 # Recent questions in Second Fundamental Theorem of Calculus

Recent questions in Second Fundamental Theorem of Calculus Andreasihf 2022-09-03

### Second Fundamental Theorem of Calculus...Let f have a continuous second derivative. Prove that$f\left(x\right)=f\left(a\right)+\left(x-a\right){f}^{\prime }\left(a\right)+{\int }_{a}^{x}\left(x-t\right){f}^{″}\left(t\right)dt.$Here is my attempt at the problem.Since f has a continuous second derivative, then the first derivative is also continuous. Therefore, by the first fundamental theorem of calculus, we have that$f\left(x\right)=f\left(a\right)+{\int }_{a}^{x}{f}^{\prime }\left(t\right)dt.$Expanding out the right-hand side of the above using integration by parts, we see that$f\left(x\right)=f\left(a\right)+{f}^{\prime }\left(t\right)t-{\int }_{a}^{x}t{f}^{″}\left(t\right)dt.$This is where I am confused. Benedict Kokeh 2022-08-19

### Give Arrhenius and Bronsted definition of acid and base. Why are Bronsted's definition more useful in describing acid base properties Carsen Patel 2022-08-11