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Expert-verified Found in: Page 777 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # q

The summary of the material for the topic "Area Accumulation". is:

• Differentiating a composition involving an area accumulation function.
• Properties of the natural logarithm function.
See the step by step solution

## Step 1. Given Information.

• The area accumulation functions
• The second fundamental theorem of calculus
• Differentiating a composition involving an area accumulation function
• Properties of the natural logarithm function

## Step 2. Explanation

If function $f$ is continuous on the interval [a,b].

The signed area between the graph of $f$ and the $x-$axis is given by the definite integral ${\int }_{a}^{b}f\left(x\right)dx$ is equal to the area accumulation function.

If $f$ is continuous in the interval [a,b] and $u\left(x\right)$ is differentiable on $\left[a,b\right]$ then for all $x\in \left[a,b\right]$

$\frac{d}{dx}{\int }_{a}^{u\left(x\right)}f\left(t\right)dt=f\left(u\left(x\right)\right)u\text{'}\left(x\right)$

Properties of the natural logarithm function:

$\mathrm{ln}\left(x\right)$ is continuous on $\left(0,\infty \right)$

$\mathrm{ln}\left(x\right)$ is differentiable on $\left(0,\infty \right)$

$\frac{d\left(\mathrm{ln}\left(x\right)}{dx}=\frac{1}{x}\phantom{\rule{0ex}{0ex}}\mathrm{ln}\left(1\right)=0\phantom{\rule{0ex}{0ex}}\mathrm{ln}\left(x\right)<0on\left(0,1\right)\phantom{\rule{0ex}{0ex}}\mathrm{ln}\left(x\right)>0on\left(1,\infty \right)\phantom{\rule{0ex}{0ex}}\mathrm{ln}\left(x\right)isincrea\mathrm{sin}gon\left(0,\infty \right)\phantom{\rule{0ex}{0ex}}\mathrm{ln}\left(x\right)isconcavedownon\left(0,\infty \right)$ ### Want to see more solutions like these? 