find equations of the tangent line and normal line to the given curve at the specified point?? `Y=2xe^x`  at the point (0,0)i can figure out tangent line but what is normal line??

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hala718 | High School Teacher | (Level 1) Educator Emeritus

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The normal line is the line that is perpendicular to the tangent line.

First we will find the tangent line at the point (0,0).

f(x)= 2xe^x

==> The slope of the tangent line is the derivative at x= 0

==> f'(x)= (2x)'e^x + 2x(e^x)'

==> f'(x)= 2e^x + 2xe^x

==> f'(0) = 2e^0 + 2(0)e^0

==> f'(0) = 2

Then the slope of the tangent line is m= 2

==> y-y1 = m(x-x1)

==> y= 2x

Then, the tangent line is : `y= 2x`

`` Now we will find the slope of the normal line.

Since the normal line is perpendicular to the tangent line, then the slope is -1/2

==> y= (-1/2)x

Then, the normal line is :`y= -1/2 x`

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