Checking differentiability \302\251 2006, Mike May, S.J. - maymk@slu.edu restart; We say a function of two variables is differentiable at a point if the graph near that point can be approximated by the tangent plane. A harder question is how to tell when a function given by a formula is differentiable. This worksheet examines how to check if a function is differentiable at a point.

A standard theorem states that a function is differentiable at a point if both partial derivatives are defined and continuous at that point. Recall that polynomials are continuous functions. Similarly we get a continuous function if we take the sum, difference, product, or composition of two continuous functions. We can also take quotients if the denominator is not 0 and Maple will compute roots if the function is not negative. We also recall that the derivative is not defined when we take a root of 0 or (for even roots) a negative number. Thus with a nice function that is defined in a single algebraic case, we easily note that the function is differentiable at all points, except perhaps at the points where we divide by 0 or take a root of 0 or a negative number.

1) The following functions are from the exercises in the section on differentiability in McCallum's Multivariable Calculus book. For each function, the reasoning above lets us say the function is "nice" everywhere, except perhaps at a small set of points. For each function identify the "difficult" points. (a) f(x, y) = x/y + y/x if x*y \342\211\240 0 and f(x, y) = 0 if x*y = 0. (b) f(x, y) = 2*x*y/(x^2 + y^2)^2 if LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEkYD9gRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn and f(0, 0) = 0. (c) f(x, y) = x*y/sqrt(x^2 + y^2) if 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 and f(0, 0) = 0. (d) f(x, y) = x^2*y/(x^4 + y^2) if 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 and f(0, 0) = 0. (e) f(x, y) = x*y^2/(x^2 + y^2) if 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 and f(0, 0) = 0. (f) f(x, y) = x*y^2/(x^2 + y^4) if 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 and f(0, 0) = 0. (g) f(x, y) = sqrt(abs(x*y)). (h) f(x, y) = x*y*(x^2 - y^2)/(x^2 + y^2) if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYlLUYjNiYtSSNtaUdGJDYmUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJStiYWNrZ3JvdW5kR1EoWzAsMCwwXUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIixGJ0Y3L0Y7USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYmUSJ5RidGNEY3RjpGQUY3RkEtRj42LlElJm5lO0YnRjdGQUZDL0ZHRkVGSEZKRkxGTkZQL0ZTUSwwLjI3Nzc3NzhlbUYnL0ZWRmpuLUkjbW5HRiQ2JVEiMEYnRjdGQUZB and f(0, 0) = 0.

A function can only be differentiable where it is continuous. You may need to review the section on continuity. Recall that a function is continuous at a point if the limit at that point equals the value of the function at that point. Since we are only worried about difficult points, the function will often be defined by a special rule at the point, and by a formula nearby. Consider the function defined in example 4 , 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, unless (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieUYnRi9GMi8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGOQ==) is the origin, in which case we use the special rule 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. The problem is at the origin. The first way to check continuity is to look at a graph to see if the function jumps. plot3d(x*y/(x^2+y^2),x=-1..1, y=-1..1,view=-1..1, axes=boxed, style=patchcontour, shading=z);

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It is clear from looking at the picture that this function is not continuous at the origin. If we look down the z-axis it is clear that all the different contours extend in to the origin. (Technical tip - When looking at functions that might be discontinuous, it is often good to specify the view, or z-range. Otherwise the graph of a function going to infinity can eliminate the other details of the graph. It is also useful to use the shading=z option the that the color scheme is determined by z-value. At discontinuous points lots of colors try to come together.) We would also like an algebraic way to look at continuity. To do that we pick two paths to the trouble spot that seem to follow different contours. We can then take old fashioned limits on the paths and see that they have different limits. (We may find a path that has no limit at all.) For our example we pick the lines y=x and y=-x to approach the origin along. Maple lets us evaluate the limits either symbolically or graphically. func:= (x, y) -> x*y/(x^2+y^2); path1 := func(x,-x); path2 := func(x,x); LimitOnPath1 := limit(path1, x=0); LimitOnPath2 := limit(path2, x=0); plot({path1,path2},x=-1..1, y=-1..1);

Zio2JEkieEc2IkkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKig5JCIiIjklRiwsJiokRisiIiNGLCokRi1GMEYsISIiRiVGJUYl

IyEiIiIiIw==

IyIiIiIiIw==

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Since the limits as we approach the origin of the two paths are different, the function cannot be continuous at the origin.

