Help with a dy/dx and d^2y/dx^2 problem!

Quote:

Originally Posted by mathask

Hey guys,

Just wondering if someone can please answer this question! Could really use the explanation guys.

http://img685.imageshack.us/img685/929/mathi.jpg

Cheers!

The first and second derivatives require continuous functions. Just individual values is not sufficient. I presume that you are really asked to estimate or approximate the derivatives.

My first thought would be to assume a linear function but then the second derivative would be 0, automatically. The next simplest thing to do is to assume a quadratic function. I would be inclined to write the function as $a(x-2)^2+ b(x-2)+ c$ so when x= 0, this is just $a(-2)^2+ b(-2)+ c= 4a- 2b+ c= 7$ , when x= 2, this is just $a(0)^2+ b(0)+ c= c= 13$ , and when x= 4, this is just [tex]a(2)^2+ b(2)+ c= 4a+ 2b+ c= 43.

That is, you have 4a- 2b+ c= 7, c= 13, and 4a+ 2b+ c= 43. It should be easy to solve those for a, b, and c and so write the quadratic function.

Of course, if $f(x)= ax^2+ bx+ c$ , then $f'(x)= 2ax+ b$ and $f"(x)= 2a$ . Evaluate those at x= 0 and x= 1.

Now, that does not use the values at x= 6, or 8 at all. You could, if you wished, write a $4^{th}$ degree polynomial passing through those 5 points.

Again, the question, as you posed it, is impossible. There exist an infinite number of twice-differentiable functions passing through those 5 points. Unless there is more to this problem than you have told us, what I am suggesting is the simplest way to get an answer.