004- USMA(WP) entrance exam question independent variable

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Discussion Overview

The discussion revolves around determining the interval of possible values for the dependent variable of the function \( W(\theta) = 2\theta^2 \) when the independent variable \( \theta \) is restricted to the interval [2, 6]. The focus is on mathematical reasoning related to function behavior and variable interpretation.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant suggests that the interval of the dependent variable is [8, 72].
  • Another participant explains that since \( 2\theta^2 \) is an increasing function for positive values of \( \theta \), the values of \( W(\theta) \) for \( \theta \) in [2, 6] will range from \( 2(2^2) = 8 \) to \( 2(6^2) = 72 \).
  • A later reply reiterates the increasing nature of the function and the calculated interval, while also questioning the interpretation of \( \theta \) as a measure of angle.
  • Another participant counters that the problem does not specify \( \theta \) as an angle or measurement, suggesting it should be treated simply as a number.

Areas of Agreement / Disagreement

Participants generally agree on the calculated interval of the dependent variable being [8, 72] based on the increasing nature of the function. However, there is disagreement regarding the interpretation of \( \theta \) as an angle or simply a number.

Contextual Notes

The discussion does not clarify assumptions regarding the nature of \( \theta \) and its implications for the function's interpretation.

karush
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ok not sure what forum this was supposed to go in,,so...

If the independent variable of $W(\theta)=2\theta^2$ is restricted to values in the interval [2,6]
What is the interval of all possible values of the dependents variable?
 
Last edited:
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[8, 72]
 
For positive values of \theta, \theta^2, and therefore 2\theta^2, is an increasing function. Values of 2\theta^2, for x between 2 and 6, lie between 2(2^2)= 8 and 2(6^2)= 72.
 
HallsofIvy said:
For positive values of \theta, \theta^2, and therefore 2\theta^2, is an increasing function. Values of 2\theta^2, for x between 2 and 6, lie between 2(2^2)= 8 and 2(6^2)= 72.

Ok I think I got ? Because $\theta$ ussually means radians or degrees!
 
Nothing is said, in this problem, about \theta being the measure of an angle or any measurement at all. It's just a number.
 

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