`x^2 + xarctany = y - 1 , (-pi/4, 1)` Use implicit differentiation to find an equation of the tangent line at the given point

Tuesday, November 1, 2011

$\displaystyle{{x}}^{{2}}+{x}{\arctan{{y}}}={y}-{1},{\left(-\frac{\pi}{{4}},{1}\right)}$ Use implicit differentiation to find an equation of the tangent line at the given point

Does the given point belongs to the given curve?

$\displaystyle{{x}}^{{2}}+{x}{\arctan{{y}}}=\frac{{\pi}^{{2}}}{{16}}-\frac{\pi}{{4}}\cdot\frac{\pi}{{4}}={0},$ and $\displaystyle{y}-{1}={0}$ also.

Then equation of the tangent line is $\displaystyle{\left({y}-{1}\right)}={\left({x}+\frac{\pi}{{4}}\right)}\cdot{y}'{\left(-\frac{\pi}{{4}}\right)}.$ Perform implicit differentiation to find $\displaystyle{y}':$

$\displaystyle{2}{x}+{\arctan{{y}}}+{x}\frac{{{y}'}}{{{1}+{{y}}^{{2}}}}={y}',$

solve this equation for $\displaystyle{y}':$

$\displaystyle{y}'{\left({x}\right)}=\frac{{{2}{x}+{\arctan{{y}}}}}{{{1}-\frac{{x}}{{{1}+{{y}}^{{2}}}}}}.$

At the given point it is equal to $\displaystyle-\frac{{{2}\pi}}{{{8}+\pi}},$ and the equation of the tangent line is

$\displaystyle{y}={1}-{\left({x}+\frac{\pi}{{4}}\right)}\cdot\frac{{{2}\pi}}{{{8}+\pi}}.$

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Tuesday, November 1, 2011

$\displaystyle{{x}}^{{2}}+{x}{\arctan{{y}}}={y}-{1},{\left(-\frac{\pi}{{4}},{1}\right)}$ Use implicit differentiation to find an equation of the tangent line 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?

Tuesday, November 1, 2011

Use implicit differentiation to find an equation of the tangent line 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?

$\displaystyle{{x}}^{{2}}+{x}{\arctan{{y}}}={y}-{1},{\left(-\frac{\pi}{{4}},{1}\right)}$ Use implicit differentiation to find an equation of the tangent line 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?