2) Use a computer plot to show that the function defined by f(x, y) = x^2*y/(x^4 + y^2) when (x, y) \342\211\240 (0, 0) and f(0, 0) = 0 is not continuous at the origin. Find two paths to the origin so that the limits on the paths are different from each other. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Having dealt with the nice points and the really bad points, we are ready to start working in from the ends. Typically, we are looking at points where the main formula would have division by 0, so the function is defined by a second rule at those points. If the function is continuous at that point, the correct value was chosen to bridge the gap. We will use the function defined by f(x, y) = x*y*(x^2 - y^2)/(x^2 + y^2) away from the origin and f(0, 0) = 0. The origin is the difficult point in this problem. A look at the graph shows that we have patched the hole at the origin with the correct value so the function is continuous. func := x*y*(x^2-y^2)/(x^2+y^2); funcx := simplify(diff(func,x)); funcy := simplify(diff(func,y));

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Since we want to determine if the partial derivatives are continuous, the obvious approach is to look at the graphs of the partial derivatives. plot3d(funcx, x=-1..1, y=-1..1,view=-1..1, axes=BOXED); plot3d(funcy, x=-1..1, y=-1..1,view=-1..1, axes=BOXED);

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Looking at the graphs it is clear that the partial derivatives are continuous at the origin. By the theorem from the book, the function is differentiable at the origin, hence everywhere.

3) Use the technique described above to show the function defined by 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 if 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNiYtRiw2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIwRidGQC1GPTYtUSIsRidGQEZCL0ZGRjhGR0ZJRktGTUZPRlEvRlVRLDAuMzMzMzMzM2VtRidGZW5GQEZARkAtRj02LVEiPUYnRkBGQkZFRkdGSUZLRk1GTy9GUlEsMC4yNzc3Nzc4ZW1GJy9GVUZjb0ZlbkZARitGQA== is differentiable everywhere.

As we start working on functions that are continuous but not differentiable, the easiest ones are those where the partial derivatives are not defined. These are function that are not differentiable when we take a cross section in x or y The easiest examples involve absolute values and roots. Let 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. The function is continuous everywhere. When we take the cross sections x=0 and y=0 we get LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYoLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GNlEnbm9ybWFsRicvSSttc2VtYW50aWNzR0YkUSRhYnNGJy8lJW9wZW5HUSkmdmVyYmFyO0YnLyUmY2xvc2VHRj9GOi8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGOA== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GNlEnbm9ybWFsRicvSSttc2VtYW50aWNzR0YkUSRhYnNGJy8lJW9wZW5HUSkmdmVyYmFyO0YnLyUmY2xvc2VHRj9GOi8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGOA== respectively. Both of these curves have corners, so they are not differentiable at 0. Hence the partial derivatives of f(x,y) don't exist. func := (x, y) -> abs(x + y); pathx := func(h,0); pathy := func(0,h); PartialOnPathx := diff(pathx, h); LimitOnPathx := limit(PartialOnPathx, h=0); PartialOnPathy := diff(pathy, h); LimitOnPathy := limit(PartialOnPathy, h=0); plot({PartialOnPathx,PartialOnPathy,pathx,pathy},h=-1..1, y=-1..1);

Zio2JEkieEc2IkkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkYWJzRyUqcHJvdGVjdGVkRzYjLCY5JCIiIjklRjBGJUYlRiU=

