`y = 3xarcsinx , (1/2, pi/4)` Find an equation of the tangent line to the graph of the function at the given point

Friday, July 15, 2016

$\displaystyle{y}={3}{x}{\arcsin{{x}}},{\left(\frac{{1}}{{2}},\frac{\pi}{{4}}\right)}$ Find an equation of the tangent line to the graph of the function at the given point

Check that the given point belongs to the given curve:

$\displaystyle\frac{\pi}{{4}}={3}\cdot\frac{{1}}{{2}}\cdot{\arcsin{{\left(\frac{{1}}{{2}}\right)}}}$ is true, because $\displaystyle{\arcsin{{\left(\frac{{1}}{{2}}\right)}}}=\frac{\pi}{{6}}$ and $\displaystyle\frac{{3}}{{2}}\cdot\frac{\pi}{{6}}=\frac{\pi}{{4}}.$

The tangent line has an equation $\displaystyle{\left({y}-\frac{\pi}{{4}}\right)}={\left({x}-\frac{{1}}{{2}}\right)}\cdot{y}'{\left(\frac{{1}}{{2}}\right)},$ so we need to find the derivative. The product rule is applicable here.

$\displaystyle{y}'{\left({x}\right)}={3}{\left({x}\cdot{\arcsin{{\left({x}\right)}}}\right)}'={3}{\left({\arcsin{{\left({x}\right)}}}+\frac{{x}}{\sqrt{{{1}-{{x}}^{{2}}}}}\right)},$

and at $\displaystyle{x}=\frac{{1}}{{2}}$ it is equal to $\displaystyle{3}\cdot{\left(\frac{\pi}{{6}}+\frac{{\frac{{1}}{{2}}}}{\sqrt{{\frac{{3}}{{4}}}}}\right)}={3}\cdot{\left(\frac{\pi}{{6}}+\frac{{1}}{\sqrt{{{3}}}}\right)}=\frac{\pi}{{2}}+\sqrt{{{3}}}.$

And the equation of the tangent line is finally

$\displaystyle{y}={\left(\frac{\pi}{{2}}+\sqrt{{{3}}}\right)}{x}-\frac{{1}}{{2}}{\left(\frac{\pi}{{2}}+\sqrt{{{3}}}\right)}+\frac{\pi}{{4}}={\left(\frac{\pi}{{2}}+\sqrt{{{3}}}\right)}{x}-\frac{\pi}{{4}}-\frac{\sqrt{{{3}}}}{{2}}+\frac{\pi}{{4}}={\left(\frac{\pi}{{2}}+\sqrt{{{3}}}\right)}{x}-\frac{\sqrt{{{3}}}}{{2}}.$

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Friday, July 15, 2016

$\displaystyle{y}={3}{x}{\arcsin{{x}}},{\left(\frac{{1}}{{2}},\frac{\pi}{{4}}\right)}$ Find an equation of the tangent line to the graph of the function at the given point

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Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

Friday, July 15, 2016

Find an equation of the tangent line to the graph of the function at the given point

No comments:

Post a Comment

Thomas Jefferson&#39;s election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

$\displaystyle{y}={3}{x}{\arcsin{{x}}},{\left(\frac{{1}}{{2}},\frac{\pi}{{4}}\right)}$ Find an equation of the tangent line to the graph of the function at the given point

Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?