Mean Value Theorem: is there a typo on the third line?
http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/part-c-mean-value-theorem-antiderivatives-and-differential-equations/session-34-introduction-to-the-mean-value-theorem/MIT18_01SCF10_ex34prb.pdf

Question

Mean Value Theorem: is there a typo on the third line?
http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/part-c-mean-value-theorem-antiderivatives-and-differential-equations/session-34-introduction-to-the-mean-value-theorem/MIT18_01SCF10_ex34prb.pdf

turingtest · Answer

The paper is drawing a parallel between linear approximations so it talks about switching out $f'(a)$ for $f'(c)$ for some specific yet undetermined c
but the last formula on the page notes that this idea extended leads to something like a quadratic approximation, which relates to Taylor series$$f(b)=[f(a)+f'(a)(b-a)]+\frac{f''(c)}2(b-a)^2$$should it not be$$f(b)=[f(a)+f'(c)(b-a)]+\frac{f''(c)}2(b-a)^2$$or is the whole point using $f'(a)$to keep it related to linear approximations?

turingtest · Answer

$$f(b)\approx f(a)+f'(a)(b-a)$$near $x=a$, whereas MVT states that$$f(b)=f(a)+f'(c)(b-a)$$for som $a<c<b$
is the whole point using $f'(a)$to keep it related to linear approximations?

turingtest · Answer

reading it again I'm thinking the answer to my own question is "no it's not a typo" (since it says the part in brackets is $exactly$ the linear approximation, which it is with f'(a) ) but if anyone has any more insight let me know

turingtest · Answer

Never.mind, I think the answer is that it is not a typo. It says the "next term" in the Taylor series will approximate error in an n-th degree approximation. I suppose only the next term in the Taylor series would require the c to get an exact equality.