Tangent Planes, Normal Lines, Extrema and Lagrange Multipliers
Tangent Planes, Normal Lines, Extrema and Lagrange Multipliers Math 2419
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This 6 page Class Notes was uploaded by Saul Cervantes on Tuesday October 20, 2015. The Class Notes belongs to Math 2419 at University of Texas at Dallas taught by Anotoly Ezlydon in Fall 2015. Since its upload, it has received 10 views. For similar materials see Accelerated Calculus II in Mathematics (M) at University of Texas at Dallas.
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Date Created: 10/20/15
TANGENT PLANES NORMAL LINES EXTREMA AND LAGRANGE MULTIPLIERS 0 Let FX y 2 be differentiable at AX0 yo zo Suppose that VF at 0 o The plane through A that is normal to VF x0 y0 20 is called the tangent plane 0 The line through A that is perpendicular to the tangent plane is called the normal line Angle of Inclination Angle between tangent plane and the xyplane Vector normal to the tangent plane is Vf Vector normal to the xyplane islt0 0 1gt 39k To nd angle of 1nc11natlon C059 Extrema of Function of Two Variables 0 Critical Points are candidates for extrema 0 Critical points exist Where V f lt 0 0 0 gt or Vf does not exist Second Derivative Test 1 fxxfyy fxy2 If 1 gt 0 and fxxa b gt 0 then f has a relative minimum at the point a b If 1 gt 0 and fxxa b lt 0 then f has a relative maximum at the point a b If 1 lt 0 then point a b fa b is a saddle point If 1 0 the test is inconclusive OOOO Qgrange Multipliers If f and g are differentiable then VfAVg gtgc fxAgx fyAgy Evaluate f at each solution obtained from the above equations The greatest value yields the maximum of f subject to the constraint gX y c and the least value yields the minimum of f subject to the constraint gX y c