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Q. 0

Expert-verifiedFound in: Page 777

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.

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

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 $[a,b]$ then for all $x\in [a,b]$

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

Properties of the natural logarithm function:

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

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

$\frac{d(\mathrm{ln}\left(x\right)}{dx}=\frac{1}{x}\phantom{\rule{0ex}{0ex}}\mathrm{ln}\left(1\right)=0\phantom{\rule{0ex}{0ex}}\mathrm{ln}\left(x\right)<0on(0,1)\phantom{\rule{0ex}{0ex}}\mathrm{ln}\left(x\right)>0on(1,\infty )\phantom{\rule{0ex}{0ex}}\mathrm{ln}\left(x\right)isincrea\mathrm{sin}gon(0,\infty )\phantom{\rule{0ex}{0ex}}\mathrm{ln}\left(x\right)isconcavedownon(0,\infty )$

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