# Second Fundamental Theorem of Calculus

• Sep 15th 2007, 10:37 AM
Fourier
Second Fundamental Theorem of Calculus
Is it possible to use the Second Fundamental Theorem of Calculus on $\displaystyle C(x)$ to find $\displaystyle C'(x)$, if

$\displaystyle C(x)=\sum_{j=1}^{n}{\bigg|N_{j}\int_{x}^{x_j}{k(s) ds}\bigg|}$,

where $\displaystyle N_j$ is some constant, and $\displaystyle k(s)$ is integrable over every interval?
• Sep 15th 2007, 12:31 PM
CaptainBlack
Quote:

Originally Posted by Fourier
Is it possible to use the Second Fundamental Theorem of Calculus on $\displaystyle C(x)$ to find $\displaystyle C'(x)$, if

$\displaystyle C(x)=\sum_{j=1}^{n}{\bigg|N_{j}\int_{x}^{x_j}{k(s) ds}\bigg|}$,

where $\displaystyle N_j$ is some constant, and $\displaystyle k(s)$ is integrable over every interval?

Yes

RonL
• Sep 15th 2007, 01:38 PM
Fourier
Would $\displaystyle C'(x)$ be

$\displaystyle C'(x)=\sum_{j=1}^{n}{\bigg|N_j k(x_j)\bigg|} \textrm{ ?}$

If this is correct, then it implies that the derivative is always nonnegative and increasing as the sum increases from 1 to n.
• Sep 17th 2007, 08:16 AM
Fourier
Can anyone check my work?

Thanks