Question

hey @parthkohli

Answer 1

when i say 30 i mean 40

Answer 2

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.

Answer 3

yea but it needs to be smooth

Answer 4

so polynomial

Answer 5

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

Answer 6

mena no rough edges and its curve

Answer 7

like continuous function

Answer 8

ahhh, better.

Answer 9

is there function that looks like the black line i drew

Answer 10

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

Answer 11

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

Answer 12

yes

Answer 13

but at the same time *

Answer 14

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

Answer 15

i have done the question let me send the link

Answer 16

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

Answer 17

sorry, my OS stopped working.

Answer 18

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

Answer 19

ok

Answer 20

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.

Answer 21

yess

Answer 22

You know how to calculate arc length using integrals right?

Answer 23

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

Answer 24

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

Answer 25

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

Answer 26

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

Answer 27

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

Answer 28

heheh

Answer 29

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

Answer 30

yea kinda forgot about e

Answer 31

thank you so much for ur help

Answer 32

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

Answer 33

what if i find an expression then too

Answer 34

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

Answer 35

ok

Answer 36

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

Answer 37

i have subscribe pro

Answer 38

wow, cool.

Answer 39

@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?

Answer 40

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.

Answer 41

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

Answer 42

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

Answer 43

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

Answer 44

ok. thanks :)

Answer 45

@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?

Answer 46

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.

Answer 47

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?

Answer 48

ah..makes sense. thanks a lot!!

Answer 49

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.

Answer 50

hey its not making sense

Answer 51

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

Answer 52

5 variables how do i get values for 5 unknown variable

Answer 53

plug it into WolframAlpha

Answer 54

i only have 4 equation

Answer 55

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

Answer 56

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

Answer 57

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

Answer 58

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

Answer 59

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

Answer 60

from here

Answer 61

it's not a 4x4 equation so

Answer 62

yea i thought so help plz

Answer 63

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

Answer 64

@ayeshaafzal221
where did you find this question?

Answer 65

it didnt

Answer 66

its my practise question for exams

Answer 67

what did it return?

Answer 68

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

Answer 69

on my ipad yes

Answer 70

even on my ipad not working

Answer 71

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

Answer 72

ok

Answer 73

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

Answer 74

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

Answer 75

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

Answer 76

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

Answer 77

Obviously. Yes.

Answer 78

alright..thanks

Answer 79

@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?