To Smart people (and to future Chemists & Mathematicians)

[SM-Count]

This is almost like teaching integral calculus to middle schoolers.

[crazyskwrls]

so since u guys are math genius help me with my stats homework...

normal range of values for blood phosphate levels is 2.6-4.8 mg of phosphate per decileter of blood. The sample mean for a kidney dialysis patient falls above range. We want to determine if this is good evidence that the patient's mean level in fact falls above 4.8

state the H0 and Ha

the data for this patient is 5.6 5.1 4.6 4.8 5.7 6.4
use data to carry out a t test, report the value of the test statistics and the p value and give a conclusion

question 2

blah blah blah ... t(46) = 4.68, p<.01 <---- wtf does this mean?




oh and i see that no one has posted an answer for number 3 that the poor kid could understand so the answer for #3 is
f(x) = 9x+2

[CrimsonKnight]

Therefore, you can subdivide the area beneath it into a rectangle with base 24 and height 8, and a triangle with base 24 and height 72.

Can you demonstrate what you mean?

The total distance traveled is the area underneath the graph of the velocity function.

The black represents the "rectangle with base 24 and height 8" and the red represents the "triangle with base 24 and height 72."

[NuclearSilo]

If u dont know integral (the inverse of derivation), there's an easier way:

d=distance
v=velocity
t=time

v=(d/t)

f(t) = 3t+8 = v(t) = d/t(t)
<=> d(t) = (3t+8)t = 3t*t + 8t + Cst

Cst=Constant how to determine it?
At t=0, we are at the origine (coordinate 0) => d(0)=0 => Cst=0

d(24) = 3*24² + 8*24 + 0

[Kazaxat]

and exactly how all of this is related to SRO?

[kome566]

and exactly how all of this is related to SRO?

It's called the Off Topic Lounge, good sir/madame.

[PicoMon]

EDIT: why did Cruor's integral give a different result?

Approximation:


Exact:

Actually, the approximation that Stress used was this - >

Stress, the reason why your answer is different is because what you did is a fancy way of simply dividing the velocity function into 24 equal rectangular subintervals and adding the area of each rectangle together, according to your calculations you show in your previous post...
[ v(1) + v(2) + v(3) + v(4).... v(23) + v(24) = 1092 ]

Anyways, a picture is worth a million words. Below is a picture of EXACTLY what you did.



As you can see, you overestimated the true area of the function. You used took the Riemann right-hand endpoints, which gives you the overestimate of the real value. Because you split it into 24 equal rectangles, and add up the area of each rectangle by multiplying length * width, your answer (1092) was over the true exact value of 1056. From the picture above, you can see that since the line has a non-zero slope, the estimation will be a bit higher.


However, if you double n to 48 equal subintervals, then you are getting closer to the real value. Example:




And if you double n again to 96 strips of equal width, the area is calculated to be 1065 (which is closer to the true value of 1056 than your original answer of 1092):




We've been getting the overestimate all this time. Similarly, we can also get the underestimate (Riemann left-hand endpoints) by shifting the x to the left by one. Ex: [ v(0) + v(1) + v(2) + v(3).... v(22) + v(23) = 1020 ]
Which is another way of saying:




So now you've got the underestimate and the overestimate, as you increase n, both the overestimate and underestimate will converge to one number (1056). As n increases, the area gets more and more accurate. So it's a very safe bet that as n approaches infinity (n -> ∞), then you will get the EXACT area. This can be mathematical interpreted to be the limit of the Riemann sum as n -> ∞ , which is simply written as the integral, both of which are shown in Cruor's post. Since you know how to do Riemann numerical integration, you can step it up a notch and use Simpson's Rule and you will get an exact approximation even when n = 2



For Quesiton 3, with calculus, you can formulate and prove an equation for the sequence to make it easier on yourself, especially with tougher sequences:
f(n) = a + d(n-1) .... plugging in the numbers...
: f(n) = 11 + 9(n-1)
: f(n) = 11 + 9n - 9
: f(n) = 9n + 2 (ANSWER)



