Question

Calculus, fundamental theorem...
how do you handle this problem?
Image coming ...

Answer 1

so would productivity be the first derivative?
f'[x]
and productivity per worker the function f[x]
then acceleration can be gleaned from f''[x] ?

guessing ...
then t is workers?
US[t] = output/t ?

Answer 2

remember that these are just slopes

Answer 3

If `us[t]`, for instance, is meant to denote US productivity, chances are `t` indeed refers to workers. This means growth rate is represented by the first derivative, i.e. `us'[t]`, and so the rate at which this changes would be `us''[t]`.

Answer 4

Does this look right then?
us[t] = U / t ( output per worker)
us'[t] = d/dx U * t^(-1) = (-1) U * t^-2 = -U/t^2 (rate of change, Growth)
us''[t] = d/dx -U t^-2 = 2U t^-3 = 2U / t^3 (acceleration)

japan[t] = J / t ( output per worker)
japan'[t] = d/dx J * t^(-1) = (-1) J * t^-2 = -J/t^2 (rate of change, Growth)
japan''[t] = d/dx -J t^-2 = 2J t^-3 = 2J / t^3 (acceleration)

germ[t] = G / t ( output per worker)
germ'[t] = d/dx G * t^(-1) = (-1) G * t^-2 = -J/t^2 (rate of change, Growth)
germ''[t] = d/dx -G t^-2 = 2G t^-3 = 2G / t^3 (acceleration)

Answer 5

GIVEN
U = US Output
J = Japanese Output
G = Germany Output

Answer 6

seems wrong though, t cant have the same population for all countries..

Answer 7

I would disagree. `u[t]` represents the production per worker in the same way some function \(f(x)\) represents a function per unit \(x\).

Answer 8

when you're analyzing outputs such as worker, terms such as acceleration is not a good way to describe profits or other economic outputs

Answer 9

here's a good example of calculus that is directly related to economics
http://www.columbia.edu/itc/sipa/math/calc_econ_interp_u.html

Answer 10

Im not sure that link is making things any clearer for me at this point.. how about some help clarifying what variables are involved, and how they are utilized?

Somewhere there has to be a part of these equations that says.
(TotalOutput / NumberOfWorkers ) = productivity. Please give me an idea how that might be utilized.

This must be different for each country, how is that factored in?
and then there is the question of
'what role does time does time play in these equations, and how is that accounted for, is time factored in as t, or is time not part of them at all?'

Answer 11

maybe t is worker hours?

Answer 12

If I find a way to answer this, I'll post it.. I have to go out for now.

Answer 13

Well I will attempt and answer, hopefully someone can confirm or correct.

Let
J = workers in Japan
G = workers in Germany
U = workers in USA
Then the productivity - output per worker - in terms of production functions are

us[t]/U
germ[t]/G
japan[t]/J

In which case us[t],germ[t],japan[t] are the total production for each country during the term.

Growth rate is total production / time
us[t]/t
germ[t]/t
japan[t]/t
Q. does this mean us'[t] = us[t]/t ?

productivity acceleration rate is given by the 2nd derivative.
us''[t]
germ''[t]
japan''[t]

Us Growth rate must have slowed and have been seen to be
us/t < 0

But both
germ''[t] >0 and japan''[t]>0 must be above zero.
since both have increased with time.

As the author has used 1st and 2nd derivative terms, he must be familiar with calculus.

Answer 14

oh and..
us[t]/U > germ[t]/G
us[t]/U > japan[t]/J

Answer 15

Thanks for any help I got here..

us[t] can't mean production per worker in the same way f[x] is some function per unit of x.
t must represent time. It makes more sense.

us[t],japan[t],germ[t] must have within them functions that are dependent on a local worker count, and then return the production over a given time interval in terms of productivity per worker. In this way t can have the same meaning no matter what country you are testing against.

Answer 16

I think this is the answer

GIVEN
national productivity as an output per worker over a given term t

us[t]
germ[t]
japan[t]

Taking the derivative of these functions with respect to time we get a rate of change, which is the growth.

d/dt us[t] = us'[t]
d/dt germ[t] = germ'[t]
d/dt japan[t] = japan'[t]

and their second derivatives of the growth with respect to time will give the acceleration (if positive) or deceleration (if negative) of the growth.

d/dt us'[t] = us''[t]
d/dt germ'[t] = germ''[t]
d/dt japan'[t] = japan''[t]

For US productivity to be higher than Japan and Germany then
us[t] > germ[t]
us[t] > japan[t]

For growth rate to slow while productivity has accelerated for the other major countries then
us''[t] < 0 meaning the growth is decelerating.
But
germ''[t] >0
japan''[t]>0
meaning these countries are accelerating.

As the author has inferred 1st derivative terms for growth
and 2nd derivative terms for acceleration, he must be familiar with calculus.

Answer 17

And I hate to be bastard but considering that I paid to get help on this one, that was pretty bad.

Answer 18

Haha, I think you have the right to say that, still need help, I could try, I just didn't want to read so much xD