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2224-Sec12_6-HWT

# 2224-Sec12_6-HWT - Mat h 22 24 Multiva ri a ble C alc Se c...

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Math 2224 Multivariable Calc–Sec. 12.6: Tangent Planes and Differentials I. Tangent Planes and Normal Line A. Definitions 1. The tangent plane at the point P 0 ( x 0 , y 0 , z 0 ) on the level surface f ( x , y , z ) = c of a differentiable function f is the plane through P 0 normal to ! f P 0 . 2. The normal line of the surface at P 0 is the line through P 0 parallel to ! f P 0 . B. Equations 1. Tangent Plane to f ( x , y , z ) = c at P 0 ( x 0 , y 0 , z 0 ) f x ( P 0 )( x ! x 0 ) + f y ( P 0 )( y ! y 0 ) + f z ( P 0 )( z ! z 0 ) = 0 OR from math 1224: < f x ( P 0 ), f y ( P 0 ), f z ( P 0 ) > ! < ( x " x 0 ),( y " y 0 ),( z " z 0 ) > , where ! n =< f x ( P 0 ), f y ( P 0 ), f z ( P 0 ) > , ! v =< ( x ! x 0 ),( y ! y 0 ),( z ! z 0 ) > (vector in the plane) 2. Normal Line to f ( x , y , z ) = c at P 0 ( x 0 , y 0 , z 0 ) x = x 0 + f x ( P 0 ) t y = y 0 + f y ( P 0 ) t z = z 0 + f z ( P 0 ) t 3. Tangent Plane to a Surface z = f ( x , y ) at ( x 0 , y 0 , f ( x 0 , y 0 )) The tangent plane to the surface z = f ( x , y ) of a differentiable function f at the point P 0 ( x 0 , y 0 , z 0 ) = ( x 0 , y 0 , f ( x 0 , y 0 )) is f x ( x 0 , y 0 )( x ! x 0 ) + f y ( x 0 , y 0 )( y ! y 0 ) ! ( z ! z 0 ) = 0 C. Examples 1. Find the equation of the tangent plane to the curve x + y + z 2 = 7 at the point P (1,2, ! 2) .

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2. Find the equation of the tangent plane to the curve xe y z 2 + ln x = 1 at the point P (1,0, ! 2) .
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