LUkkYWJzRyUqcHJvdGVjdGVkRzYjSSJoRzYi

LUkkYWJzRyUqcHJvdGVjdGVkRzYjSSJoRzYi

LUkkYWJzRyUqcHJvdGVjdGVkRzYkIiIiSSJoRzYi

SSp1bmRlZmluZWRHJSpwcm90ZWN0ZWRH

LUkkYWJzRyUqcHJvdGVjdGVkRzYkIiIiSSJoRzYi

SSp1bmRlZmluZWRHJSpwcm90ZWN0ZWRH

NiktJSdDVVJWRVNHNiQ3VTckJCEjNSEiIiQiIzUhIiI3JCQhMW5tbTtwMGsmKiEjOyQiMW5tbTtwMGsmKiEjOzckJCExTEwkMzxYWj0qISM7JCIxTEwkMzxYWj0qISM7NyQkITFubW1UJXAiZSgpISM7JCIxbm1tVCVwImUoKSEjOzckJCExbW1tIjRtKEckKSEjOyQiMW1tbSI0bShHJCkhIzs3JCQhMU1MJDNpLjkheiEjOyQiMU1MJDNpLjkheiEjOzckJCExbm07L1I9MHYhIzskIjFubTsvUj0wdiEjOzckJCEsdkxAXDQoISM2JCIsdkxAXDQoISM2NyQkITFubTsvc2lxbSEjOyQiMW5tOy9zaXFtISM7NyQkISx2eSRwWmkhIzYkIix2eSRwWmkhIzY3JCQhMUxMTCR5YUUiZSEjOyQiMUxMTCR5YUUiZSEjOzckJCExbm1tIj5zJUhhISM7JCIxbm1tIj5zJUhhISM7NyQkITIvKysrTio0KSpcISM8JCIyLysrK04qNCkqXCEjPDckJCEqRGJcYyUhIiokIipEYlxjJSEiKjckJCEqbFN2OSUhIiokIipsU3Y5JSEiKjckJCExbm07LyMpW29QISM7JCIxbm07LyMpW29QISM7NyQkITJPTExMPWV4SiQhIzwkIjJPTExMPWV4SiQhIzw3JCQhMU1MTEwyJGYkSCEjOyQiMU1MTEwyJGYkSCEjOzckJCEyMCsrdmp1PFwjISM8JCIyMCsrdmp1PFwjISM8NyQkITJWTExMQkAnKTQjISM8JCIyVkxMTEJAJyk0IyEjPDckJCEyJyoqKipcUCdwc20iISM8JCIyJyoqKipcUCdwc20iISM8NyQkISxEIjRfYzchIzYkIixEIjRfYzchIzY3JCQhMUpMTCQzeCV6IykhIzwkIjFKTEwkM3gleiMpISM8NyQkITFNTEwzcyRRTSUhIzwkIjFNTEwzcyRRTSUhIzw3JCQhMjkrK0RjeDdBIyEjPSQiMjkrK0RjeDdBIyEjPTckJCExXm9tbTt6cikqISM+JCIxXm9tbTt6cikqISM+NyQkIjJVS0wzLURnNSMhIz0kIjJVS0wzLURnNSMhIz03JCQiMTxMTGV6dzVWISM8JCIxPExMZXp3NVYhIzw3JCQiMCoqKipcUFEjXCIpISM7JCIwKioqKlxQUSNcIikhIzs3JCQiMkJMTCRlIipbSDchIzwkIjJCTEwkZSIqW0g3ISM8NyQkIjIkKioqKioqKnB2eGwiISM8JCIyJCoqKioqKipwdnhsIiEjPDckJCIxKSoqKipcX3FuMiMhIzskIjEpKioqKlxfcW4yIyEjOzckJCIyJSkqKipcaSZwQFsjISM8JCIyJSkqKipcaSZwQFsjISM8NyQkIjEpKioqKlwyJ0hLSCEjOyQiMSkqKioqXDInSEtIISM7NyQkIjFsbW1tWnZPTCEjOyQiMWxtbW1adk9MISM7NyQkIjInKioqKioqXDJnb1AhIzwkIjInKioqKioqXDJnb1AhIzw3JCQiMkNMTGVSPCpmVCEjPCQiMkNMTGVSPCpmVCEjPDckJCIyKCoqKioqKlwpSHhlJSEjPCQiMigqKioqKipcKUh4ZSUhIzw3JCQiMlltbSJIIW8tKlwhIzwkIjJZbW0iSCFvLSpcISM8NyQkIjEqKioqXDdrLjZhISM7JCIxKioqKlw3ay42YSEjOzckJCIxbW1tO1dUQWUhIzskIjFtbW07V1RBZSEjOzckJCIxKioqKlxpISozYGkhIzskIjEqKioqXGkhKjNgaSEjOzckJCIxTUxMTCp6eW0nISM7JCIxTUxMTCp6eW0nISM7NyQkIjFMTEwzTjEjNCghIzskIjFMTEwzTjEjNCghIzs3JCQiMW1tO0hZdDd2ISM7JCIxbW07SFl0N3YhIzs3JCQiKXhHKip5ISIpJCIpeEcqKnkhIik3JCQiMW1tbVQ2S1UkKSEjOyQiMW1tbVQ2S1UkKSEjOzckJCIxTExMTGJkUSgpISM7JCIxTExMTGJkUSgpISM7NyQkIixET2w1OyohIzYkIixET2w1OyohIzY3JCQiLHYuVWFjKiEjNiQiLHYuVWFjKiEjNjckJCIjNSEiIiQiIzUhIiItJSZDT0xPUkc2JiUkUkdCRyQiIzUhIiIkIiIhISIiJCIiISEiIi0lJ0NVUlZFU0c2JDdobjckJCEjNSEiIiQhIzUhIiI3JCQhMW5tbTtwMGsmKiEjOyQhIzUhIiI3JCQhMUxMJDM8WFo9KiEjOyQhIzUhIiI3JCQhMW5tbVQlcCJlKCkhIzskISM1ISIiNyQkITFtbW0iNG0oRyQpISM7JCEjNSEiIjckJCExTUwkM2kuOSF6ISM7JCEjNSEiIjckJCExbm07L1I9MHYhIzskISM1ISIiNyQkISx2TEBcNCghIzYkISM1ISIiNyQkITFubTsvc2lxbSEjOyQhIzUhIiI3JCQhLHZ5JHBaaSEjNiQhIzUhIiI3JCQhMUxMTCR5YUUiZSEjOyQhIzUhIiI3JCQhMW5tbSI+cyVIYSEjOyQhIzUhIiI3JCQhMi8rKytOKjQpKlwhIzwkISM1ISIiNyQkISpEYlxjJSEiKiQhIzUhIiI3JCQhKmxTdjklISIqJCEjNSEiIjckJCExbm07LyMpW29QISM7JCEjNSEiIjckJCEyT0xMTD1leEokISM8JCEjNSEiIjckJCExTUxMTDIkZiRIISM7JCEjNSEiIjckJCEyMCsrdmp1PFwjISM8JCEjNSEiIjckJCEyVkxMTEJAJyk0IyEjPCQhIzUhIiI3JCQhMicqKioqXFAncHNtIiEjPCQhIzUhIiI3JCQhLEQiNF9jNyEjNiQhIzUhIiI3JCQhMUpMTCQzeCV6IykhIzwkISM1ISIiNyQkITFNTEwzcyRRTSUhIzwkISM1ISIiNyQkITI5KytEY3g3QSMhIz0kISM1ISIiNyQkITFeb21tO3pyKSohIz4kISM1ISIiNyQkIjEuIjNGcF55IVIhIz4kIiM1ISIiNyQkIjJlSTNfXVwobzwhIz4kIiM1ISIiNyQkIjEsZTlUUXJZSiEjPSQiIzUhIiI3JCQiMShIJDN4InlZXyUhIz0kIiM1ISIiNyQkIjEpR2UqW29nIUcoISM9JCIjNSEiIjckJCIyeUskM19ObC41ISM9JCIjNSEiIjckJCIxRSRla0dSW2IiISM8JCIjNSEiIjckJCIyVUtMMy1EZzUjISM9JCIjNSEiIjckJCIyMUskZSpbJ1IzSyEjPSQiIzUhIiI3JCQiMTxMTGV6dzVWISM8JCIjNSEiIjckJCIxYG1tbUorSWkhIzwkIiM1ISIiNyQkIjAqKioqXFBRI1wiKSEjOyQiIzUhIiI3JCQiMkJMTCRlIipbSDchIzwkIiM1ISIiNyQkIjIkKioqKioqKnB2eGwiISM8JCIjNSEiIjckJCIxKSoqKipcX3FuMiMhIzskIiM1ISIiNyQkIjIlKSoqKlxpJnBAWyMhIzwkIiM1ISIiNyQkIjEpKioqKlwyJ0hLSCEjOyQiIzUhIiI3JCQiMWxtbW1adk9MISM7JCIjNSEiIjckJCIyJyoqKioqKlwyZ29QISM8JCIjNSEiIjckJCIyQ0xMZVI8KmZUISM8JCIjNSEiIjckJCIyKCoqKioqKlwpSHhlJSEjPCQiIzUhIiI3JCQiMlltbSJIIW8tKlwhIzwkIiM1ISIiNyQkIjEqKioqXDdrLjZhISM7JCIjNSEiIjckJCIxbW1tO1dUQWUhIzskIiM1ISIiNyQkIjEqKioqXGkhKjNgaSEjOyQiIzUhIiI3JCQiMU1MTEwqenltJyEjOyQiIzUhIiI3JCQiMUxMTDNOMSM0KCEjOyQiIzUhIiI3JCQiMW1tO0hZdDd2ISM7JCIjNSEiIjckJCIpeEcqKnkhIikkIiM1ISIiNyQkIjFtbW1UNktVJCkhIzskIiM1ISIiNyQkIjFMTExMYmRRKCkhIzskIiM1ISIiNyQkIixET2w1OyohIzYkIiM1ISIiNyQkIix2LlVhYyohIzYkIiM1ISIiNyQkIiM1ISIiJCIjNSEiIi0lJkNPTE9SRzYmJSRSR0JHJCIyJ1xfJ2VNJXlnPiEjPCQiMWlqOS5lQFIhKSEjOyQiMidcXydlTSV5Zz4hIzwtJSVWSUVXRzYkOyQhIzUhIiIkIiM1ISIiOyQhIzUhIiIkIiM1ISIiLSUrQVhFU0xBQkVMU0c2J0kiaEc2IkkieUc2Ii0lJUZPTlRHNiUlIUclIUciIzUlK0hPUklaT05UQUxHJStIT1JJWk9OVEFMRy0lKkFYRVNTVFlMRUc2IyUnTk9STUFMRy0lKFNDQUxJTkdHNiMlLlVOQ09OU1RSQUlORURHLSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIkIT4hIiItJSlCT1VORFNfWUc2IyQiJD8iISIiLSUtQk9VTkRTX1dJRFRIRzYjJCIlcU8hIiItJS5CT1VORFNfSEVJR0hURzYjJCIlXVAhIiItJSlDSElMRFJFTkc2Ig==

