Undergrad Differentiation formula: Is this a typo?

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The discussion centers on a potential typo in the differentiation formula notes, where an initially stated positive error term becomes negative after substituting positive values for A and C. Participants question whether this discrepancy is a typographical error or if there is a theoretical basis for it. The conversation suggests that the formula may relate to estimating the error term in the Taylor series. Clarification is sought regarding the notation used, which some find confusing. Overall, the thread highlights uncertainty about the accuracy of the notes and the implications of the error term's sign.
maistral
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Resource found in the 'net, trying to know if this is a typo or not.
Red arrows.
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The notes initially say that the error term is positive. After substitution of A and C which are clearly positive, the term suddenly became negative...? Is this a typo, or is there a theory behind this?
 
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It seems OK observing that the third term of (5.14) is f”(x).
 
Last edited:
anuttarasammyak said:
It seems OK observing that the third term of (5.14) is f”(x).
What does this imply? That either positive or negative will work?
 
maistral said:
Summary:: Resource found in the 'net, trying to know if this is a typo or not.

Red arrows.
View attachment 287254

The notes initially say that the error term is positive. After substitution of A and C which are clearly positive, the term suddenly became negative...? Is this a typo, or is there a theory behind this?
It would seem to be the formula for an estimate of the error term in the Taylor series. Is this what your question was?
 
forlmula(5.14).jpg


Transfer the second term of RHS to LHS.
 
Last edited:
I finally got it! The notation was so confusing, sorry.

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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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