write LOL. Without simplifying it's
(3^((-1/10)(3^5)))^(3^(-2x)) = -5x-10
And the above just isn't solvable... there is some trick I'm missing. I'm assuming it has to do with the fact that both bases are 3 in each equation.... but I just can't see it.
(3^((-1/10)(3^5)))^(3^(-2x)) = -5x-10
And the above just isn't solvable... there is some trick I'm missing. I'm assuming it has to do with the fact that both bases are 3 in each equation.... but I just can't see it.
Tried it... yeah its unsolvable by hand so far. I ended up having a exponential = logarithmic equation. Tried all the tricks I could do, from transforms to exponents and log properties manipulation and nothing goes out. If I try to transform one, I end up having a log within a log... LOGCEPTION...
Last resort is to go to wolframalpha... the thing gave me a number for x thats ugly... Maybe I can come up with something after this kid sleeps though lol.
But yeah I ain't giving up yet...
Sketch the region bounded by the graphs of the equations and find its area.
x = 4y - y^3 ; x = 0
1) Without a graphing utility or point-plotting, how would I even graph x = 4y - y^3?
2) The function is symmetric with respect to the origin so does that guarantee that it is odd?
3) Why can we double the area when integrating? I know that their sum would be zero if we didn't, but there's a theorem stating that if a function is odd on an interval [-a,a], it's integral is 0.
The question seems relatively simple, but for the life of me I feel that I may be forgetting some basic algebra and that's tripping me.
The answer:
x = 4y - y^3 ; x = 0
1) Without a graphing utility or point-plotting, how would I even graph x = 4y - y^3?
2) The function is symmetric with respect to the origin so does that guarantee that it is odd?
3) Why can we double the area when integrating? I know that their sum would be zero if we didn't, but there's a theorem stating that if a function is odd on an interval [-a,a], it's integral is 0.
The question seems relatively simple, but for the life of me I feel that I may be forgetting some basic algebra and that's tripping me.
The answer:
Area = 8
Not me... but in general tip you can always go to your algebra problem solving and practice making "models", so it's something that would be practiced...
If you got a problem, just show it to us and we'll come up with something...
1. Instead of y(x) (y is a function of x), it's a x
(x is a function of y). To make it easy to imagine, get your graphing paper, draw x and y axis like you normally do, then tilt it 90 degrees to the left, and treat the y axis like the x axis like we usually do (but a bit inverted, but you get the drift).If you look at it that way, this problem just becomes your typical integration problem with a little twist.
2. Depends. As a function of y, it is odd, but on x it is not. It depends on what axis you refer it too.
3. It depend on what kind of problem you are dealing with. If the equation is just the boundary of a land you are surveying, and the axes just happened to be there, yeah double it. But sometimes these kind of equations are modeled with profit and losses, with areas above the axis are the profit and under it are losses. Of course you can't declare your losses as profit, so in that case, they subtract each other, which is the case on many integration problems.
I think though that he copied the question wrong... Yeah, that is a little bit too complicated for exponentials and logarithms...
DEATH;97424513 said:Not me... but in general tip you can always go to your algebra problem solving and practice making "models", so it's something that would be practiced...
If you got a problem, just show it to us and we'll come up with something...
1. Instead of y(x) (y is a function of x), it's a x
If you look at it that way, this problem just becomes your typical integration problem with a little twist.
2. Depends. As a function of y, it is odd, but on x it is not. It depends on what axis you refer it too.
3. It depend on what kind of problem you are dealing with. If the equation is just the boundary of a land you are surveying, and the axes just happened to be there, yeah double it. But sometimes these kind of equations are modeled with profit and losses, with areas above the axis are the profit and under it are losses. Of course you can't declare your losses as profit, so in that case, they subtract each other, which is the case on many integration problems.
I think though that he copied the question wrong... Yeah, that is a little bit too complicated for exponentials and logarithms...
Thanks for the help, I worked it out with a friend of mine.
Partial Gamification
Banned
I think its just a matter of getting the parametric form of the cylinders.
x^2 + y^2 = 25 to < 5*cos(t), 5*sin(t), z >
and z = (+/-)sqrt(20-y^2) from y^2 + z^2 = 20
so the vector function is
< 5*cos(t), 5*sin(t), (+/-)sqrt(20 - (5*sin(t))^2) >
I think that is correct, its been awhile.
cpp_is_king
Member
Usually it involves trig substitutions so that you end up with something that leads to a nice simplification due to standard trig identities.
