`y = e^sinhx , (0, 1)` Find an equation of the tangent line to the graph of the function at the given point

Thursday, June 25, 2009

$\displaystyle{y}={{e}}^{{\sinh{{x}}}},{\left({0},{1}\right)}$ Find an equation of the tangent line to the graph of the function at the given point

The tangent line must go through the given point $\displaystyle{\left({x}_{{0}},{y}_{{0}}\right)}$ and have the slope of $\displaystyle{y}'{\left({x}_{{0}}\right)}.$ Thus the equation of the tangent line is

$\displaystyle{\left({y}-{y}_{{0}}\right)}={\left({x}-{x}_{{0}}\right)}\cdot{y}'{\left({x}_{{0}}\right)}.$

To find the derivative we need the chain rule and the derivative of $\displaystyle{\sinh{{\left({x}\right)}}},$ which is $\displaystyle{\cosh{{\left({x}\right)}}}.$ Therefore $\displaystyle{y}'{\left({x}\right)}={\left({{e}}^{{\sinh{{\left({x}\right)}}}}\right)}'={{e}}^{{\sinh{{\left({x}\right)}}}}\cdot{\cosh{{\left({x}\right)}}}.$ For $\displaystyle{x}={x}_{{0}}={0}$ it is $\displaystyle{{e}}^{{0}}\cdot{1}={1}.$

So the equation of the tangent line is $\displaystyle{y}-{1}={\left({x}-{0}\right)}\cdot{1},$ or simply y = x + 1.

We have to check that $\displaystyle{y}{\left({x}_{{0}}\right)}={x}_{{0}}.$ Yes, $\displaystyle{y}{\left({x}_{{0}}\right)}={y}{\left({0}\right)}={1}$ and $\displaystyle{x}_{{0}}={1}.$

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Thursday, June 25, 2009

$\displaystyle{y}={{e}}^{{\sinh{{x}}}},{\left({0},{1}\right)}$ 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?

Thursday, June 25, 2009

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?

$\displaystyle{y}={{e}}^{{\sinh{{x}}}},{\left({0},{1}\right)}$ 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?