On a side note Stress, since you know how to take the derivatives, the integral is kinda like the antiderivative of a function, so you can reverse the process by differentiating the integral. So if you know the velocity of a function, you can find the displacement function (distance traveled) by taking the antiderivative and evaluating it with the end position and subtracting the starting position. Similarly, you can find the acceleration function by taking the derivative (d/dt) of the velocity function. All of these computation can be done easily in a few seconds with only a pencil, without the tedious work as you may already know from taking derivatives using the power rule. You have excellent math skills for your age. I'm certain you will be a great mathematician one day. Let me know if you have any quesiton or need any clarification.

[PicoMon]

and exactly how all of this is related to SRO?

It's called the Off Topic Lounge, good sir/madame.

Mathematics can be applied to every anything, including Silkroad.

Unlike in grade school, where you learn algebra, trig, and other elementary mathematics which is barely used in real life. Calculus, on the other hands, is the study of changes. The whole whole is a dynamic place that changes all the time, so that is where Calculus kicks in, not a static place where things can be plotted on a graph by a straight line like you learn in Algebra and other grade-school math classes.

You can use advanced mathematics to model pretty much anything in real life (or in the virtual world of Silkroad). For instance, you can use Calculus to model an equation for the amount of HP you get a level in Silkroad:



The picture above is the model of the exact amount of HP points you get get STR point, per level. Notice that it's not a straight line, which pretty much in the real world, almost nothing is.

Similarly, you can use multivariable calculus, numerical analysis, and other advanced mathematical techniques to mathematically prove what combination of STR/INT points will give you the most melee damage output in Silkroad, so you can decide which build you'd like to make to maximize the damage you do on others. It can be modeled by the following:



Since there are multiple variables involve, you get a 3D graph. From there, you can calculate for each build of the Euro/Chinese, what combination of STR/INT ponts will maximize the damage output by using the differential to find the maxima. Or you can take the absolute maxima to get the highest dmg build, if you don't mind what build/class you'd like to play on.

[crazyskwrls]

umm wtf?!?

[NuclearSilo]

@PicoMon: Stop BS, just tell me what highest dmg with highest HP, therefore the strongest build?

[[SD]Kratos]

@PicoMon: Stop BS, just tell me what highest dmg with highest HP, therefore the strongest build?

He must know it better than anyone else...

[Kazaxat]

sum1 please interpret the two models that pico posted please.

[ThiefzV2]

EDIT: why did Cruor's integral give a different result?

Approximation:


Exact:

Actually, the approximation that Stress used was this - >

Stress, the reason why your answer is different is because what you did is a fancy way of simply dividing the velocity function into 24 equal rectangular subintervals and adding the area of each rectangle together, according to your calculations you show in your previous post...
[ v(1) + v(2) + v(3) + v(4).... v(23) + v(24) = 1092 ]

Anyways, a picture is worth a million words. Below is a picture of EXACTLY what you did.



As you can see, you overestimated the true area of the function. You used took the Riemann right-hand endpoints, which gives you the overestimate of the real value. Because you split it into 24 equal rectangles, and add up the area of each rectangle by multiplying length * width, your answer (1092) was over the true exact value of 1056. From the picture above, you can see that since the line has a non-zero slope, the estimation will be a bit higher.


However, if you double n to 48 equal subintervals, then you are getting closer to the real value. Example:




And if you double n again to 96 strips of equal width, the area is calculated to be 1065 (which is closer to the true value of 1056 than your original answer of 1092):




We've been getting the overestimate all this time. Similarly, we can also get the underestimate (Riemann left-hand endpoints) by shifting the x to the left by one. Ex: [ v(0) + v(1) + v(2) + v(3).... v(22) + v(23) = 1020 ]
Which is another way of saying:




