Finding a Mystery Curve

[Crozius]

I need to find a function that can be proved using calculus techniques. I have used every local source of information to find a method to work this out but alas... the answer eludes me.

 

Heres the question:

 

Following 5 point lie on a function: (1,20) (2,4) (5,3) (6,2) (10,1)

Find an equation that passes through these points and has all the following:

 

3 points of inflection

At least 1 local minimum and local maximum

At least one Critical point (eg max/min/intercept etc) is not at a given point

The curve is continuous and differentiable throughout

The equation is not a single polynomial, but must be a piecewise-defined function

 

Now I know that theres gonna be many different functions. All I need is a method to work one out for myself. I am at a loss of how im meant to get this...

 

Thanks for any help provided

 

Crozius

[xceL]

Looks like you and I have the same maths class

Lake Tuggers ?

 

Either way, I'm looking for the exact same question. So looks like we won't have unique functions Crozius .

 

For the first, pick the first three values and arbitrarily decide the derivative at the third point should be -1.

 

For the second, start on the other end and match the derivative, since the problem statement says it has to be differentiable everywhere.

 

Really, just pick some functions out of a hat and make them do what you want. The derivative of -1 was 100% arbitrary.

 

The starting place is a function with arbitrary coefficients. Then you solve for the coefficients that make it do what you want.

Edited May 31, 2008 by xceL

[Riogho]

Plug that baby into a calculator.

[Cap'n Refsmmat]

If you can find me a calculator that can do that, I would be very surprised.

[Klaynos]

I'm probably reading far to much into this, but it looks like you want to do a polynomail fit, and decent maths package will do it...

 

Even some like excel (ewwwwwwwwwwww) http://facultystaff.richmond.edu/~cstevens/301/Excel4.html

 

If it where I I'd use R... http://www.r-project.org/

[Riogho]

I use a TI-84 to find polynomials to fit through a bunch of random points all the time.

[Cap'n Refsmmat]

It won't satisfy all the requirements of the first post, though.

 

The equation is not a single polynomial, but must be a piecewise-defined function

[Riogho]

He doesn't always have to go for the 'A'

[Crozius]

All of that doesnt answer my question...

 

I use a TI-84 plus calculator and it doesnt give the graph that im expecting, nor does it have the qualities that is required.

 

Any method you guys can come up with that i can use on paper then check with the calculator would be better so i can show working.

 

Thanks.

 

Xcel yes, lake tuggers, which teacher you got?

[Bignose]

So, here's my particular thought....

 

why not think a little outside the box.

 

You have several requirements here:

(A) goes through (1,20) (2,4) (5,3) (6,2) and (10,1)

(B) 3 points of inflection

© At least 1 local minimum and local maximum

(D) At least one Critical point (eg max/min/intercept etc) is not at a given point

(E) The curve is continuous and differentiable throughout

(F) The equation is not a single polynomial, but must be a piecewise-defined function

 

1) Fit a polynomial to go through those 6 points. This takes care of (A). Limit the polynomial to the range x=(0,11)

2) Then piecewise append functions in the negative quadrant of x, and x>11 to fulfill the other requirements.

 

That is, if you've written out the entire problem statement, there isn't a given limit on how much of x you can use. So, use the rest of [math] (-\infty,0)[/math] and [math](11,+\infty)[/math] Put the necessary points of inflection at x = 100,200,500. Or something like that. The only real difficulty here is to pick good functions at the piecewise breaks to fulfill the continuity and differentiability requirements.

[xceL]

Ms. R... something with an R.

Rietta? Yeh that's the one. Line 1 with rietta. lulz.

 

I already got an answer on another forum but it has like 9 decimal places LOL.

I'll have to use that. ;<

It's like 12.2203213983x + 52.1092312029^x + ln(x+5) + 44.210931^whatever lol.

Eitehr way i;x'm drunk at the moment and have no idea how to do the asisngment LOL!

gud luck!

Edited June 1, 2008 by xceL