{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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) .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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) .
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}