## TuringTest 3 years ago 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

1. TuringTest

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?

2. TuringTest

$f(b)\approx f(a)+f'(a)(b-a)$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?

3. TuringTest

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

4. TuringTest

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.