Finding Polynomial Graphs Through Given Points with Linear Algebra

[frasifrasi]

Ok, so I grasped how to some versions of this question, but one question in the book is
asking to find all the polynomials of degree <= 2 whose graphs run through the points (1,1) and (2,0) such that integral (from 1 to 2) of f(t) dt = -1.


I have never done anything like this, so if anyone can help, thank you

 

[jhicks]

Don't let the integral fool you! It's more of the same. What is the definite integral of $$ax^2+bx+c$$ from 1 to 2? Do the definite integral and you will see that you still have the same three unknowns.

 

[frasifrasi]

ok, but at what point do I apply the integral?

 

[Dick]

You have three conditions. Applying those conditions will still give you three linear equation in three unknowns. Compute the integral in terms of a,b and c.

 

[HallsofIvy]

Knowing that you can write the polynomial y= ax²+ bx+ c, what equation does x=1, y=1? x= 2, y= 0? x= 2, y= 0? And, of course, what equation, for a, b, and c, do you get from $$\int_1^2 (ax^2+ bx+ c)dx= -1$$?

 

[frasifrasi]

I am doing the matrix for

a + b +c =1
and
4a + 2b + c =1

but het infinitely many solutions. Can you help me by saying if this is the correct matrix?

 

[Dick]

I am doing the matrix for

a + b +c =1
and
4a + 2b + c =1

but het infinitely many solutions. Can you help me by saying if this is the correct matrix?

You have a third equation. Work out the integral Halls was kind enough to write out.

 

[frasifrasi]

I know this is a stupid question, but I am getting

7/3a + 3/2b + c = -1 for the integral. Can anyone confirm this? It is just the answer doesn't seem right.

 

[Dick]

I know this is a stupid question, but I am getting

7/3a + 3/2b + c = -1 for the integral. Can anyone confirm this? It is just the answer doesn't seem right.

That's right.

 

[frasifrasi]

Dick or anyone,

My book is terrible so I am having to research a lot of topics.

For the dot product of the col matrix
1
2
3

and

1
-2
1

I am getting
1
-4
3

just by multiplying, is this the correct way? how does this differ from matrix multiplication(cross product) once you are dealing with larger matrices?

 

[Dick]

The dot product is the SUM of the products of the vector components. It's a scalar. In this case 1-4+3=0. If you work through matrix multiplication, you'll see you are building a matrix by taking dot products of row vectors and transposed column vectors.