Math 241 Spring 2014 Exam Information Time and Date Thursday February 20th 6 45 8pm Location Will be sent to you by Tuesday February 18th Pencil BRING A NUMBER 2 PENCIL Conflict Exams Will be scheduled to minimize number of exams Website http www math illinois edu jathreya 241 exam1 html Math 241 Spring 2014 Level sets of the circle Math 241 Spring 2014 Level sets Claim f is orthogonal to tangent vectors to level sets That is for a function f R2 R f is perpendicular to tangent lines For f R3 R f is perpendicular to level surfaces Math 241 Spring 2014 Level sets Idea Since directional derivatives of f along level sets are 0 we have for any tangent vector v to a level set of f Dv f f v 0 Math 241 Spring 2014 Tangent planes to level surfaces We can use this idea to compute tangent planes to level surfaces of f f x y z Idea The normal vector to the level surface x y z g x y z L at the point x0 y0 z0 is given by the gradient vector g x0 y0 z0 Math 241 Spring 2014 Tangent planes to graphs Note that the graph of a surface z f x y can be viewed as the 0 level surface of the function g x y z f x y z g fx fy 1 Math 241 Spring 2014 Maxima and Minima 14 7 Local maxima Definition f Rn R has an local maximum at the point p if there is a 0 such that f p f x for 0 kx pk Math 241 Spring 2014 Maxima and Minima 14 7 Local minima Definition f Rn R has an local minimum at the point p if there is a 0 such that f p f x for 0 kx pk Math 241 Spring 2014 Maxima and Minima 14 7 Critical Points Idea If any component of the gradient vector is non zero the function will change increase decrease in that direction So local maxima and minima of a function f happen at places where f 0 Math 241 Spring 2014 Maxima and Minima 14 7 Critical Points Definition The critical points of the function f are exactly the places where f 0 Question How can we tell when a critical point is a minimum or a maximum Math 241 Spring 2014 Maxima and Minima 14 7 Saddle Points In higher dimensions other things can happen Math 241 Spring 2014 Maxima and Minima 14 7 Potato Chips http freakonomics com 2011 04 15 the math of pringles Math 241 Spring 2014 Maxima and Minima 14 7 Second derivative test 2 variables Like in 1 variable we have an second derivative test For f f x y at a critical point a b we consider Second partials fxx a b fyy a b fxy a b fyx a b Discriminant D fxx a b fyy a b fxy a b 2 Math 241 Spring 2014 Maxima and Minima 14 7 Second derivative test 2 variables Theorem Let a b be a critical point for the function f R2 R i e f a b 0 If D 0 fxx a b 0 f a b is a local minimum for f D 0 fxx a b 0 f a b is a local maximum for f D 0 f a b is neither a local maximum or minimum for f Math 241 Spring 2014 Maxima and Minima 14 7 Examples
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