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Calculus 3 When should we have Test 3? Monday, November 10 DISCUSS Critical points & second-derivative test COMPARE 61(1,3),63(1,2)Tangent planes,normal lines PRESENT 61(1), 63(1) ASSIGNMENT Write up 61(3),63(2) Work 75, 80 critical points NOTEBOOKS November 11: November 18: Anonymous, Pizza Resubmissions Critical points, Extreme points, and 2nd derivative tests If a surface has a max or min point (hilltop or valley dip), what is the equation of the tangent plane like there? What does that tell you about the partials there? critical values y = f(x) a where f’(a)=0 or f’(a) undefined z=f(x,y) (a,b) where BOTH fx(a,b)= 0 and fy(a,b)=0 Example: Find the critical values for z = xy2-6x2-3y2 fx(a,b)= 0 y2-12x = 0 fy(a,b)=0 2xy-6y=0, so 2y(x-3)=0, so y=0 or x=3 If y=0, then -12x=0, so x=0. If x=3, y2-36=0, so y= 6, or -6. So the three points are (0,0), (3, 6), and (3, -6). How do you tell if a critical point is a max, min, or neither? Examples: Max Min Neither y = -x2 at 0 y = x2 at 0 y = x3 at 0 z = -x2-y2 at (0,0) z = x2+y2 at (0,0) z = x2-y2 at (0,0) For y=f(x), if f’’(a) > 0, then (a, f(a)) is a minimum point. Why? What happens if f’’(a) < 0? What can you conclude if f’’(a)=0? For y=f(x,y), the second derivative test is given in theorem 78. Let’s use it on our four examples.