Using Maclaurin's theorem to estimate second derivative

[Roomie]

Homework Statement

Use Maclaurin's theorem to estimate d$$^{2}$$y/dx$$^{2}$$ at x=0

It's the deflection of a 2m beam, where x is the distance along the beam, y is the deflection in mm.

Homework Equations

The Attempt at a Solution

I didn't know where to start so I tried to solve it a different way, by figuring out the formula for the sequence and then just differentiating it twice.. I got the answer of -200 or -0.2 depending upon if I needed to convert the mm to m?

 

[vela]

You have y=f(x), so y''(0)=f''(0). If you can find the coefficient of x², you can solve for the second derivative.

The expansion f(x) = f(0) + f'(0)x + f''(0)/2 x² is the equation of a parabola. Fit your data points to a parabola. The coefficients you get will correspond to f(0), f'(0), and f''(0)/2.

 

[Roomie]

You have y=f(x), so y''(0)=f''(0). If you can find the coefficient of x², you can solve for the second derivative.

The expansion f(x) = f(0) + f'(0)x + f''(0)/2 x² is the equation of a parabola. Fit your data points to a parabola. The coefficients you get will correspond to f(0), f'(0), and f''(0)/2.

So the answer is -200 then? Since y = -100x^2 + 100

So f''(0)/2 = -100

so f''(0) = -200

?? Or should I convert the mm to m?

 

[vela]

The answer isn't a pure number. It has to have units. The numerical value will depend on what units you decide to use.

 

[Roomie]

The answer isn't a pure number. It has to have units. The numerical value will depend on what units you decide to use.

Does that mean the -200 was right :P?

Well if I converted it to metres then it would be a pure number wouldn't it? As it'd be m/m otherwise it's mm/m?

Thanks a lot for your help!

 

[vela]

No, it's not right because you left off the units! You always have to include units on physical measurements. For example, the length of the bar in this problem is 2 meters or 200 cm. The answers are equivalent, but you only know that because the units were included. A pure number is typically meaningless for physical measurements.

Regarding the units of y'', if y is measured in millimeters and x is in meters, you'd have the following units:

y => mm
y' => mm/m
y'' => mm/m²

Every time you differentiate, you're essentially dividing by x, so you get another power of meters on the bottom.

 

[Roomie]

No, it's not right because you left off the units! You always have to include units on physical measurements. For example, the length of the bar in this problem is 2 meters or 200 cm. The answers are equivalent, but you only know that because the units were included. A pure number is typically meaningless for physical measurements.

Regarding the units of y'', if y is measured in millimeters and x is in meters, you'd have the following units:

y => mm
y' => mm/m
y'' => mm/m²

Every time you differentiate, you're essentially dividing by x, so you get another power of meters on the bottom.

Thanks for all your help!