Linear Approximation: Check Your Answer

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SUMMARY

The discussion centers on the linear approximation of the function \( f(x) \) at the point \( (3, -2) \). The user confirms that \( f(2.8) \approx -2.8 \) and explores how to determine if this value is an overestimate or underestimate using the derivative \( f'(x) = \sqrt{x^2 + 7} \). The user concludes that the second derivative \( f''(3) \) provides insight into the concavity of \( f(x) \) at \( x = 3 \), which is crucial for understanding the approximation's accuracy.

PREREQUISITES
  • Understanding of linear approximation concepts
  • Familiarity with derivatives and their applications
  • Knowledge of second derivatives and concavity
  • Basic proficiency in calculus, specifically with functions and their graphs
NEXT STEPS
  • Study the concept of linear approximation in calculus
  • Learn how to compute and interpret first and second derivatives
  • Explore the implications of concavity on function behavior
  • Practice problems involving linear approximations and their accuracy
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Students studying calculus, educators teaching mathematical concepts, and anyone interested in understanding linear approximations and their applications in real-world scenarios.

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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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