Second Fundamental Theorem of Calculus

**Fourier** · September 15th 2007, 10:37 AM

Is it possible to use the Second Fundamental Theorem of Calculus on $C(x)$ to find $C'(x)$ , if

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

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

**CaptainBlack** · September 15th 2007, 12:31 PM

Originally Posted by Fourier

Is it possible to use the Second Fundamental Theorem of Calculus on $C(x)$ to find $C'(x)$ , if

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

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

Yes

RonL

**Fourier** · September 15th 2007, 01:38 PM

Would $C'(x)$ be

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

If this is correct, then it implies that the derivative is always nonnegative and increasing as the sum increases from 1 to n.

**Fourier** · September 17th 2007, 08:16 AM

Can anyone check my work?

Thanks

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