`y = sinh(1-x^2) , (1, 0)` Find an equation of the tangent line to the graph of the function at the given point

Thursday, July 18, 2013

$\displaystyle{y}={\sinh{{\left({1}-{{x}}^{{2}}\right)}}},{\left({1},{0}\right)}$ Find an equation of the tangent line to the graph of the function at the given point

First check that the given point $\displaystyle{\left({x}_{{0}},{y}_{{0}}\right)}$ belongs to the given graph, i.e. that $\displaystyle{y}{\left({x}_{{0}}\right)}={y}_{{0}}.$ Yes, $\displaystyle{\sinh{{\left({1}-{{1}}^{{2}}\right)}}}={0}.$

Then the equation of the tangent line is

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

(this line goes through $\displaystyle{\left({x}_{{0}},{y}_{{0}}\right)}$ and has the required slope).

The derivative of $\displaystyle{y}$ is $\displaystyle{\cosh{{\left({1}-{{x}}^{{2}}\right)}}}\cdot{\left(-{2}{x}\right)},$ here we use the chain rule and the known derivative of $\displaystyle{\sinh{.}}$ Thus $\displaystyle{y}'{\left({x}_{{0}}\right)}={y}'{\left({1}\right)}=-{2}\cdot{\cosh{{\left({0}\right)}}}=-{2}.$

So the equation of the tangent line is $\displaystyle{y}={\left({x}-{1}\right)}\cdot{\left(-{2}\right)}=-{2}{x}+{2}.$

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Thursday, July 18, 2013

$\displaystyle{y}={\sinh{{\left({1}-{{x}}^{{2}}\right)}}},{\left({1},{0}\right)}$ 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, July 18, 2013

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?

$\displaystyle{y}={\sinh{{\left({1}-{{x}}^{{2}}\right)}}},{\left({1},{0}\right)}$ 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?