Math 241 Spring 2014 Jayadev S Athreya Spring 2014 Jayadev S Athreya Math 241 Spring 2014 Approaching along lines We need limits from all directions to exist In 2 variables lines through 0 0 are of the form x at y bt for some a b R For lim x y p1 p2 f x y L we need ALL that is for all a b of the following limits to exist lim f p1 at p2 bt t 0 and to ALL equal L Jayadev S Athreya Math 241 Spring 2014 Approaching along lines In n variables lines through 0 0 0 are of the form x1 a1 t x2 a2 t xn an t for some a1 an R For lim f x L x p we need ALL that is for all a1 an of the following limits to exist lim f p t a1 an t 0 and to ALL equal L Jayadev S Athreya Math 241 Spring 2014 Non existence In particular to show a limit DOESN T exist all you need to show is that One of these directional limits doesn t exist Two of these directional limits are not equal Example the h x y we used above Jayadev S Athreya Math 241 Spring 2014 Polar coordinates In 2 dimensions we can use polar coordinates to compute directional limits Magnitude and Direction For each x y 6 0 0 we can write x y r cos r sin p with r x 2 y 2 tan 1 yx 0 2 Each ray Can be expressed associated to an angle The ray associated to angle is r cos r sin r 0 Jayadev S Athreya Math 241 Spring 2014 Limits and Polar Coordinates Limit along a ray For a function f and a point p p1 p2 R2 and angle we can define the directional limit lim f p1 r cos p2 r sin r 0 For limit to exist All directional limits must exist and be equal Warning This is NECESSARY but NOT SUFFICIENT for limit to exist It s a good way to check that limit DOESN T exist 2r Example h x y x 22xy y 2 limit as r 0 depends on Jayadev S Athreya 2 cos sin r2 2 cos sin So the Math 241 Spring 2014 Important Example Can have limit exist and be equal along all lines but still not exist Consider x 3y M x y 6 x y2 as x y 0 0 Along axes x 0 or y 0 yields M x y 0 so limit along axes is 0 Along other lines x at y bt a b 6 0 a3 bt 2 a3 bt 4 lim 0 t 0 a6 t 4 b 2 t 0 a6 t 6 b 2 t 2 lim M at bt lim t 0 Jayadev S Athreya Math 241 Spring 2014 Important Example Can have limit exist and be equal along all lines but still not exist Consider x 3y M x y 6 x y2 as x y 0 0 Along cubic y x 3 M x x 3 Jayadev S Athreya x6 1 2 x6 x6 Math 241 Spring 2014 Limit Rules The usual limit rules do work suppose f g Rn R and lim f x L lim g x M x p x p Then Sums limx p f x g x L M Products limx p f x g x LM Quotients If M 6 0 limx p f x g x L M Jayadev S Athreya Math 241 Spring 2014 Continuity f Rn R is continuous at p if lim f x f p x p Most functions we look at will be continuous but we need to be careful Jayadev S Athreya Math 241 Spring 2014 Examples None of the below are continuous at 0 f x y g x y h x y sin x 2 y 2 x 2 y 2 x 2 y 2 x 2 y 2 2xy x 2 y 2 Jayadev S Athreya Math 241 Spring 2014 Examples None of the below are continuous at 0 f x y g x y h x y sin x 2 y 2 x 2 y 2 x 2 y 2 x 2 y 2 2xy x 2 y 2 If we define f 0 0 1 we make f continuous Jayadev S Athreya Math 241 Spring 2014 How functions change We will be interested in how functions change and how to approximate them by functions we can understand In 1 variable the derivative played two roles Rate of change f 0 x tells us how fast the function f was changing at x Linear approximation f 0 x is the slope of the tangent line at x f x which gives us a good approximation to the function Jayadev S Athreya Math 241 Spring 2014 Partial derivatives Given a function of 2 variables f x y we can hold one of the variables fixed and let the other one change to get functions of one variable and can compute derivatives of those functions x The partial derivative with respect to x at a b f denoted x a b is given by if the limit exists f f a h b f a b a b lim x h h 0 Other notation Sometimes we write Dx f a b Jayadev S Athreya Math 241 Spring 2014 Partial derivatives Given a function of 2 variables f x y we can hold one of the variables fixed and let the other one change to get functions of one variable and can compute derivatives of those functions y The partial derivative with respect to y at a b denoted f h a b is given by if the limit exists f f a b h f a b a b lim y h h 0 Other notation Sometimes we write Dy f a b Jayadev S Athreya Math 241 Spring 2014 Example Jayadev S Athreya Math 241 Spring 2014 More Examples f x y ex y Jayadev S Athreya Math 241 Spring 2014 More Examples f x y ex y f x y sin xy Jayadev S Athreya Math 241 Spring 2014 More Examples f x y ex y f x y sin xy f x y ex 2 Jayadev S Athreya Math 241 Spring 2014 More Examples f x y ex y f x y sin xy f x y ex 2 f x y x 3 y 2 Jayadev S Athreya Math 241 Spring 2014 Slopes of Tangent Lines Jayadev S Athreya Math 241 Spring 2014 Partial derivatives 3D Given a function of 3 variables f x y z we can hold all but one of the variables fixed and let the other one change to get functions of one variable and can compute derivatives of those functions x The partial derivative with respect to x at a b c f denoted x a b c is given by if the limit exists f f a h b c f a b c a b c lim x h h 0 Other notation Sometimes we …
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