Marginal Analysis: At a factory, the daily output is Q(K) =

[Becky4paws]

At a factory, the daily output is Q(K) = 600K^1/2 units, where K is th capital investment measured in units of $1,000. The current capital investment is $900,000. Estimate the effect that an additional capital investment of $800 will have on the daily output.

I'm not sure I am setting this up correctly.

Q'(K) = 300K^-1/2
K=900,000/1,000
Change in K=800

Is the problem set up by finding Q'(900) and Q'(900.8) and then subtracting the two totals? With Q'(K) being raised to a negative power, should I have a negative answer?

 

[pka]

Use linear approximation.
\(\displaystyle \L
\begin{array}{rcl}
Q(x + h) & \approx & hQ'(x) + Q(x) \\
Q(900.8) & = & Q(900 + 0.8) \\
& \approx & \left( {0.8} \right)Q'(900) + Q(900) \\
\end{array}\)

 

[Becky4paws]

0.8(2/3) + 270,000=
1.6/3 +270,000 =
270,005.33

If I was a betting woman, I'd bet against myself.

 

[pka]

Good lord, what did you do?

Q(900)=734.847
Q'(900)=0.408

 

[Becky4paws]

That's a great question, what did I do?

Q(900) = 600(900^1/2) = 600(450) = 270,000

I am obviously messing up when it comes to raising a number to a power. Could you show me the steps of solving something like above?

 

[pka]

I may be reading the question wrong.
Is it \(\displaystyle \sqrt {600K}
\mbox{ or }
600\sqrt K ?\)

If it is the second Q(900)=18000 and Q'(900)=10.

 

[Becky4paws]

The problem is actually written:
Q(K) = 600K^1/2

Read as 600K to the 1/2, no seperation is made between the 600, the K or the 1/2.

 

[Becky4paws]

Here's one mistake...

I have been taking 'to the 1/2 power' as a division of 1/2. So what order do I take when trying to solve Q(K) = 600(K)^1/2. Read as '600K to the 1/2 power'. And when K=900, do I multiply the 600(900) and then take it to the nth power, or do I take the variable K=900 to the nth power and then multiply that answer by 600?

Now that I realize I've been dividing by 1/2 instead of taking it the the 1/2 power, what would any variable be if it was raised to the 1/2 power?