So now you've got the underestimate and the overestimate, as you increase n, both the overestimate and underestimate will converge to one number (1056). As n increases, the area gets more and more accurate. So it's a very safe bet that as n approaches infinity (n -> ∞), then you will get the EXACT area. This can be mathematical interpreted to be the limit of the Riemann sum as n -> ∞ , which is simply written as the integral, both of which are shown in Cruor's post. Since you know how to do Riemann numerical integration, you can step it up a notch and use Simpson's Rule and you will get an exact approximation even when n = 2



For Quesiton 3, with calculus, you can formulate and prove an equation for the sequence to make it easier on yourself, especially with tougher sequences:
f(n) = a + d(n-1) .... plugging in the numbers...
: f(n) = 11 + 9(n-1)
: f(n) = 11 + 9n - 9
: f(n) = 9n + 2 (ANSWER)



On a side note Stress, since you know how to take the derivatives, the integral is kinda like the antiderivative of a function, so you can reverse the process by differentiating the integral. So if you know the velocity of a function, you can find the displacement function (distance traveled) by taking the antiderivative and evaluating it with the end position and subtracting the starting position. Similarly, you can find the acceleration function by taking the derivative (d/dt) of the velocity function. All of these computation can be done easily in a few seconds with only a pencil, without the tedious work as you may already know from taking derivatives using the power rule. You have excellent math skills for your age. I'm certain you will be a great mathematician one day. Let me know if you have any quesiton or need any clarification.

finally something that is written in english instead of alien symbols. if you were a teacher teaching math, many students wouldn't hate math as much.

[Stress]

I understood the graph posted above. The integral is, in this case, essentially, a Riemann sum, but with infinite numbers of subdivisions of the interval. With the integral, the left-hand Riemann endpoints and the right-hand ones become essentially the same. That's why it gives the exact result, I think.

Right-hand Riemann endpoints are an overestimate, whereas left-hand Riemann endpoints are an underestimate. I also furthered my study regarding the Riemann sums, and the way they relate to the integral.

I observed that in the case of a linear function, the calculated integral for a certain interval is precisely the arithmetic middle between the left-hand endpoint Riemann sum and the right-hand endpoint Riemann sum.

If you look at the graph posted, you will observe that the area of the surface above the graph, and below it is the same. The reason is, that the function is linear. It divides each rectangle into two triangles of the same area, as seen in:



On the other hand, it's different with an exponential (or, a non-linear) function, as the graph is no longer symmetrical regarding the two axes of the coordinate system, so the integral is no longer the arithmetic middle of the two approximations.


That's the research I did on my own, and I wanted to share it, hoping for it's accuracy.

[Kazaxat]

umm wtf?!?

@PicoMon: Stop BS, just tell me what highest dmg with highest HP, therefore the strongest build?

+2

@PicoMon: Stop BS, just tell me what highest dmg with highest HP, therefore the strongest build?

He must know it better than anyone else...

thats bullshit.


edit - pico if u claim to have the damage formula what is my dmg, lets say if i use a soul spear truth on a penon fighter, if this is my stats

and this is my spear

i will be waitin

[crazyskwrls]

i am astound how a 13 yr old could ask such a profound question...

anyways anyone wanna answer my stats question its due tommorrow

oh and what math u guys taking/took?

i still wonder how u guys got all those math equation image thingys

and stop bullshitting about silkroad, save your amazing finding about highest hp and highest damage for the general discussion

[SM-Count]

Remedial algebra.

[PicoMon]

umm wtf?!?

@PicoMon: Stop BS, just tell me what highest dmg with highest HP, therefore the strongest build?

+2

@PicoMon: Stop BS, just tell me what highest dmg with highest HP, therefore the strongest build?

He must know it better than anyone else...

thats bullshit.


edit - pico if u claim to have the damage formula what is my dmg, lets say if i use a soul spear truth on a penon fighter, if this is my stats

and this is my spear

i will be waitin

Damage: 14,163

[PicoMon]

I understood the graph posted above. The integral is, in this case, essentially, a Riemann sum, but with infinite numbers of subdivisions of the interval. With the integral, the left-hand Riemann endpoints and the right-hand ones become essentially the same. That's why it gives the exact result, I think.

