hey @parthkohli

- anonymous

hey @parthkohli

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- anonymous

when i say 30 i mean 40

- ParthKohli

what does "reducing the length of the curve" mean?
I mean you can draw the shortest possible path from (10,10) to (30,0) that also happens to be a straight line.

- anonymous

yea but it needs to be smooth

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## More answers

- anonymous

so polynomial

- ParthKohli

what does that mean? a linear function is a polynomial too.
what exactly do you mean by smooth?

- anonymous

mena no rough edges and its curve

- anonymous

like continuous function

- ParthKohli

ahhh, better.

- anonymous

is there function that looks like the black line i drew

- ParthKohli

so you need a polynomial function that makes a smooth transition and is also the shortest

- anonymous

so it has curve to smothly attach to another track but at the smae shorter length

- anonymous

yes

- anonymous

but at the same time *

- ParthKohli

what's the context of this question? it's really specific

- anonymous

i have done the question let me send the link

- anonymous

but what i am trying to figure out is the shorter length than this

- ParthKohli

sorry, my OS stopped working.

- anonymous

so on xy plane i want a function that give me length <27.37

- anonymous

|dw:1443516744812:dw|

- anonymous

ok

- ParthKohli

You're permitted to use calculators right?\[g(x) = ax^4 + bx^3 + cx^2 + dx + e\]\[10^4 a + 10^3 b + 10^2 c + 10 d + e = 10\]\[30^4 a + 30^3 b + 30^2 c + 30 d + e=0\]\[\int_{10}^{30} g(x)dx=200\]\[4a(10)^3 + 3b(10)^2 + 2c(10) + d(10) =0\]Now the last one is tricky.

- anonymous

yess

- ParthKohli

You know how to calculate arc length using integrals right?

- anonymous

yea but i use wolfram alpha normaly u mind using it sending the link plz

- ParthKohli

Because you really are looking to minimise arc-length given the above constraints.

- anonymous

ok wait do i do the same steps as i did for my previous function find a,b,c,d

- anonymous

and then put them in an expression and use arc length to get length

- ParthKohli

I think Wolfram is gonna have some good fun with this one.

- anonymous

heheh

- ParthKohli

Here, you have a, b, c, d and e.

- anonymous

yea kinda forgot about e

- anonymous

thank you so much for ur help

- ParthKohli

I'm curious to see how you'll input so much into Wolfram. It's scary.

- anonymous

what if i find an expression then too

- ParthKohli

sorry, the last equation is\[4a(10^3) + 3b(10^2 ) + 2c(10) + d = \color{red}1\]

- anonymous

ok

- ParthKohli

wow the expression is way too long... you'll have to be subscribe to pro

- anonymous

i have subscribe pro

- ParthKohli

wow, cool.

- baru

@ParthKohli
for arguments sake, (specifics not necessary)
i would write the function for arc length squared
lets call it L=f(a b c d e)
now what?to minimize, we need to substitute variables in terms of each other so that they represent their inter-dependencies right? how will you do that? you have four other equations ?
or am i way off, and talking crap?

- ParthKohli

yes - we have four equations. in the arc length function, we'll eliminate four variables and will be left with a function that is in terms of only one variable.

- baru

is that a general thing, as in if there are n unknowns and (n-1) equations we can bring it down to 1 variable?

- ParthKohli

I suppose. Let's take a smaller example.\[f(x,y,z) = x+y-z\]\[x=y+1\]\[z = -x\]\[f(x,y,z) =x+(x-1) - (-x) = 3x - 1\]

- ParthKohli

Three variables, two equations. Now we can minimise the function with one-variable very easily.

- baru

ok. thanks :)

- baru

@ParthKohli
one other thing,
i get why you have picked an equation with 5 unknowns since we have 5 pieces of information,
but the equation could just as easily been \[ax^7 + bx^6 +cx^5 +dx +e\]
or something else equally arbitrary
so why specifically have you chosen the one that you have?

- ParthKohli

that is a very good observation, one that I did give some thought to but skipped.
unfortunately, it may be impossible to find *the* polynomial that satisfies the above constraints because the degree may go as high as... anything we want it to reach. the degree-seven polynomial you wrote may not be the best degree-seven polynomial to satisfy the conditions because it's missing some terms and unless we're sure while solving for the seventh-degree that those terms turn out to be zero, we'll never be sure.
so choose a degree-four polynomial adds certainty to what we get - we can be sure that we'll get the best fourth-degree polynomial. of course there are polynomials much better than that one, but we can only go as far as we can look.

- ParthKohli

ultimately, we're looking for a polynomial that resembles a straight-line the most in the region x = 10 to x = 30, so we can add as many terms as we like, but we can't go that far - can we?

- baru

ah..makes sense. thanks a lot!!

- ParthKohli

by the way, we still can approach this problem through multivariable calculus for polynomials that have more terms than five. that way, we can get an even closer polynomial.

- anonymous

hey its not making sense

- ParthKohli

thanks for the question - I really had to think about it to come up with an answer.

- anonymous

5 variables how do i get values for 5 unknown variable

- ParthKohli

plug it into WolframAlpha

- anonymous

i only have 4 equation

- anonymous

i found a ,b, c ,d but not e

- ParthKohli

were you able to find a, b, c, d in terms of e?

- anonymous

\[0.0001852x ^{4}-0.01246x ^{3}+0.2076x ^{2}-0.01555x+e\]

- anonymous

http://math.bd.psu.edu/~jpp4/finitemath/4x4solver.html

- ParthKohli

wait what? you were able to uniquely determine a, b, c, d?!

- anonymous

from here

- ParthKohli

it's not a 4x4 equation so

- anonymous

yea i thought so help plz

- ParthKohli

plug the stuff into Wolfram and it should give you all other variables in terms of one variable

- baru

@ayeshaafzal221
where did you find this question?

- anonymous

it didnt

- anonymous

its my practise question for exams

- ParthKohli

what did it return?

- ParthKohli

you don't have a Wolfram pro subscription do you?

- anonymous

on my ipad yes

- anonymous

even on my ipad not working

- ParthKohli

there's this data entry or something feature which you can use

- anonymous

ok

- anonymous

or u know the values i have found cant i just sub it into any equation and get the e value

- ParthKohli

no, you can't do that. it's not a 4x4 equation so you most likely input the wrong equations.

- anonymous

yea true i am gonna go old fashion , and do em manually

- baru

@ParthKohli , when you say multi-variable calculus approach: are you talking about Lagrange multipliers?(i've just reached this topic)

- ParthKohli

Obviously. Yes.

- baru

alright..thanks

- anonymous

@ParthKohli srry i am disturbing u again when u say we have five variables and one of them was minimum arc length what do u mean?

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