Mathematics: Find the slope of the normal line to y=x+cos(xy) at (0,1)

Find the slope of the normal line to y=x+cos(xy)

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skywil8981

18.01.2022

Derivating the equation:

$y=x+\cos(xy)\Longrightarrow y'=1+(-y\sin(xy)-xy'\sin(xy))\iff \\\\y'=1-y\sin(xy)-xy'\sin(xy)\iff y'+xy'\sin(xy)=1-y\sin(xy)\iff \\\\y'=\dfrac{1-y\sin(xy)}{1+x\sin(xy)}$

So the slope of the tangent in the point (0,1) is:

$y'=\dfrac{1-y\sin(xy)}{1+x\sin(xy)}\Longrightarrow y'=\dfrac{1-1\sin(0\cdot1)}{1+0\sin(0\cdot1)}\Longrightarrow y'=1$

Then, the slope of the normal (n) line in the point (0,1) is:

$n\cdot y'=-1\Longrightarrow n\cdot1=-1\iff\boxed{n=-1}$

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Ответ:

noah2o2o

04.04.2023

Mathematics

Emilias inequality, x<6 is correct because the number of hours is less than it equal to 6. Emilias number line is correct. The endpoint is correctly filled in and the number line should be shaded to the left of the endpoint.

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Find the slope of the normal line to y=x+cos(xy) at (0,1)

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