New Notes Blog: `y = arctan(x/2) , (2, pi/4)` Find an equation of the tangent line to the graph of the function at the given point

Wednesday, 30 October 2013

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

Equation of a tangent line to the graph of function $\displaystyle{f{}}$ at point $\displaystyle{\left({x}_{{0}},{y}_{{0}}\right)}$ is given by $\displaystyle{y}={y}_{{0}}+{f{'}}{\left({x}_{{0}}\right)}{\left({x}-{x}_{{0}}\right)}.$

The first step to finding equation of tangent line is to calculate the derivative of the given function. To calculate this derivative we will have to use the chain rule $\displaystyle{\left({u}{\left({v}\right)}\right)}'={u}'{\left({v}\right)}\cdot{v}'$

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

Now we calculate the value of the derivative at the given point.

$\displaystyle{y}'{\left({2}\right)}=\frac{{1}}{{{2}{\left({1}+{{\left(\frac{{2}}{{2}}\right)}}^{{2}}\right)}}}=\frac{{1}}{{2}}{\left({1}+{1}\right)}=\frac{{1}}{{4}}$

We now have everything needed to write the equation of the tangent line.

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

$\displaystyle{y}=\frac{{x}}{{4}}+\frac{{\pi-{2}}}{{4}}$

...

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

Now we calculate the value of the derivative at the given point.

$\displaystyle{y}'{\left({2}\right)}=\frac{{1}}{{{2}{\left({1}+{{\left(\frac{{2}}{{2}}\right)}}^{{2}}\right)}}}=\frac{{1}}{{2}}{\left({1}+{1}\right)}=\frac{{1}}{{4}}$

We now have everything needed to write the equation of the tangent line.

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

$\displaystyle{y}=\frac{{x}}{{4}}+\frac{{\pi-{2}}}{{4}}$

Graph of the function along with the tangent line can be seen in the image below.

New Notes Blog

Wednesday, 30 October 2013

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

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

Wednesday, 30 October 2013

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

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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?

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

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