Limits - Did I do the questions correctly?

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Homework Help Overview

The discussion revolves around a problem related to exponential growth, specifically the value of an investment in stock over time, modeled by the function f(x) = 5000(1.12)^x. Participants are tasked with finding the average value of the function between two points and the instantaneous value at a specific point.

Discussion Character

  • Conceptual clarification, Assumption checking, Mixed

Approaches and Questions Raised

  • Participants discuss the calculation of the average value and question the appropriateness of calculating f(3) as an average. There is a focus on the standard definition of "average" for continuous functions. Additionally, there is confusion regarding the request for an instantaneous value and the relevance of differentiation in this context.

Discussion Status

The discussion is exploring different interpretations of the terms "average value" and "instantaneous value." Some participants express uncertainty about the original poster's approach and raise questions about the definitions and assumptions underlying the problem. Clarifications about the derivative and its relevance to the problem are also being discussed.

Contextual Notes

There is mention of a potential typo in the formulas provided by the original poster, which may have contributed to misunderstandings. Participants are also noting that the problem may not align with the concept of limits, as initially suggested.

Macleef
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Homework Statement


If you invest $5000 in a stock that is increasing in value at the rate of 12% per year, then the value of your stock is given by:

f(x) = 5000(1.12)^x, where x is measured in years

a) find average value from x = 2 to x = 3
b) find instantaneous value at x = 3


Homework Equations


y = b^x
y' = (b^x)(ln^x)


The Attempt at a Solution


a) f(x) = 5000(1.12)^{5 - 2} = 5000(1.12)^3 = 7024.64
b) f'(x) = (5000(1.12)^3)(ln 1.12) = (7024.64)(0.113) = 796.093
 
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Macleef said:

Homework Statement


If you invest $5000 in a stock that is increasing in value at the rate of 12% per year, then the value of your stock is given by:

f(x) = 5000(1.12)^x, where x is measured in years

a) find average value from x = 2 to x = 3
b) find instantaneous value at x = 3


Homework Equations


y = b^x
y' = (b^x)(ln^x)
No. the derivative of b^x is ln(b) b^x. I guess that was a typo.


The Attempt at a Solution


a) f(x) = 5000(1.12)^{5 - 2} = 5000(1.12)^3 = 7024.64
b) f'(x) = (5000(1.12)^3)(ln 1.12) = (7024.64)(0.113) = 796.093[/QUOTE]
I have no idea why you did either of those things. You are asked for the "average value" of f(x) between x= 2 and x= 3 and you calculated f(3)? In what sense is that an average? What is the standard definition of "average" of a continuous function? Is it at all likely that the "average" value of an incerasing function will be the very last value?

For (b), why did you differentiate? there is nothing said obout a derivative, only the "instantaneous" value of the function. Do you really think the value at x= 3 is lower than the amount you started with?
 
HallsofIvy said:
No. the derivative of b^x is ln(b) b^x. I guess that was a typo.



b) f'(x) = (5000(1.12)^3)(ln 1.12) = (7024.64)(0.113) = 796.093
I have no idea why you did either of those things. You are asked for the "average value" of f(x) between x= 2 and x= 3 and you calculated f(3)? In what sense is that an average? What is the standard definition of "average" of a continuous function? Is it at all likely that the "average" value of an incerasing function will be the very last value?

For (b), why did you differentiate? there is nothing said obout a derivative, only the "instantaneous" value of the function. Do you really think the value at x= 3 is lower than the amount you started with?

You may be completely misunderstanding the question- I cannot see that it has anything to do with "limits".
 
I thought instantaneous value means the same thing as slope and slope means the same thing as finding the derivative?

My teacher wanted the class to use those formulas mentioned and yes it was a typo..it's y' = (b^x)(lnb)
 

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