Exactly... that's one way to put it

Right-hand Riemann endpoints are an overestimate, whereas left-hand Riemann endpoints are an underestimate. I also furthered my study regarding the Riemann sums, and the way they relate to the integral.

That is not necessarily always the case. Sometimes, the Riemann right-hand endpoints can be the underestimate and the left-hand endpoints can be the overestimate. You may be wondering why that is so. When a linear function has a negative slope, then it will reverse the role of the Riemann left-and-right endpoints. In the case of non-linear functions (e.g. polynomials, power functions, exponential functions, hyperbolic functions, logarithmic functions, periodic functions, etc), then it is also the case depending on the concavity of the curve (when it concaves up). However, if you take the Riemann endpoints, one of them will always be the underestimate and the other will always be the overestimate.

I observed that in the case of a linear function, the calculated integral for a certain interval is precisely the arithmetic middle between the left-hand endpoint Riemann sum and the right-hand endpoint Riemann sum.

If you look at the graph posted, you will observe that the area of the surface above the graph, and below it is the same. The reason is, that the function is linear. It divides each rectangle into two triangles of the same area, as seen in:



On the other hand, it's different with an exponential (or, a non-linear) function, as the graph is no longer symmetrical regarding the two axes of the coordinate system, so the integral is no longer the arithmetic middle of the two approximations.


That's the research I did on my own, and I wanted to share it, hoping for it's accuracy.

That is a very clever observation in your research. However, as you stated, it only applies to linear functions. By intuition, it is clear that you can get a more accurate approximation by averaging the both the left and right Riemann endpoints, which is EXACTLY the same thing as using the Trapezium Numerical Intregration. Using calculus, it can be proven that the formula is:

Don't let that scare you though. It's relatively simple. Let's try to integrate the same linear function [ v(t)=3x+8 ] using the trapezium method. Instead of splitting the graph into 24 pieces (n=24) using Riemann endpoints like you did earlier in this topic, we will let n=1 this time and see how good the trapezium integration method is for linear functions.



Delta x is the difference of the ending and starting position of what you are trying to integrate, divided by n (the number of intervals you want to split it up into). The larger the intervals, the more accurate the answer is.

As you can see, with the trapezium method you can let n=1 and get an exact approximation without all the tedious calculation of the Riemann endpoints. You didn't have to do the calculation for 24 intervals and add them all together. Furthermore, you had to calculate both of the Riemann endpoints to add insult to the injury, for a total of 48 calculations. With the Trapezium method, you only have to do one very simple and small calculation to get an exact answer for linear functions. Whereas, the only time you can get an exact approximation with the Riemann endpoint when n=1 is when the function has a 0 slope, which is essentially a quadrilateral, so you might as well get the area by multiplying length times width.



I would leave off here, but since you've dwell much into the topic, I'll go a little bit further. Like I mention in my previous posts, in the real world almost nothing can be model by a straight line since we live in a dynamic place. It would be a lot more accurate to model the velocity of a car by something like this that I generated:



Obviously, that is a real world portrayal of the velocity of a car. In the real world, a car don't go at a constant velocity from the beginning to the end of the trip. It increases and decreases smoothly, since a car can't go from 20mph to 30mph without reaching 25mph at some point in time and etc. Now the question is how the heck do you find the distance traveled since it's not a straight line and there's no equation for the model?


You can try and use the Riemann left-hand endpoint. Let's try that with n=6:



Obvious, the method above is a good attempt but it's far from being accurate.


Similarly, we can try and get the Riemann right-hand endpoint with n=6:



The method above is also not a good way to solve the problem.


Now, instead of using rectangles, let's try using trapezoids with n=6 (which again is the average of the Riemann left and right endpoints).:



Clearly, the method above of using trapezoids instead of rectangles is a lot better than the previous two, but it wouldn't do justice if you want high precision. So now you are probably wondering to yourself, "How the heck can you get the exact distance travel then?"

