New Notes Blog: `g(r) = int_0^r(sqrt(x^2 + 4))dx` Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

Friday, 10 July 2015

$\displaystyle{g{{\left({r}\right)}}}={\int_{{0}}^{{r}}}{\left(\sqrt{{{{x}}^{{2}}+{4}}}\right)}{\left.{d}{x}\right.}$ Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

Hello!

Part 1 of the Fundamental Theorem of Calculus states that for a continuous function $\displaystyle{f{}}$ $\displaystyle{F}'_{{a}}{\left({x}\right)}={f{{\left({x}\right)}}},$ where $\displaystyle{F}_{{a}}{\left({x}\right)}={\int_{{a}}^{{x}}}{f{{\left({t}\right)}}}{\left.{d}{t}\right.}.$

Here $\displaystyle{f{{\left({t}\right)}}}=\sqrt{{{{t}}^{{2}}+{4}}}$ and $\displaystyle{g{{\left({x}\right)}}}={F}_{{0}}{\left({x}\right)}.$

Therefore

$\displaystyle{g{'}}{\left({x}\right)}={F}'_{{0}}{\left({x}\right)}={f{{\left({x}\right)}}}=\sqrt{{{{x}}^{{2}}+{4}}}.$

or $\displaystyle{g{'}}{\left({r}\right)}=\sqrt{{{{r}}^{{2}}+{4}}}$ (identical variable replacement).

Hello!

Part 1 of the Fundamental Theorem of Calculus states that for a continuous function $\displaystyle{f{}}$
$\displaystyle{F}'_{{a}}{\left({x}\right)}={f{{\left({x}\right)}}},$ where $\displaystyle{F}_{{a}}{\left({x}\right)}={\int_{{a}}^{{x}}}{f{{\left({t}\right)}}}{\left.{d}{t}\right.}.$

Here $\displaystyle{f{{\left({t}\right)}}}=\sqrt{{{{t}}^{{2}}+{4}}}$ and $\displaystyle{g{{\left({x}\right)}}}={F}_{{0}}{\left({x}\right)}.$

Therefore

$\displaystyle{g{'}}{\left({x}\right)}={F}'_{{0}}{\left({x}\right)}={f{{\left({x}\right)}}}=\sqrt{{{{x}}^{{2}}+{4}}}.$

or $\displaystyle{g{'}}{\left({r}\right)}=\sqrt{{{{r}}^{{2}}+{4}}}$ (identical variable replacement).

New Notes Blog

Friday, 10 July 2015

$\displaystyle{g{{\left({r}\right)}}}={\int_{{0}}^{{r}}}{\left(\sqrt{{{{x}}^{{2}}+{4}}}\right)}{\left.{d}{x}\right.}$ Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

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In "By the Waters of Babylon," under the leadership of John, what do you think the Hill People will do with their society?

Friday, 10 July 2015

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

No comments:

Post a Comment

In &quot;By the Waters of Babylon,&quot; under the leadership of John, what do you think the Hill People will do with their society?

$\displaystyle{g{{\left({r}\right)}}}={\int_{{0}}^{{r}}}{\left(\sqrt{{{{x}}^{{2}}+{4}}}\right)}{\left.{d}{x}\right.}$ Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

In "By the Waters of Babylon," under the leadership of John, what do you think the Hill People will do with their society?