MHB Linear Approximation: Check Your Answer

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The discussion focuses on verifying a linear approximation for the function f(x) at the point 2.8, with an initial approximation of -2.8. Participants explore how to determine if this value is an overestimate or underestimate by using the derivative formula f'(x)=sqrt(x^2+7). The linear approximation is identified as the tangent line to f(x) at the point (3,-2). Additionally, the sign of the second derivative f''(3) is questioned for its implications on the behavior of f(x) at x=3. Understanding these derivatives is crucial for assessing the accuracy of the approximation.
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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