4) Show that the function 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 is not differentiable at the origin.

The text points out that a function can be differentiable even if the partial derivatives are not continuous. We want some way to show that a function is not differentiable. The trick is to notice that for a differentiable function, all the tangent vectors at a point lie in a plane. Suppose that F(x,y) is differentiable at (a,b) with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNictSSVtc3ViR0YkNiUtRiw2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYsNiVRInhGJ0Y5RjwvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y9USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGRy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGUi8lKXN0cmV0Y2h5R0ZSLyUqc3ltbWV0cmljR0ZSLyUobGFyZ2VvcEdGUi8lLm1vdmFibGVsaW1pdHNHRlIvJSdhY2NlbnRHRlIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0Zbby1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRImFGJ0Y5RjwtRk02LVEiLEYnRkdGUC9GVEY7RlVGV0ZZRmVuRmduRmluL0Zdb1EsMC4zMzMzMzMzZW1GJy1GLDYlUSJiRidGOUY8RkRGR0ZHRkRGRy1GTTYtUSI9RidGR0ZQRlNGVUZXRllGZW5GZ24vRmpuUSwwLjI3Nzc3NzhlbUYnL0Zdb0ZjcC1GLDYlUSJjRidGOUY8RkRGR0YrRkRGRw== and 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. Let U=[u_1, u_2] be a unit vector. Then the directional derivative in direction U must be the same whether we compute it from the definition of the directional derivative or by taking the dot product of U with the gradient vector. Consider the function defined by f(x,y) = (x^2*y)/(x^2+y^2). The comments about easy points tell us that this function is differentiable everywhere except perhaps at the origin. Look at the graph of the function and its partial derivatives. func:= (x, y) -> x^2*y/(x^2+y^2); funcx := simplify(diff(func(x,y),x)); funcy := simplify(diff(func(x,y),y)); plot3d(func(x,y), x=-1..1, y=-1..1,view=-1..1, axes=BOXED); plot3d(funcx, x=-1..1, y=-1..1,view=-1..3, axes=BOXED); plot3d(funcy, x=-1..1, y=-1..1,view=-3..1, axes=BOXED);

Zio2JEkieEc2IkkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKig5JCIiIzklIiIiLCYqJEYrRixGLiokRi1GLEYuISIiRiVGJUYl

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We see from the graphs that the function is continuous and that the partial derivatives are discontinuous.

We can simply have Maple compute the directional derivatives by both methods and compare them. func:= (x, y) -> x^2*y/(x^2+y^2); funcxorg := limit(diff(func(x,0),x),x=0); funcyorg := limit(diff(func(0,y),y),y=0); dirderiv1 := theta -> cos(theta)*funcxorg+sin(theta)*funcyorg; dirderiv2 := theta -> limit((func(h*cos(theta), h*sin(theta))-0)/h,h=0); plot([dirderiv1(theta), dirderiv2(theta)], theta=0..2*Pi, axes=boxed, legend=["Using partials", "Directional Dirivative"]);

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IiIh

IiIh

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Since the two ways of computing directional derivatives do not always agree (and in fact disagree everywhere but on the axes) we conclude that the function is not differentiable. We can also look at where we expect points to be based on the two methods of finding the derivative. Red points are computed using the gradient method. Blue points use directional derivatives. del := 2; surf:= plot3d(func(x,y), x=-del..del, y=-del..del, view=-del..del, axes=boxed, color=green, style = patchcontour): tanring1 :=plots[spacecurve]([cos(theta), sin(theta), dirderiv1(theta)], theta=0..2*Pi, color=red, thickness=2): tanring2 :=plots[spacecurve]([cos(theta), sin(theta), dirderiv2(theta)], theta=0..2*Pi, color=blue, thickness=2): plots[display3d]({surf, tanring1, tanring2}, view=[-2..2, -2..2, -4..4]);

IiIj

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One interesting feature of this function is that the graph looks the same no matter how far you zoom in or out. (The picture looks the same if del is 1 or 0.000001.) del := 0.0001; plot3d(func(x,y), x=-del..del, y=-del..del, view=-del..del, axes=normal, style=patchcontour, orientation=[16,39]);

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We don't need to compute the directional derivative by both methods in every direction to show that a function is not differentiable. We simply need to show that the two methods give different answers in some direction. Consider the function that sends (x,y) to 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 and sends the origin to 0. Once again, the comments about easy points tell us that this function is differentiable everywhere except perhaps at the origin. We compute the partial derivatives at the origin. func := (x, y) -> x^2*y/(x^2+y^2) + sin(x) -2*y; funcxorg := eval(diff(func(x,0),x),x=0); funcyorg := eval(diff(func(0,y),y),y=0);