For example, a circle is x^2 + y^2 = r^2. So if you let x = r sin(theta), y = r cos(theta), and plug those into the left hand side, you end up with (r sin(theta))^2 + (r cos(theta))^2 = r^2 (sin^2(theta) + cos^2(theta) = r^2. So the equation is true.
There is no "systematic way", however. As with most trig identities and simplification, it's 1/3 intuition, 1/3 building off of previous patterns and identities you've learned, and 1/3 experimentation until you find something that works.
Generally though, if you see a bunch of squareds in the equation, chances are you're looking for something with sin and cos that will allow you to make use of the fact that sin^2 + cos^2 = 1
How about for sketching an equation such as this: r(t) = <t^3, t^6> (I made this question up).
In order to sketch this, I would do a substitution such as x = t^3. Therefore, r(x^(1/3)) = <x, x^2>. Then it becomes a question of recognizing that r(x^1/3) = <a variable, a function of the variable>?
I realize I have quite a bit of review to do. I haven't touched Calc III for more than a year and half now.
Edit: I would like to clarify that my school did NOT provide a good introduction to the concept of parametric equations and parametrization. In our Calc III class, they told us this: if r(t) = <a, b, c> + t<c,d,e>, then the parametric equations of the line are x(t) = a + tc, y(t) = b + td, etc...
I'm not even kidding.
In order to sketch this, I would do a substitution such as x = t^3. Therefore, r(x^(1/3)) = <x, x^2>. Then it becomes a question of recognizing that r(x^1/3) = <a variable, a function of the variable>?
I realize I have quite a bit of review to do. I haven't touched Calc III for more than a year and half now.
Edit: I would like to clarify that my school did NOT provide a good introduction to the concept of parametric equations and parametrization. In our Calc III class, they told us this: if r(t) = <a, b, c> + t<c,d,e>, then the parametric equations of the line are x(t) = a + tc, y(t) = b + td, etc...
I'm not even kidding.
Is there a way to evaluate the integral of sqrt(x^2 + 9) that does not involve using a hyperbolic function?
I have never been taught a hyperbolic function, so I don't know how to use it.
Edit: Mother of god. The arc length function requires me to evaluate sqrt(-----) integrals. I'm so boned.
Edit-Edit: Woops, forgot to differentiate the parametric functions before finding their length. 0.0. Please disregard!
I have never been taught a hyperbolic function, so I don't know how to use it.
Edit: Mother of god. The arc length function requires me to evaluate sqrt(-----) integrals. I'm so boned.
Edit-Edit: Woops, forgot to differentiate the parametric functions before finding their length. 0.0. Please disregard!
Memento_Mori15
Member
Pretty sure you can use trig substitution. It's been a while since I did integrals but looking at wiki, let x=3 tan(u) dx= 3sec^2(u) and go from there. I would do the rest if I was sure. Sorry if this is not helpful, but I'm sure someone else can post the solution or verify this is how you do it.
Edit: Okay, nvm then
thesaucetastic
Unconfirmed Member
Hey, was hoping someone here would be able to help me out.
Basically, if I is the identity matrix and A is a square complex matrix (i.e., its dimensions are nXn and it can have complex numbers as entries), I'm supposed to show that:
I − A^(m+1) = (I − A)(I + A + A^2 + · · · + A^m).
I'm not entirely sure where to start. I mean, when I see a matrix raised to an indefinite power, my first response would be to try to diagonalize it, but I don't see how that would help in this context. Anyone got any ideas on where to begin?
Basically, if I is the identity matrix and A is a square complex matrix (i.e., its dimensions are nXn and it can have complex numbers as entries), I'm supposed to show that:
I − A^(m+1) = (I − A)(I + A + A^2 + · · · + A^m).
I'm not entirely sure where to start. I mean, when I see a matrix raised to an indefinite power, my first response would be to try to diagonalize it, but I don't see how that would help in this context. Anyone got any ideas on where to begin?
Expand the right term. That reduces to the left term. The identity times the items will be (I+A+A^2+...+A^m) while -A times the terms will be (-A-A^2-...-A^(m+1)). The sum will then reduce to cancel out all terms but I and -A^(m+1).
thesaucetastic
Unconfirmed Member
Right, I thought that was one way to do it. I was just wondering if it was possible to expand the left side to the right side, which I guess would be too much work for little reason. Regardless, thank you so much!
You can do the other way to. Just add and subtract the terms A through A^m then refactor. Not much point though.
Maybe I'm wrong, but shouldn't that just tend to infinity? That denominator of 3x is not bounding the approach to infinity of tan(3x/2) when x is tending to pi.
It could just be that he/she wants you to see that sin(150) is the same as sin(30). Graphing a 150 degree arc might show you that.
