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 $x=3$ ?
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
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