Determining a function given points and the values of derivatives at points

[Liquidessa]

Homework Statement

Using Maple, I'm asked to create a quintic function, in the form of ax^5+bx^4+cx^3+dx^2+ex+f given the following data:

It will pass through the points (-5,15), (-5/2, 100), and (10, -5)
-f'(5)=(-1)
-f''(5)=1

Homework Equations

How would I go about doing this? I'm allowed to use Maple. If anyone could demonstrate, that'd be great.

The Attempt at a Solution

I'm not sure where to start. I can't solve for anyone coefficient in particular, because then I just end up with the value of it in relation to the other coefficients. ANy help would be greatly appreciated! Thanks!

Edit: I think I could even do it if I had one more point, I just don't know how to find a fourth one.

 

[micromass]

It will pass through the points (-5,15), (-5/2, 100), and (10, -5).

What does that mean? What equation must hold in order that this is true?

 

[Liquidessa]

I have the basic quintic function f(x)=ax^5+bx^4+cx^3+dx^2+ex+f
I need to find the values of the coefficients a,b,c,d,e and f, creating a function that satisfies these conditions:

It will pass through the points (-5,15), (-5/2, 100), and (10, -5). (If I graphed the function I'm trying to find, it would pass through those points)

-f'(5)=(-1)
-f''(5)=1

 

[micromass]

It will pass through the points (-5,15), (-5/2, 100), and (10, -5). (If I graphed the function I'm trying to find, it would pass through those points)

What does this mean for f? What equation must hold?

 

[micromass]

If I give you the function f(x)=x²+x+1. How do you check that the function goes through (1,1)??

 

[Liquidessa]

Sorry, but I'm not really sure what you mean. Here is a link to the assignment, I'm doing question 1, and I used a value of 5 for 'k', so I figured out my points from that. I hope that's clearer.

http://people.stfx.ca/pkeizer/assignments/a4.pdf

 

[micromass]

Your assignment is clear to me. I'm just asking questions to guide you to a solution.

If I give you f(x)=x²+x+1, then how do check that this curve goes through (1,3)? How do you know that?

 

[Liquidessa]

Wouldn't I just substitute the x-coordinate into my original function, and solve? If it equals the y-coordinate, it would pass through that point. I'm just not sure how to go about doing it with a function this complex. Apparently there is some way I am supposed to use maple to do it, but I don't know.

 

[micromass]

Well, maple will come in later. We will first construct a system of 5 equations with 5 unknowns. You will have to use maple to solve this.

So (-5,15) has to lie on f. Thus the following must hold:

a(-5)^5+b(-5)^4+c(-5)^3+d(-5)^2+e(-5)+f=15

This is our first equation. Can you use the other two points to obtain other equations??

 

[Liquidessa]

Aha! Ok, I understand this much at least. Yes, I have my three equations now. AM I supposed to set them up a system, and ask maple to solve for the variables?

 

[Liquidessa]

Oh! Also, there was a fourth point give that I missed, (5/2, 50).

 

[micromass]

Well, you could try. But maple won't give you many useful answers. The reason is that three equations just isn't enough. If you've got 5 variables, then you need to have (at least) 5 equations. We only have 3 equations, so we need two other ones.

These equations will come from f'(5)=-1 and f''(5)=1.
So all you need to do now is to calculate the first and second derivative and substitute the values. This will give you two other equations...

 

[Liquidessa]

They gave a fourth point, I've got a fourth equation now, but how can I get my fifth from the first and second derivatives?

 

[micromass]

First, find the first derivative. Then use the information that f'(5)=-1, this will give you a fifth equation...

 

[Liquidessa]

Haha, thanks, I'm almost done. Once I have my fifth equation, how do I use Maple to solve for the coefficients? I'm fairly new to Maple, but maybe I should know this.

 

[micromass]

I'm sorry, I really don't know anything about maple

Check the Help file of maple. It should have a lot of tools to solve equations...

 

[Liquidessa]

I now have the equations:

-5 = a(10)^5+b(10)^4+c(10)^3+d(10)^2+e(10)+h;
15 = a(-5)^5+b(-5)^4+c(-5)^3+d(-5)^2+e(-5)+h
100 = a(-5/2)^5+b(-5/2)^4+c(-5/2)^3+d(-5/2)^2+e(-5/2)+h
50= a(5/2)^5+b(5/2)^4+c(5/2)^3+d(5/2)^2+e(5/2)+h
-1 = 5*a(5)^4+4*b(5)^3+3*c(5)^2+eCan anyone tell me how I would determine the coefficients with Maple?