See my notes. I'm not sure f(x) = sqrt(x-.75) is correct for your data, as opposed to, say, f(x) = .8 sqrt(x - .75) or f(x) = 1.3 sqrt(x - .75). The coefficient 1 doesn't seem likely, though it's possible.
Also I believe that your values for the derivative are incorrect. See my note and let me know if you disagree with the values I show.
15-87-310
For these follow up questions, I had to determine the integral with respect to length for several segments of the graph. Using the same graph as the one from the original experiment, it is clear that the graph of the Force (y axis) is equal to the square root of the distance minus .75 N, which is how far the graph is shifted to the right. In other words, f(x)=`sqrt(x-.75).
To find the integral of force with respect to length for each segment, I used this basic equation, and found its integral. I ended up with this formula as the integral for the graph:
I question the coefficient 1 of your function. The curve fit looks good. The problem is that I can’t read the vertical scale on the graph and you didn’t include the data, which I need in order to be sure your function is correct. It seems too much of a coincidence that the coefficient of your square root is 1, but it’s certainly possible and does give answers in the expected range. Otherwise the function does check out, with correct x intercept, etc..
For the next part, I had to plug in the different increments of distance to determine its integral. This is the solution for the first distance.
Distance Integral
9-9.5 cm 1.46 kg cm /s^2
9.5-10 cm 1.5 kg cm/s^2
10-10.5 cm 1.541 kg cm/s^2
10.5-11 cm 1.581 kg cm/s^2
11-11.5 cm 1.62 kg cm/s^2
11.5-12 cm 1.659 kg cm/s^2
9-10 cm 2.957 kg cm/s^2
9-10.5 cm 4.498 kg cm/s^2
9-11 cm 6.079 kg cm/s^2
9-11.5 cm 7.699 kg cm/s^2
9-12 cm 9.358 kg cm/s^2
As the formula for the first distance shows, the Integral is 1.46. Using this same formula for each increment, I determined the integral for each distance. As the table shows, the integral increases as the lengths increase.
Next, I found the derivative of the equation of the graph.
*For this part, the answers I got were negative. I think I am messing this one up. I thought I knew what you were looking for, but now I am starting to wonder. Could you let me know what I am messing up on this one before I go any farther! Thank you.
I’ve evaluated derivative, which looks correct, and got the following:
x .5(x-7.5)^(-.5)
9.5 0.353553
10 0.316228
10.5 0.288675
11 0.267261
11.5 0.25
12 0.235702
I suspect that you are evaluating your expression incorrectly. Again, your derivative function itself looks fine.
Distance Derivative
9-9.5 cm
9.5-10 cm
10-10.5 cm
10.5-11 cm
11-11.5 cm
11.5-12 cm
9-10 cm
9-10.5 cm
9-11 cm
9-11.5 cm
9-12 cm