Memento_Mori15
Member
I thought it wouldn't exist since on one side, you approach positive infinity, and the other side, you approach negative infinity.
Ya, you're probably right.
youngwerther
Banned
We are working with complex numbers in my engineering class and are just now learning about some procedures with them that I am unfamiliar with.
In the past I've learned only how to convert between polar and rectangular forms and how to perform arithmetic operations.
Right now I have a problem in the book that says
"Evaluate the following by reducing the answer to rectangular form"
j^3
I'm not really sure what it is asking me to do. I'm also not aware of what j^3 even means. Any help?
In the past I've learned only how to convert between polar and rectangular forms and how to perform arithmetic operations.
Right now I have a problem in the book that says
"Evaluate the following by reducing the answer to rectangular form"
j^3
I'm not really sure what it is asking me to do. I'm also not aware of what j^3 even means. Any help?
YianGaruga
Banned
Since it's enginieering the j is the imaginiary unit since you're weird like that. Rectangular should be of the type z = Re(z) + j*Im(z), so the problem seems to be that you need to figure out how to get from j^3 to an expression containing j^1, which is easy.
The result is then just -j, since j^2 = -1?
I'm not sure what else it could be asking for.
I'm not sure what else it could be asking for.
youngwerther
Banned
Thanks, I suppose that is what they want, I guess I was overthinking it. I guess this section of questions is just meant to be like brain puzzles to see if you are following along.
Take the next one for example and tell me if I'm on the right track.
"Evaluate the following by reducing the answer to rectangular form"
e^j(pi + 2pim) where m is an integer.
So then
z = cos(pi + 2 pi m) + jsin(pi + 2 pi m)
so then our trigonometric functions will always be evaluated at odd multiples of pi
and since cos(pi) = -1 and sin(pi) = 0
z = -1 + j0
I guess?? I'm questioning it because if there is no complex component then I'm not sure what the point is of expressing such a thing in complex form.
YianGaruga
Banned
That's correct. A real number is still part of the set of complex numbers, and being able to evaluate the imaginary exponential function is a useful skill to have in many applications. And as you just said, you evaluated a function at a complex number and got a number that does not have an imaginary part as the result. Isn't that already somewhat interesting?
I guess the point is to show you the connection between strictly real and complex numbers? Both examples are quite easy and you're probably overthinking things
That's a great question, it illustrates that any 2pi rotation of a complex number returns the same number, and that the complex numbers contain the real numbers. Or to put it another way, that the real numbers are a subset of the complex numbers.
Maybe every real number you've worked with your whole life was actually secretly a complex number.
youngwerther
Banned
Maybe every real number you've worked with your whole life was actually secretly a complex number.
NOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOoooooooooooo........
cpp_is_king
Member
What is the point of expressing the point (4,0) with a y-component of 0, when you could simply say it was the number 4?
The real numbers are simply a line on the complex plane, just like the line y=0 is a line on the real plane.
Can you help me guys? I got a guy in circuits class that didn't really look at the class prerequisites, and kinda got surprised that he needs to learn a couple of math and physics stuff... I want him to stay in the class as I am scared of getting another class I got enrolled in gets cancelled again...
So can you help me get resources/lesson plan that would kinda wrap differentiation, integration, linear algebra and differential equations that people need in a circuits class? I would try to teach him deeper stuff later on, but I just need something that he can watch/read in the weekend that atleast he can understand what those things really mean...
So can you help me get resources/lesson plan that would kinda wrap differentiation, integration, linear algebra and differential equations that people need in a circuits class? I would try to teach him deeper stuff later on, but I just need something that he can watch/read in the weekend that atleast he can understand what those things really mean...
He's only at Precalc this semester (just starting), and yeah he didn't look at prerequisites. My teacher isn't strict about prerequisites though (I skipped one engineering graphics class thanks to him. He covered all we need to know on the first days of class).
Don't worry, I told him he's swimming in the middle of the Pacific Ocean with no land in sight on freezing temperatures while being surrounded by sharks. (no, not really, but I said he's swimming in deep waters). He looks like a hard worker though.
But yeah... It seems that his algebra skills is solid enough and he knows trig. Its just that he doesn't know ANYTHING about limits, derivatives, integrals and matrices, let alone diffeq... So I am trying to think on what resources he can get to study some bits just to have his foot set at least and just teach him more later...
YianGaruga
Banned
Yeah, it'll be tough to learn all of that and follow the course material at the same time. It's not that the material is extremely complicated (it's not trivial either though), it's that it requires LOTS of practice to get comfortable with it and get a feeling for how any of it works and why.
