Thread:

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

Hey guys,

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

Cheers!

Hey guys,

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

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.
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