Functions of several variables (§14.1)Functions from R2 to RFunctions from R3 to RLimits and Continuity (§14.2)One-variable limitsHigher-dimensional limitsFunctions of several variables (§14.1)Limits and Continuity (§14.2)Math 241, Spring 2014Jayadev S. AthreyaSpring 2014Jayadev S. Athreya Math 241, Spring 2014Functions of several variables (§14.1)Limits and Continuity (§14.2)Functions from R2to RFunctions from R3to RInputs and outputsA function from R2to R is a machine that takes in two inputsand spits out one output.We write f : R2→ R.Examples:f (x, y) = 2x + 3yf (x, y) = x2+ y2.There are different geometric ways of looking at functions f .Jayadev S. Athreya Math 241, Spring 2014Functions of several variables (§14.1)Limits and Continuity (§14.2)Functions from R2to RFunctions from R3to RGraphsLike in one-variable, we can look at the graph.For a function g : R → R, the graph is a curve in R2, given by{(x, y) : y = f (x)}.For f : R2→ R, it will plotted in 3-space, and will be an surfacesitting in R3:{(x, y, z); z = f (x, y)}3D PlotterJayadev S. Athreya Math 241, Spring 2014Functions of several variables (§14.1)Limits and Continuity (§14.2)Functions from R2to RFunctions from R3to RLevel setsThe level sets of a function f : R2→ R are the sets of points inR2where f has a constant value. That is, given c ∈ R, the levelset of level c is given by{(x, y) ∈ R2: f (x, y) = c}Jayadev S. Athreya Math 241, Spring 2014Functions of several variables (§14.1)Limits and Continuity (§14.2)Functions from R2to RFunctions from R3to RCirclesExample: f (x, y ) = x2+ y2.What are level sets of f ?For c >0For c =0For c<0Jayadev S. Athreya Math 241, Spring 2014Functions of several variables (§14.1)Limits and Continuity (§14.2)Functions from R2to RFunctions from R3to RCirclesExample: f (x, y ) = x2+ y2.What are level sets of f ?For c >0 Circle, centered at 0, of radius√c.For c =0 the origin (0, 0)For c<0 EmptyJayadev S. Athreya Math 241, Spring 2014Functions of several variables (§14.1)Limits and Continuity (§14.2)Functions from R2to RFunctions from R3to RLinesExample: f (x, y ) = 2x + 3yFor any c ∈ R, the level curve is the line 2x + 3y = c, ory =13(c − 2x)Jayadev S. Athreya Math 241, Spring 2014Functions of several variables (§14.1)Limits and Continuity (§14.2)Functions from R2to RFunctions from R3to ROther examplesExponential f (x, y) = exey= ex+y. Level sets are again lines,since they are level sets of x + y.Polynomial f(x, y) = x2− y. Level sets are parabolasy = c − x2.Jayadev S. Athreya Math 241, Spring 2014Functions of several variables (§14.1)Limits and Continuity (§14.2)Functions from R2to RFunctions from R3to RExamplesWhat if we want to consider functions with 3 inputs? That is,functions f : R3→ R?f (x, y, z) = x2+ y2+ z2f (x, y, z) = x + y + zNow, if we wanted to draw the graph, we’d have to work in 4dimensions! Can’t draw this.Jayadev S. Athreya Math 241, Spring 2014Functions of several variables (§14.1)Limits and Continuity (§14.2)Functions from R2to RFunctions from R3to RLevel surfacesBut we can consider level surfaces, places where the functionis constant. That is, given c ∈ R, consider the set of points{(x, y, z) ∈ R3: f (x, y, z) = c}.f (x, y, z) = x2+ y2+ z2. The level sets{(x, y, z) : f (x, y , z) = c} arec > 0 Sphere of radius√c centered at 0.c = 0 The point (0, 0, 0)c < 0 emptyf (x, y, z) = x + y + z For any c ∈ R, the level set is the planex + y + z − c = 0 (which has normal vectorn = (1, 1, 1)).Jayadev S. Athreya Math 241, Spring 2014Functions of several variables (§14.1)Limits and Continuity (§14.2)Functions from R2to RFunctions from R3to RQuadric surfaces§(12.6)Jayadev S. Athreya Math 241, Spring 2014Functions of several variables (§14.1)Limits and Continuity (§14.2)Functions from R2to RFunctions from R3to RQuadric surfaces (§12.6)Jayadev S. Athreya Math 241, Spring 2014Functions of several variables (§14.1)Limits and Continuity (§14.2)Functions from R2to RFunctions from R3to RQuadric surfaces (§12.6)Quadric Surfaces ToolMore in section!Jayadev S. Athreya Math 241, Spring 2014Functions of several variables (§14.1)Limits and Continuity (§14.2)One-variable limitsHigher-dimensional limitsLimitsThe most important concept of calculus is the limit:Informally, we saylimx→af (x) = Lif no matter how close () you want f(x) to get to L, you can finda small neighborhood (of size δ) so that x being within δ of aguarantees that f (x) is within of L.All this relies on is a notion of distance.We have to be careful in higher dimensions, because you canapproach a point in many different ways.Jayadev S. Athreya Math 241, Spring 2014Functions of several variables (§14.1)Limits and Continuity (§14.2)One-variable limitsHigher-dimensional limitsOne-variable limitsThe sentence limx→af (x) = L means the following:For every > 0 (acceptable error)There is δ > 0 (how close you need to be)Such that 0 < |x − a| < δ implies |f (x) −L| < .Jayadev S. Athreya Math 241, Spring 2014Functions of several variables (§14.1)Limits and Continuity (§14.2)One-variable limitsHigher-dimensional limitsOne-variable limits: repriseGiven h ∈ R, h 6= 0, let E(h) = |f (a + h) − L|.The sentence limx→af (x) = L means the following:For every > 0 (acceptable error)There is δ > 0 (how close you need to be)Such that 0 < |h| < δ implies |E(h)| < .Jayadev S. Athreya Math 241, Spring 2014Functions of several variables (§14.1)Limits and Continuity (§14.2)One-variable limitsHigher-dimensional limitsOne-sided limitsIn one-variable, a limit exists if and only if the one-sided limitsexist and are equal.Limit from the left limx→a−f (x) = L ifFor every > 0 (acceptable error)There is δ > 0 (how close you need to be)Such that 0 < a − x < δ implies |f (x) − L| < .Limit from the right limx→a+f (x) = L ifFor every > 0 (acceptable error)There is δ > 0 (how close you need to be)Such that 0 < x − a < δ implies |f (x) −L| < .Jayadev S. Athreya Math 241, Spring 2014Functions of several variables (§14.1)Limits and Continuity (§14.2)One-variable limitsHigher-dimensional limitsOne-sided limits, repriseIn one-variable, a limit exists if and only if the one-sided limitsexist and are equal.Given h ∈ R, h 6= 0, let E(h) = |f (a + h) − L|.Limit from the left limx→a−f (x) = L ifFor every > 0 (acceptable error)There is δ > 0 (how close you need to be)Such that 0 < h < δ implies |E(−h)| < .Limit from
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