Partial Derivatives (§14.3)Tangent Planes and Differentiability (§14.4)Math 241, Spring 2014Jayadev S. AthreyaSpring 2014Jayadev S. Athreya Math 241, Spring 2014Approaching along linesWe need limits from all directions to exist.In 2-variables, lines through (0, 0) are of the formx = at, y = bt(for some a , b ∈ R). Forlim(x,y)→(p1,p2)f (x, y) = L,we need ALL (that is, for all a, b) of the following limits to exist:limt→0f (p1+ at, p2+ bt)and to ALL equal L.Jayadev S. Athreya Math 241, Spring 2014Approaching along linesIn n-variables, lines through (0, 0, . . . , 0) are of the formx1= a1t, x2= a2t, . . . , xn= ant(for some a1, . . . an∈ R). Forlimx→pf (x) = L,we need ALL (that is, for all a1, . . . , an) of the following limits toexist:limt→0f (p + t(a1, . . . , an))and to ALL equal L.Jayadev S. Athreya Math 241, Spring 2014Non-existenceIn particular, to show a limit DOESN’T exist, all you need toshow 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 2014Polar coordinatesIn 2 dimensions, we can use polar coordinates to computedirectional limits:Magnitude and Direction For each (x, y) 6= (0, 0) we can write(x, y) = (r cos θ, r sin θ),with r =px2+ y2, θ = tan−1yx∈ [0, 2π)Each ray Can be expressed associated to an angle θ. Theray associated to angle θ is{(r cos θ, r sin θ) : r > 0}.Jayadev S. Athreya Math 241, Spring 2014Limits and Polar CoordinatesLimit along a ray For a function f and a point p = (p1, p2) ∈ R2,and angle θ we can define the directional limitlimr→0f (p1+ r cos θ, p2+ r sin θ).For limit to exist All directional limits must exist and be equal.Warning This is NECESSARY, but NOT SUFFICIENT forlimit to exist. It’s a good way to check that limitDOESN’T exist.Example: h(x, y) =2xyx2+y2=2r2cos θ sin θr2= 2 cos θ sin θ. So thelimit as r → 0 depends on θ!Jayadev S. Athreya Math 241, Spring 2014Important ExampleCan have limit exist and be equal along all lines, but still notexist: ConsiderM(x, y) =x3yx6+ y2as (x, y) → (0, 0).Along axes x = 0 or y = 0 yields M(x, y) = 0, so limit alongaxes is 0.Along other lines x = at, y = bt(a, b 6= 0),limt→0M(at, bt) = limt→0a3bt4a6t6+ b2t2= limt→0a3bt2a6t4+ b2= 0Jayadev S. Athreya Math 241, Spring 2014Important ExampleCan have limit exist and be equal along all lines, but still notexist: ConsiderM(x, y) =x3yx6+ y2as (x, y) → (0, 0).Along cubic y = x3,M(x, x3) =x6x6+ x6= 1/2Jayadev S. Athreya Math 241, Spring 2014Limit RulesThe usual limit rules do work: suppose f , g : Rn→ R, andlimx→pf (x) = L, limx→pg(x) = M.Then:Sums limx→pf (x) + g(x) = L + MProducts limx→pf (x)g(x) = LMQuotients If M 6= 0, limx→pf (x)/g(x) = L/MJayadev S. Athreya Math 241, Spring 2014Continuityf : Rn→ R is continuous at p iflimx→pf (x) = f (p).Most functions we look at will be continuous, but we need to becareful.Jayadev S. Athreya Math 241, Spring 2014ExamplesNone of the below are continuous at 0:f (x, y) =sin(x2+y2)x2+y2g(x, y) =x2−y2x2+y2h(x, y ) =2xyx2+y2If we define f (0, 0) = 1, we make f continuous.Jayadev S. Athreya Math 241, Spring 2014ExamplesNone of the below are continuous at 0:f (x, y) =sin(x2+y2)x2+y2g(x, y) =x2−y2x2+y2h(x, y ) =2xyx2+y2If we define f (0, 0) = 1, we make f continuous.Jayadev S. Athreya Math 241, Spring 2014How functions changeWe will be interested in how functions change, and how toapproximate them by functions we can understand. In1-variable, the derivative played two roles:Rate of change f0(x) tells us how fast the function f waschanging at x.Linear approximation f0(x) is the slope of the tangent line at(x, f (x)), which gives us a good approximation tothe function.Jayadev S. Athreya Math 241, Spring 2014Partial derivativesGiven a function of 2 variables f (x, y), we can hold one of thevariables fixed and let the other one change, to get functions ofone-variable, and can compute derivatives of those functions:∂∂xThe partial derivative with respect to x at (a, b),denoted∂f∂x(a, b) is given by (if the limit exists)∂f∂x(a, b) = limh→0f (a + h, b) − f (a, b)hOther notation Sometimes we write (Dxf )(a, b)Jayadev S. Athreya Math 241, Spring 2014Partial derivativesGiven a function of 2 variables f (x, y), we can hold one of thevariables fixed and let the other one change, to get functions ofone-variable, and can compute derivatives of those functions:∂∂yThe partial derivative with respect to y at (a, b),denoted∂f∂h(a, b) is given by (if the limit exists)∂f∂y(a, b) = limh→0f (a, b + h) − f (a, b)hOther notation Sometimes we write (Dyf )(a, b)Jayadev S. Athreya Math 241, Spring 2014ExampleJayadev S. Athreya Math 241, Spring 2014More Examplesf (x, y) = ex+yf (x, y) = sin(xy)f (x, y) = ex2f (x, y) = x3y2Jayadev S. Athreya Math 241, Spring 2014More Examplesf (x, y) = ex+yf (x, y) = sin(xy)f (x, y) = ex2f (x, y) = x3y2Jayadev S. Athreya Math 241, Spring 2014More Examplesf (x, y) = ex+yf (x, y) = sin(xy)f (x, y) = ex2f (x, y) = x3y2Jayadev S. Athreya Math 241, Spring 2014More Examplesf (x, y) = ex+yf (x, y) = sin(xy)f (x, y) = ex2f (x, y) = x3y2Jayadev S. Athreya Math 241, Spring 2014Slopes of Tangent Lines
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