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Given $y\sin(8x) = x\cos(2y)$ find the tangent line at the point ($\pi\over2$, $\pi\over4$).

I got $y = 2x - 2.36$, but my teacher wants a fraction. Can somebody help me get the answer in fraction form?

  • 0
    See [here](https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/implicitdiffdirectory/) and [here](http://tutorial.math.lamar.edu/Classes/CalcI/ImplicitDiff.aspx).2013-11-19

4 Answers 4

1

You have $y\sin(8x)=x\cos(2y)$

Then

$y'\sin(8x)+8y \cos(8x)=\cos(2y)-2y'x\sin(2y)$

Thus

$y'(\sin(8x)+2x\sin(2y))=\cos(2y)-8y\cos(8x)$

$y'=\frac{\cos(2y)-8y\cos(8x)}{\sin(8x)+2x\sin(2y)}$

We look at $(\pi/2,\pi/4)$

$y'_{(\pi/2,\pi/4)}=\frac{\cos(2 \pi/4)-8 \pi/4\cos(8 \pi/2)}{\sin(8 \pi/2)+2 \pi/2\sin(2 \pi/4)}$

$\eqalign{ & {{y'}_{(\pi /2,\pi /4)}} = \frac{{\cos (\pi /2) - 2\pi \cos (4\pi )}}{{\sin (4\pi ) + \pi \sin (\pi /2)}} \cr & {{y'}_{(\pi /2,\pi /4)}} = \frac{{0 - 2\pi }}{{0 + \pi }} = - 2 \cr} $

Since $y(\pi/2)=\pi/4$, we get

$y_T=-2(x-\pi/2)+\pi/4$

$y_T=-2x+\pi+\pi/4=-2x+5\pi/4$

1

General and short formula:

If $F(x,y)=0$ defines $y$ as a function of $x$ implicity, then we always have $y'=\frac{-F_x}{F_y}$.

Here we have $y\sin(8x) = x\cos(2y)$ so $F(x,y)=y\sin(8x) - x\cos(2y)=0$ and then $F_x=8y\cos(8x)-\cos(2y),~~~F_y=\sin(8x)+x\sin(2y)$

0

Differentiate (implicitly). We get $8y\cos(8x)+\sin(8x)y'=-2x\sin(2y)y'+\cos(2y).$ When we substitute the given values of $x$ and $y$, the numbers become very simple, since $\cos 2y=0$ and $\sin 8x=0$. We get that at our target point, $2\pi =-\pi y',$ and our slope is $-2$. The tangent line therefore has equation $y-\frac{\pi}{4}=-2\left(x-\frac{\pi}{2}\right).$ This simplifies to $y=-2x +\dfrac{5\pi}{4}.$

0

Differentiating implicitly the equation $\,y\sin 8x=x\cos 2y\,$ ,we get

$\sin 8x\,dy+8y\cos 8x\,dx=\cos 2y\,dx-2x\sin 2y\,dy\Longrightarrow$

$(\sin 8x+2x\sin 2y)dx=(\cos 2y-8y\cos 8x)dx\Longrightarrow$

$\frac{dy}{dx}=\frac{\cos 2y-8y\cos 8x}{\sin 8x+2x\sin 2y}\Longrightarrow$

$\left.\frac{dy}{dx}\right|_{\left(\frac{\pi}{2},\frac{\pi}{4}\right)}=\frac{0-2\pi}{0+\pi}=-2$

Thus, the tangent line's given by

$y-\frac{\pi}{4}=-2\left(x-\frac{\pi}{2}\right)\Longleftrightarrow y=-2x+\frac{5\pi}{4}$