zatajuxoqj

2022-03-22

What is the equation of the tangent line of $f\left(x\right)=\frac{{\mathrm{cos}}^{2}x}{{e}^{x}}-x\mathrm{tan}x$ at $x=\frac{\pi}{4}$ ?

shvatismop1rj

Beginner2022-03-23Added 10 answers

First, find the point the tangent line will intersect. I'll be using decimals since there is no clean way to simplify this by any means.

$f\left(\frac{\pi}{4}\right)=-0.5574$

The tangent line will intersect the point $(0.7854,-0.5574).$

(Replacing $\frac{\pi}{4}$ with $0.7854.$)

To find the slope of the tangent line, find ${f}^{\prime}\left(\frac{\pi}{4}\right)$.

To find f'(x), use quotient rule for the first term and product rule for $x\mathrm{tan}x$.

${f}^{\prime}\left(x\right)=\frac{2\mathrm{cos}x(-\mathrm{sin}x){e}^{x}-{e}^{x}{\mathrm{cos}}^{2}x}{{e}^{2x}}-\mathrm{tan}x-x{\mathrm{sec}}^{2}x$

$=\frac{-\mathrm{cos}x(2\mathrm{sin}x+\mathrm{cos}x)}{{e}^{x}}-\mathrm{tan}x-x{\mathrm{sec}}^{2}x$

Plug in $\frac{\pi}{4}$.

${f}^{\prime}\left(\frac{\pi}{4}\right)=-3.2547$

Plug the point and slope into an equation in point-slope form:

$y+0.5574=-3.2547(x-0.7854)$

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

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