critical - CRITICAL POINTS Math21a, O. Knill HOMEWORK:...

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Unformatted text preview: CRITICAL POINTS Math21a, O. Knill HOMEWORK: 13.8: 2, 18, 20, 30, 40 CRITICAL POINTS. A point ( x ,y ) in the plane is called a critical point of f ( x,y ) if f ( x ,y ) = (0 , 0). Critical points are also called stationary points . Critical points are candidates for extrema because at critical points, all directional derivatives D v f = f v are zero. EXAMPLE 1. f ( x,y ) = x 4 + y 4 4 xy +2. The gradient is f ( x,y ) = (4( x 3 y ) , 4( y 3 x )) with critical points (0 , 0) , (1 , 1) , ( 1 , 1). EXAMPLE 2. f ( x,y ) = sin( x 2 + y ) + y . The gradient is f ( x,y ) = (2 x cos( x 2 + y ) , cos( x 2 + y ) + 1). For a critical points, we must have x = 0 and cos( y ) + 1 = 0 which means + k 2 . The critical points are at (0 , ) , (0 , 3 ) ,... . EXAMPLE 3. (volcano) f ( x,y ) = ( x 2 + y 2 ) e x 2 y 2 . The gradient F = (2 x 2 x ( x 2 + y 2 ) , 2 y 2 y ( x 2 + y 2 )) e x 2 y 2 vanishes at (0 , 0) and on the circle x 2 + y 2 = 1. There are many critical points. EXAMPLE 4 (pendulum) f ( x,y ) = g cos( x ) + y 2 / 2 is the energy of the pendulum. The gradient F = ( y, g sin( x )) is (0 , 0) for x = 0 ,, 2 ,...,y = 0. These points are equilibrium points, where the pendulum is at rest. EXAMPLE 5 (Volterra Lodka) f ( x,y ) = a log( y ) by + c log( x ) dx . (This function is left invariant by the flow of the Volterra Lodka differential equation x = ax bxy, y = cy + dxy which you might have seen in Math1b.) The point ( c/d,a/b ) is a critical point.) is a critical point....
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