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molybdenum42
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The problem is from an online homework assignment. I know it's probably fairly simple, but my brain isn't grasping it right now for some reason.[The Problem]
We know:
r(t) = <3t2 - 8t + 3, -9t2 + 2t + 7>
And we are asked to find d2y/dx2.[Background Information]
My understanding of d2y/dx2 is that it is the second derivative with respect to x (the first derivative of the function having been with respect to both x and y).
In other words, it's broken down like this:
(d/dx)(dy/dx) = d2y/dx2
The derivative (with respect to x) of the first derivative of the parametric function, r(t), is equal to that mess on the right hand side of the equation.
[Attempt at a Solution]
So we know that:
(dy/dx) = (dy/dt) / (dx/dt) (From the textbook.)
And the above function, r(t), can be broken down into two parts:
x(t) = -9t2 + 2t + 7
y(t) = 3t2 - 8t + 3
Therefore:
(dx/dt) = 6t - 8
(dy/dt) = -18t + 2
And
(dy/dx) = (-18t + 2) / (6t - 8)So, now, here's where I feel like I'm guessing a little bit. Following that logic, would d2y/dx2 = (6) * [(-18t + 2)/(6t - 8)]?
Heh, this is probably a silly question, but thanks very much in advance for any help!
We know:
r(t) = <3t2 - 8t + 3, -9t2 + 2t + 7>
And we are asked to find d2y/dx2.[Background Information]
My understanding of d2y/dx2 is that it is the second derivative with respect to x (the first derivative of the function having been with respect to both x and y).
In other words, it's broken down like this:
(d/dx)(dy/dx) = d2y/dx2
The derivative (with respect to x) of the first derivative of the parametric function, r(t), is equal to that mess on the right hand side of the equation.
[Attempt at a Solution]
So we know that:
(dy/dx) = (dy/dt) / (dx/dt) (From the textbook.)
And the above function, r(t), can be broken down into two parts:
x(t) = -9t2 + 2t + 7
y(t) = 3t2 - 8t + 3
Therefore:
(dx/dt) = 6t - 8
(dy/dt) = -18t + 2
And
(dy/dx) = (-18t + 2) / (6t - 8)So, now, here's where I feel like I'm guessing a little bit. Following that logic, would d2y/dx2 = (6) * [(-18t + 2)/(6t - 8)]?
Heh, this is probably a silly question, but thanks very much in advance for any help!