Linear Approximation: Check Your Answer

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    Approximation Linear
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Discussion Overview

The discussion revolves around a linear approximation problem, specifically checking the correctness of a solution and determining whether the approximation is an overestimate or underestimate. Participants explore the implications of the derivative and the behavior of the function at a specific point.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant confirms that the linear approximation yields $f(2.8) \approx -2.8$.
  • Another participant questions how to determine if this approximation is an overestimate or underestimate.
  • There is a suggestion to use the derivative formula $f'(x)=\sqrt{x^2+7}$ to analyze the approximation further.
  • One participant states that using the derivative leads to the conclusion that $-2.8$ is approximately equal to $3.852$, which is claimed to be the same as $2.8$.
  • A participant notes that the linear approximation is a tangent line to $f(x)$ at the point $(3,-2)$ and asks about the significance of the second derivative $f''(3)$ in understanding the behavior of $f(x)$ at that point.

Areas of Agreement / Disagreement

Participants express agreement on the linear approximation result but have differing views on how to assess whether it is an overestimate or underestimate. The discussion remains unresolved regarding the implications of the second derivative.

Contextual Notes

There are missing assumptions regarding the function $f(x)$ and its derivatives, as well as unresolved steps in the reasoning about the relationship between the approximation and the actual function values.

jaredjjj
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Have I solved this linear approximation question correctly?
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yes, $f(2.8) \approx -2.8$

now, how to determine if that value is an over or under estimate?
 
skeeter said:
yes, $f(2.8) \approx -2.8$

now, how to determine if that value is an over or under estimate?
To answer the second half do I have to use the formula f'(x)=sqrt(x^2+7) which would mean -2.8 would be approximately equal to 3.852 which is the same as 2.8.
 
jaredjjj said:
To answer the second half do I have to use the formula f'(x)=sqrt(x^2+7) which would mean -2.8 would be approximately equal to 3.852 which is the same as 2.8.

?

The linear approximation is a line tangent to $f(x)$ at the point $(3,-2)$.

What does the sign of $f''(3)$ tell you about the behavior of $f(x)$ at $x=3$ ?
 

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