Differentiation, integration, linear algebra and differential equations were spread over 1 1/2 years at my uni.
Maybe he's a genius and he'll manage, I doubt I would've been able to do it in addition to a normal semester's workload.
DEATH;98409122 said:
Quite simply hes fucked.
But for calculus, Patrick JMT is your guy.
http://patrickjmt.com/
Memento_Mori15
Member
Concept wise, I would recommend giving him khan academy but he goes into deprh and that might waste time.DEATH;98409122 said:
Patrickjmt gives you the basis and it's good enough to do well in those classes with shorter videos.
But yeah...that's like 2 years of math. Good luck to him.
Partial Gamification
Banned
Another online resource is from the USN Academy (ODEs and circuits), contains examples and a schedule. I'm not sure how comprehensive this is as it is a subject I've not studied.DEATH™;98393597 said:
Something else to consider is acquiring a CAS calculator and just learning how to use it, if allowed on exams. This is a huge crutch and setting up the problem is something the calculator won't do for the user. This might hinder a deeper understanding of the systems but I'm not sure what would be required to get the needed information out of this class.
That's how you find local/abs max and mins.
Curvature involves derivatives, but it's a special formula. I'm on my phone and can't write it up, but try googling a curvature formula. Hopefully it's in your notes somewhere.
The formula for curvature is |f ' (x)|/(1+[f '' (x)]^2)^3/2. Isn't this a function of which we can evaluate the derivative to find its maximum point?
Also, is there a shortcut to evaluating the following derivative: T(t) = 1/f(x)<g(x), h(x), i(x)>.
I hate having to use the quotient rule just to find the normal vector, but I'll do it if I have no choice.
Edit: I couldn't do this question because I copied the curvature formula incorrectly. >.> Fyi, it's |f '' (x)| on top and f ' (x) on the bottom.
Also, does +/- infinity count as a valid point when trying to determine where a function equals zero? For example, f ' (x) = 1/x^2. Subbing in either infinities would yield zero.
thesaucetastic
Unconfirmed Member
Yeah, I'm back again. This might be more of a physics topic, but it's still partial differential equations, so whatev.
My book has the following derivation for the heat equation. I have to use a similar method to derive the convection diffusion equation (in 1D):
So what I did was set the change in mass = -Du_x(x,t) + vu(x,t), according to Fick's 1st Law and arbitrarily assigning v as moving in the positive direction, since I want u_xx and u_x to have differing signs at the end. From there, I basically followed the method Farlow used; total mass inside is the integral from x to x + dx of u(s,t)ds. Then I take the derivative of this in terms of t, set it equal to -D[u_x(x + dx,t) - u_x(x,t)] + v[u(x + dx,t) - u(x,t)], use the mean value theorem, let dx -> 0, and end up with u_t(x,t) = -D*u_xx(x,t) + vu_x(x,t).
I think this is the part where I'm supposed to apply the conservation of mass thing (matter is not created or destroyed). I initially thought that the law implied the change in mass in the interval [x + dx, x] = 0, but my teacher told me that it only meant the total mass everywhere is conserved; in other words, mass can flow out of the region [x + dx, x] by diffusion/transport.
I'm at a loss, so I'd greatly appreciate any push in the right direction. Also let me know if this isn't articulated very well, I'll try to clarify it.
My book has the following derivation for the heat equation. I have to use a similar method to derive the convection diffusion equation (in 1D):
So what I did was set the change in mass = -Du_x(x,t) + vu(x,t), according to Fick's 1st Law and arbitrarily assigning v as moving in the positive direction, since I want u_xx and u_x to have differing signs at the end. From there, I basically followed the method Farlow used; total mass inside is the integral from x to x + dx of u(s,t)ds. Then I take the derivative of this in terms of t, set it equal to -D[u_x(x + dx,t) - u_x(x,t)] + v[u(x + dx,t) - u(x,t)], use the mean value theorem, let dx -> 0, and end up with u_t(x,t) = -D*u_xx(x,t) + vu_x(x,t).
I think this is the part where I'm supposed to apply the conservation of mass thing (matter is not created or destroyed). I initially thought that the law implied the change in mass in the interval [x + dx, x] = 0, but my teacher told me that it only meant the total mass everywhere is conserved; in other words, mass can flow out of the region [x + dx, x] by diffusion/transport.
I'm at a loss, so I'd greatly appreciate any push in the right direction. Also let me know if this isn't articulated very well, I'll try to clarify it.
From wikipedia it looks like u_t + J_x = 0 is conservation of mass. If you assume v does not change with x, you get an equation like yours above.
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