revenue R = f(x): meaning of f'? units of f'? etc

[Sophie]

I would be greatful if this could be confirmed right or wrong and given some direction with units. Here is the question and what I have done:

A company's revenue R (in thousands od dollars) is given by R = f(x) where x is its advertising budget (also in thousands of dollars).

a) What does the sign of f' mean? What are the units of f'?

I put: Rate of change in the companies revenue with respect to the advertising budget.

My problem the units? I guess $ per thousand

b) What does f'(100)=2 mean?

I put: The company's revenue is increasing by 2 units as the advertising budget reaches 100 000.

c) Suppose the company plans to spend $100 000 on advertising and f'(100) = 0.5. Should the company spend more or less on advertising?

i put: The company could expect to increase their revenue by 0.5 units for every 100,000 spent in advertising. However it would be expected that at some point this would plateau.

My biggest problem is not understanding the units for this. Could the above be confirmed right or wrong?

Thanks!

 

[stapel]

Some ideas:

a) If f(x) is the income $1000R given $1000x in advertizing, and if f is differentiated in terms of x, then what would be the units on dR/dx? (Hint: What are the units on R? What are the units on x?)

Remember to be specific. For instance, what does "$ per thousand" mean? "One dollar per thousand items sold"? "Dollar bills burned per thousand hours of advertising broadcast"?

b) Shouldn't the budget have units? ("A hundred thousand" whats?) And what are the units? Yes, the revenue is increasing at a rate of two somethings per something else, but what? (Hint: What are the units on R? What are the units on x?)

c) Why do you think the revenues will increase? (Give the reasoning.) What do you mean by "at some point"? (Define this, or rewrite.) Do you think they're at that point? Past that point? How would one know?

What does it mean for something to be "cost effective"? With f' = 0.5, is advertising cost effective?

Eliz.

 

[Sophie]

Ok round 2, Thanks for the help so far I am certainly feeling more confidant on this subject, but it is still a week spot. If someone could check if I am going in the right direction I would be very grateful.

a. f’ rate of change in the advertising budget with respect to the advertising budget.
Units. When $ per 1000 is spent in advertising $per 1000 is created in revenue.
Therefore $per 1000/$per 1000

b. When the advertising budget spends 100, 000 the following applies:
For every $1 per 1000 spent in advertising $2 per 1000 is gained in revenue.

c. f’(100) = 0.5
When the advertising budget spends 100, 000 the following applies:
For every $1 per 1000 spent in advertising $0.5 per 1000 is gained in revenue.

Therefore the revenue gained is less than the advertising. There are 2 options:
The company could spend more on advertising, which could raise the ratio that is gained for example f’(150) = 1.5.
The company could spend less on advertising, which could raise the ratio that is gained for example f’(75) = 2.
It would depend what impact the advertising would have on the consumer.

 

[stapel]

When $ per 1000 is spent....

What does "$" mean? "Per thousands" of what? This still needs to be defined.

$per 1000/$per 1000

So you are saying that this is actually unit-free (since the "units" you've given will of course cancel, being identical)...?

For every $1 per 1000 spent in advertising $2 per 1000 is gained in revenue.

Since we don't know what you mean by "$1 per 1000", it is difficult to assess if your "definition" is correct. Sorry.

Eliz.

 

[Sophie]

Thanks again for your help, I have to say I am feeling more confused than ever. I have 2 text books in front of me with worked examples and the unit thing confuses me more and more. I understand what you are saying about the units cancelling out, but I do not understand how they can be different. I have tired again, but do not feel it is right. I would be grateful for further direction.

a. f’ is the instantaneous rate of change of R with respect to x. f’ is the rate of change of the advertising budget with respect to the companies revenue.

Now for the units:
Since f’(x) = lim (rate of change x=0) rate of change Revenue/Rate of change advertising budget

And rate of change revenue is measured in dollars which is 1 dollar per $1000 made in revenue.
And rate of change advertising budget is measured in dollars 1 dollar per $1000 spent in the advertising budget.

It follows the units for f’ are 1 dollar per $1000

(I still think from what I have said it should be 1 dollar per $1000/1 dollar per $1000, which is obviously wrong as they cancel each other out.)

b. f’(100) = 2 means that after $100 000 made in revenue the rate at which this is increasing is 2 dollars per $1000 been spent in the advertising budget. So the cost of producing a $100 000 revenue is about $2000.

c. f’(100) = 0.5 Therefore the cost of producing 100 000 revenue is about $500. If this was a linear relationship when $100 000 is spent on advertising the revenue would increase to $2 000 000. Assuming the relationship stayed linear it is advisable for the company to spend more on advertising as revenue would increase.


Please please please tell me I am at least partly right!!!

Thanks