DAY 20 Quadratic approximations Plan: We want to find p2 (x, y) = a0 + a1 (x x0 ) + b1 (y y0 ) + a2 (x x0 )2 + b2 (x x0 )(y y0 ) + c2 (y y0 ) 2 which matches f to second order at (x0 , y0 ). This can be done by choosing the coefficients appropriately. The formula: Near (x0 , y0 ) we have f (x, y) ⇡ f (x0 , y0 ) + @x f (x0 , y0 ) (x x0 ) + @y f (x0 , y0 ) (y y0 ) 1 1 + @x2 f (x0 , y0 ) (x x0 )2 + @x @y f (x0 , y0 ) (x x0 )(y y0 ) + @x2 f (x0 , y0 ) (y y0 )2 2 2 Interpretation: View above formula in three parts: Constant term, linear contributions to form tangent plane, quadratic correction to include some bending. p Example: Write out second-order approximation of cos (x2 + y 2 ) near ( 2⇡ , 0). Mathematica: We can easily view the graph of a function together with various approximations. It is helpful to give each piece a name. Consider this code for the function cos (x2 + y 2 ). . . graph = Plot3D[Cos[x^2 + y^2], {x, -3.5, 3.5}, {y, -3.5, 3.5}, RegionFunction -> Function[{x, y}, x^2 + y^2 < 5], PlotRange -> {-2, 5}, PlotStyle -> Opacity[.5], Mesh -> None, ColorFunction -> Function[{x, y}, Red], AxesLabel -> {"x", "y", "z"}]; plane = Plot3D[ 1/Sqrt[2] - Sqrt[Pi/2] (x - Sqrt[Pi]/2), {x, -3.5, 3.5}, {y, -3.5, 3.5}, PlotStyle -> Opacity[.5],ColorFunction -> Function[{x, y}, Yellow]]; quadratic = Plot3D[ 1/Sqrt[2]-Sqrt[Pi/2](x - Sqrt[Pi]/2)-((2+Pi)/Sqrt[2])(x-Sqrt[Pi]/2)^2-Sqrt[2] y^2, {x, -3.5, 3.5}, {y, -3.5, 3.5}, ColorFunction -> Function[{x, y}, Green]]; point = ListPointPlot3D[{{Sqrt[Pi]/2, 0, 1/Sqrt[2]}}, PlotStyle -> Directive[Blue, PointSize[Large]]]; Show[graph, plane, point] Show[graph, quadratic, point] 48 HOMEWORK PROBLEMS 49 Homework problems (1) Find the following approximations: (a) Linear approximation of the function f (x, y) = x3 3xy + y 3 near the point (2, 1). (b) Linear approximation of the function f (x, y) = arctan xy near the point (1, 1). p (c) Linear approximation of the function f (x, y, z) = x2 + y 2 + z 2 near the point (3, 4, 12). (d) Quadratic approximation of the function f (x, y) = ln(x2 + y 2 ) near the point (1, 0); (e) Quadratic approximation of the function f (x, y, z) = x2 y 2 z 2 near the point (1, 1, 1). (2) This problem concerns the function f1 (x, y) = e x2 +y 2 2 . (a) Use Mathematica to graph this function in the vicinity of the domain point (0, 0). (b) Based on the appearance of this graph predict the equation of the tangent plane to the graph of the function f1 at the domain point (0, 0). (c) Confirm your prediction with computation. That is, find the equation of the tangent plane to the graph of (or find the linearization of) the function f1 at the domain point (0, 0). (d) Use Mathematica to graph the function and its tangent plane (linearization) in the vicinity of the domain point (0, 0). (e) Find the equation of the quadratic (second-order) approximation of the function f1 at the domain point (0, 0). (f) Use Mathematica to graph the function, its tangent plane (linearization), and its quadratic (Taylor) approximation in the vicinity of the domain point (0, 0). (g) Based on the quadratic approximation you found estimate the value of f1 (0.1, 0.2) without using your calculator. Compare with the answer one gets for f1 (0.1, 0.2) using the actual rule for it, and your calculator. 4 4 (3) This problem concerns the function f2 (x, y) = xy x16 y16 . (a) Use Mathematica to graph this function in the vicinity of the domain point (2, 2). (b) Based on the appearance of this graph predict the equation of the tangent plane to the graph of the function f2 at the domain point (2, 2). (c) Confirm your prediction with computation. That is, find the equation of the tangent plane to the graph of (or find the linearization of) the function f2 at the domain point (2, 2). (d) Use Mathematica to graph the function and its tangent plane (linearization) in the vicinity of the domain point (2, 2). (e) Find the equation of the quadratic (Taylor) approximation of the function f2 at the domain point (2, 2). (f) Use “technology” to graph the function, its tangent plane (linearization), and its quadratic (second order) approximation in the vicinity of the domain point (2, 2). p (4) This problem concerns the function f (x, y) = x2 + y 2 nearby the point (3, 4). (Think about what this function represents!) HOMEWORK PROBLEMS 50 @f (a) Compute @f @x (3, 4) and @y (3, 4). What do these two quantities represent? (b) Complete the following sentence. “The distance between a particle and the origin as the particle leaves the point (3, 4) increases at the rate of per each unit of displacement in the x-direction and in the y-direction. (c) A particle is leaving the point (3, 4) with velocity vector h1, 2i. At what rate is the particle’s distance away from the origin changing? (d) Complete the following sentences: “A particle is leaving the point (3, 4) with velocity vector hv1 , v2 i. The distance between a particle and the origin as the particle leaves the . This example is meant to illustrate point (3, 4) changes at the rate of the concept we call .” (e) Find the linear approximation of the function f at the point (3, 4). (f) Judging by the linear approximation you just found, how far way from the origin is the point (3.1, 3.9)? Please do not use a calculator to answer this question. (g) Express the total di↵erential of the function f at the point (3, 4) in terms of dx and dy, and interpret it in words. p (5) Repeat the previous problem for the function f (x, y, z) = x2 + y 2 + z 2 and the point ( 5, 0, 12).

© Copyright 2022 Paperzz