functionvalues = f[x, y, z] /. points
Reading QR In NoneUsing Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications.
Printing QR Code In NoneUsing Barcode maker for Software Control to generate, create QR-Code image in Software applications.
0,0,0,0,0,0,0,0, 2 , 2 , 2 , 2 , 2 , 2 , 2 , 2 ,0,0,0,0 3 3 3 3 3 3 3 3
Decode QR Code JIS X 0510 In NoneUsing Barcode decoder for Software Control to read, scan read, scan image in Software applications.
QR Code 2d Barcode Creation In Visual C#.NETUsing Barcode maker for .NET framework Control to generate, create Denso QR Bar Code image in .NET applications.
Max[functionvalues] 2 3 Min[functionvalues] 2 3
Printing QR Code In VS .NETUsing Barcode maker for ASP.NET Control to generate, create QR Code 2d barcode image in ASP.NET applications.
QR Creator In .NETUsing Barcode printer for Visual Studio .NET Control to generate, create QR Code image in .NET applications.
SOLVED PROBLEMS
Printing Denso QR Bar Code In VB.NETUsing Barcode creation for VS .NET Control to generate, create QR Code 2d barcode image in Visual Studio .NET applications.
Print DataMatrix In NoneUsing Barcode maker for Software Control to generate, create ECC200 image in Software applications.
10.8 Find all relative extrema of the function f(x, y) = x2 y2. Sketch the surface.
EAN13 Creator In NoneUsing Barcode maker for Software Control to generate, create GTIN - 13 image in Software applications.
Generating ANSI/AIM Code 128 In NoneUsing Barcode printer for Software Control to generate, create Code 128 Code Set B image in Software applications.
SOLUTION
Drawing Bar Code In NoneUsing Barcode encoder for Software Control to generate, create barcode image in Software applications.
EAN128 Generator In NoneUsing Barcode creation for Software Control to generate, create UCC - 12 image in Software applications.
f[x_, y_] = x2 y2; Solve[{ xf[x, y] 0, yf[x, y] 0}, {x, y}] {{x 0, y 0}} 2 d[x_, y_]= {x, 2} f[x, y] (y, 2} f[x, y] ( x, y f[x, y]); d[0, 0] 4
Paint Identcode In NoneUsing Barcode generator for Software Control to generate, create Identcode image in Software applications.
Data Matrix ECC200 Scanner In C#Using Barcode reader for .NET framework Control to read, scan read, scan image in Visual Studio .NET applications.
Negative number; saddle point at (0, 0).
Recognizing Code128 In Visual Basic .NETUsing Barcode scanner for .NET framework Control to read, scan read, scan image in .NET framework applications.
Barcode Creation In Objective-CUsing Barcode generation for iPhone Control to generate, create barcode image in iPhone applications.
Plot3D[f[x, y], {x, 5, 5}, {y, 5, 5}, BoxRatios {1, 1, 1}]
Barcode Creation In NoneUsing Barcode creation for Font Control to generate, create barcode image in Font applications.
ANSI/AIM Code 39 Drawer In JavaUsing Barcode generator for Java Control to generate, create Code 39 Extended image in Java applications.
5 20
DataMatrix Drawer In NoneUsing Barcode generator for Microsoft Word Control to generate, create Data Matrix image in Word applications.
Code-39 Creation In Visual Studio .NETUsing Barcode drawer for VS .NET Control to generate, create Code 39 Full ASCII image in Visual Studio .NET applications.
20 5 0 5
Multivariate Calculus
10.9 Find all relative extrema of the function f ( x , y) = xye x
SOLUTION
2 y2
. Sketch the surface.
f[x_, y_] = x y Exp[ x2 y2] pdx = x f[x, y] //Factor x x
2 y2
( 1 + 2 x2)y x( 1 + 2 y2)
pdy = y f[x, y] //Factor
2 y2
If we try to use Solve to find where the partial derivatives are 0, we will get an error message due 2 2 to the presence of the (non-algebraic) exponential. However, since x y cannot equal zero, we can ignore its presence.
Solve[{ 1 + 2 x2) y 0, x ( 1 + 2 y2) 0, {x, y}]
x 0, y 0}, x 1 , y 1 , x 1 , y 1 , 2 2 2 2 x 1 ,y 1 , x 1 ,y 1 2 2 2 2
2 d[x_, y_] = {x,2} f[x, y] (y,2} f[x, y] ( x,y f[x, y]);
d[0, 0] 1 d[ 1 / 2, 1 / 2] 4 2 {x,2} f[x, y]/.{ x 1 / 2, y 1 / 2 } 2 d[ 1 / 2, 1 / 2] 4 2 {x,2} f[x, y]/.{ x 1 / 2, y 1 / 2 } 2 d[1 / 2, 1 / 2] 4 2 {x,2} f[x, y]/.{ x 1 / 2, y 1 / 2 } 2 d[1/ 2, 1/ 2] 4 2 {x,2} f[x, y]/.{ x 1 / 2, y 1 / 2 } 2
Relative maximum at Relative minimum at Relative minimum at Relative maximum at Negative number; no relative extremum.
1 1 , . 2 2
1 1 , . 2 2
1 1 , . 2 2
1 1 , . 2 2
Multivariate Calculus
We sketch the surface showing two views.
Plot3D[f[x, y], {x, 3, 3}, {y, 3, 3}, PlotPoints 30, ViewPoint {1.391, 3.001, 0.713}, PlotRange All] Plot3D[f[x, y], {x, 3, 3}, {y, 3, 3}, PlotPoints 30, ViewPoint {0.617, 3.318, 0.245}, PlotRange All]
2 2 0.1 0.0 0.1 2 0 2 2 0 2 0 2
2 0.1 0.0 0.1
10.10 Use Lagrange multipliers to find the points on the circle x2 + y2 2x 4y = 0 closest to and farthest from P(4, 4).
SOLUTION
circle = ContourPlot[x2 + y2 2 x 4 y 0, {x, 5, 5 }, {y, 1, 5}]; point = Graphics[{PointSize[.01], Point[{4, 4}]}]; Show[circle, point, Axes True, Frame False, AspectRatio Automatic]
5 4 3 2 1 4 2 1 2 4
f[x_, y_] = (x 4)2 + (y 4)2; g[x_, y_] = x2 + y2 2 x 4 y;
We minimize the square of the distance from P.
conditions = Eliminate[{ xf[x, y] k x g[x, y], yf[x, y] k y g[x, y], g[x, y] 0}, k] 2 x 4 + 3 y & & 52 y + 13 y 2 32 points = Solve[conditions]
x 1 (13 3 65 ), y 2 (13 65 ) , x 1 (13 + 3 65 ), y 2 (13 + 65 ) 13 13 13 13 Based upon the computed distances, the first point is farthest from P and the second point is closest.
f[x,y] /. points //N {5.84162, 1.36948}
Multivariate Calculus
10.11 Find the points on the sphere x2 + y2 + z2 = 1 that are closest to and farthest from (1, 2, 3).
SOLUTION
f[x_, y_, z_] = (x 1)2 + (y 2)2 + (z 3)2; g[x_, y_, z_] = x2 + y2 + z2 1; conditions = Eliminate[{ x f[x, y,z] x g[x, y,z], y f[x, y,z] k y g[x, y,z], z f[x, y,z] k z g[x, y,z], g[x, y,z] 0}, k]