Zio2JEkieEc2IkkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgqKDkkIiIjOSUiIiIsJiokRixGLUYvKiRGLkYtRi8hIiJGLy1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRixGL0YuISIjRiVGJUYl

IiIi

ISIj

Now we compute the directional derivative along the line y=x (theta=Pi/4) using both methods. dirderiv1 := theta -> cos(theta)*funcxorg+sin(theta)*funcyorg: dirderiv2 := theta -> limit((func(h*cos(theta), h*sin(theta))-0)/h,h=0): "Deriv from partials" = dirderiv1(Pi/4); "Deriv along path" = dirderiv2(Pi/4);

L1E0RGVyaXZ+ZnJvbX5wYXJ0aWFsczYiLCQqJCIiIyMiIiJGJyMhIiJGJw==

L1ExRGVyaXZ+YWxvbmd+cGF0aDYiLCQqJCIiIyMiIiJGJyMhIiIiIiU=

Since we get different answers, we conclude that the function is not differentiable at the origin.

5) Show that the function in 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 is not differentiable at the origin even though it is continuous there. 6) Determine the status of the function defined by f(x, y) = x*y^2/(x^2 + y^4) if (x,y) \342\211\240 (0, 0) and f(0, 0) =0 at the origin. Is it not continuous, continuous but not differentiable, or differentiable at the origin?

The exercises for this section have been set up with the difficult point at the origin and with the functions symmetric about the origin. Such functions are often easier to describe in terms of polar coordinates. We can use Maple to convert functions to polar coordinates. makepolar := g -> simplify(subs({x=r*cos(theta), y=r*sin(theta)}, g));

Zio2I0kiZ0c2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2Iy1JJXN1YnNHRiw2JDwkL0kieEdGJSomSSJyR0YlIiIiLUkkY29zR0YrNiNJJnRoZXRhR0YlRjcvSSJ5R0YlKiZGNkY3LUkkc2luR0YrRjpGNzkkRiVGJUYl

Now look at the function from problem 3 func := x*y/sqrt(x^2+y^2); polarfunc := makepolar(func);

KihJInhHNiIiIiJJInlHRiRGJSwmKiRGIyIiI0YlKiRGJkYpRiUjISIiRik=

KipJInJHNiIiIiItSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0kmdGhldGFHRiRGJS1JJHNpbkdGKEYrRiUtSSVjc2duR0YoNiNGI0Yl

The function csgn(r) is a sign function that is either 1 or -1. Looking at the polar form, it is clear the the function is continuous at the origin. We also can see that the function is zero everywhere on the x or y axis. Time do check differentiability as r approaches 0. drfunc := diff(polarfunc,r); drfunc0 := limit(drfunc, r=0);

LCYqKC1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSZ0aGV0YUdGKSIiIi1JJHNpbkdGJkYqRiwtSSVjc2duR0YmNiNJInJHRilGLEYsKipGMkYsRiRGLEYtRiwtRjA2JEYsRjJGLEYs

SSp1bmRlZmluZWRHJSpwcm90ZWN0ZWRH

The problem we are running into is the csgn(r), which is 1 in one direction and -1 in the other. drfunc0r := limit[dir](drfunc, r=0, right); drfunc0l := limit[dir](drfunc, r=0, left);

KiYtSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kmdGhldGFHRigiIiItSSRzaW5HRiVGKUYr

LCQqJi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSZ0aGV0YUdGKSIiIi1JJHNpbkdGJkYqRiwhIiI=

Thus for most values of theta we get a corner, not a tangent line. (The right hand and left hand limits for the derivative exist, but they are different from each other.) The function is clearly not differentiable at the origin.

7) Convert the functions in parts (b)-(h) of exercise 1 into polar form. Comment about continuity and differentiability at the origin.