How about instead of using rectangles or trapezoidals and etc, we can try splitting it into 6 equal intervals and use curves instead . Here's a representation of the Simpson's Method:




As you can see from the simulation above, it is very accurate even though n=6 is a very small number. For every additional subintervals we divide it into, the accuracy increases exponentially. Simpson's Rule is one of the best integration methods of all. There are more advance methods and algorithms out there, like Bode's Rule, which is study only in post-graduate mathematics, but it's similar to Simpson's Rule because it also uses curves to find the area of curves lol. Using calculus, you can prove the formula for the Simpson's Rule as:




You may also be wondering why would you care about advanced techniques of numerical integration, when you can do symbolic integration of the definite integral and make it easier on yourself. Well, the fact is there are some problems in the real world can't be modeled by a function. So you are left with no choice but to use numerical integration. You can still use Riemann R&L endpoints to calculate the exact area, but you will have to let n=999,999,999,9... (infinity), which would not be a very wise thing to do since you wouldn't complete the computation in your lifetime. The same thing can be say the Trapezium Method and other less advanced numerical integration methods.


Stress, you've only seen the tip of the iceberg of what Calculus can do. Calculus is a very enormous area. If you can yield its power, it's a very powerful tool to have in your arsenal. Many of the greatest problems in the world and in the sciences have been solved using different branches or variations of Calculus. It's very difficult to exxagerate how powerful it is. Let me know if anything needs clarication.

[Kazaxat]

Damage: 14,163

Wrong:

[Cruor]

Damage: 14,163

Wrong:

One percent is a big difference, eh? Never stopped to think that maybe the estimate was the maximum for your gear/weapon? I've had a word with PicoMon, and that damage output model is the real deal.

[[SD]Kratos]

I was serious when i said he must know it better then anyone else...

[XemnasXD]

why is Cruor using summations and Sigma's....i thought he was like 12, way too smart.....

[Cruor]

why is Cruor using summations and Sigma's....i thought he was like 12, way too smart.....

I suppose I look younger than my age, but a 12 year old?

[NuclearSilo]

Damage: 14,163

Wrong:

One percent is a big difference, eh? Never stopped to think that maybe the estimate was the maximum for your gear/weapon? I've had a word with PicoMon, and that damage output model is the real deal.

It doesnt matter the build. Im pretty sure pure int has the highest dmg output as well as his nuke. Although, dmg on a player is different on a monster. If your balance go below 80% phy and 70% mag, it's become very weak.

[Phix]

3) Write the equation of the function if:
f(1)=11
f(2)=20
f(3)=29
f(4)=38
f(5)=47
...

F(6)=56
f(7)=65

F=2+(NUMBERHERE Times 9)

[Kazaxat]

I was serious when i said he must know it better then anyone else...

i went to the penon fighter and killed a little over 100 of them and 14,161 was the max dmg i could inflict. ok u guys win it is the maximum dmg for my "gear/build"

but how would u know it from someone with 3 posts mr kratos? u must know him.

[NuclearSilo]

I was serious when i said he must know it better then anyone else...

i went to the penon fighter and killed a little over 100 of them and 14,161 was the max dmg i could inflict. ok u guys win it is the maximum dmg for my "gear/build"

but how would u know it from someone with 3 posts mr kratos? u must know him.

I must be a GM, or JM's coder
Or D2U with reverse engineering skill

[YangKang]

I was serious when i said he must know it better then anyone else...

i went to the penon fighter and killed a little over 100 of them and 14,161 was the max dmg i could inflict. ok u guys win it is the maximum dmg for my "gear/build"

but how would u know it from someone with 3 posts mr kratos? u must know him.

Because you can judge from the way he is talking? Its not only about post count.. As example BlackFox tlaked in 6k+ posts BULLSHIT that doesnt mean hes smart eh?

[[SD]Kratos]

I was only reffering to all those maths calculus and stuffs