(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2"; MacintoshStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8, 24, "Times"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, "Times"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6, 14, "Times"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20, 18, "Times"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15, 14, "Times"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12, 12, "Times"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-4, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-4, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R32768, L-4, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-4, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B32768, L-4, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = leftheader, inactive, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, 12, "Times"; fontset = leftfooter, inactive, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; paletteColors = 128; showRuler; currentKernel; ] :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] Math 225: Calculus III Assignment 5 :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] Name: :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] Section: :[font = text; inactive; preserveAspect; startGroup] I affirm that the solutions presented in this assignment are entirely my own work :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; endGroup] Signature: :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Instructions :[font = text; inactive; preserveAspect] This assignment contains problems involving functions of several variables: the chain rule, tangent planes, the gradient of a function, local maxima/minima & saddle points on the graph of a function, see Chapter 13 in Finney & Thomas or Chapter 3 in the Lecture Notes. Before starting this assignment you may want to review the notebooks Derivatives, TangentsNormals, and AdvancedCalculusDemo (in the Notebook folder or the back of the Lecture Notes). The functions Grad[], CriticalPoints[], and Discriminant[] are used for the first time in this assignment. Recall that to substitute a value into an expression use: expr /. var -> value. For example, x^2 - y^2 /. x->2 4 - y^2 If an expression has already been defined as a function, then just use the function notation to get values. For example f[x_,y_] = x^2 - y^2; f[2,y] 4 - y^2 To plot the level curve f[x,y] == c, use ContourPlot[f[x,y],{x,minx,maxx},{y,miny,maxy}, Contours->{c}, ContourShading->False] To plot a level surface f[x,y,z] == c, use ContourPlot3D[f[x,y,z],{x,minx,maxx},{y,miny,maxy}, {z,minz,maxz}, Contours->{c}] To get a line segment from p to q representing a 3D vector use Vector[p,q] and combine it with other plots with the Show[] command. Be sure to type in comments explaining what you are doing. Remember to uncheck Show In/Out Names in the File menu and close this group before printing. ;[s] 44:0,0;339,3;350,0;351,3;367,0;372,3;393,0;467,1;473,0;475,1;491,0;496,1;511,0;619,1;641,0;654,1;675,0;676,1;680,2;687,0;808,1;849,2;856,1;857,0;882,1;893,0;899,1;988,0;1013,1;1026,0;1031,1;1117,0;1145,1;1146,0;1150,1;1151,0;1181,1;1192,0;1235,1;1241,0;1332,1;1349,0;1357,1;1361,0;1405,-1; 4:21,13,9,Times,0,12,0,0,0;18,13,10,Courier,1,12,0,0,0;2,13,10,Courier,0,12,0,0,0;3,13,9,Times,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] Initialization :[font = input; initialization; noKeepOnOnePage; preserveAspect] *) <1, s->2} :[font = output; output; inactive; preserveAspect; endGroup] (5*Cos[1/2 + 2*(1 + E^2)])/2 ;[o] 1 2 5 Cos[- + 2 (1 + E )] 2 --------------------- 2 :[font = input; preserveAspect; startGroup] N[%] :[font = output; output; inactive; preserveAspect; endGroup] -0.001618492093354384 ;[o] -0.00161849 :[font = input; preserveAspect; startGroup] D[w[x[r,s],y[r,s]],s] :[font = output; output; inactive; preserveAspect; endGroup] (2*E^s - r/s^2)*Cos[2*(E^s + r) + r/s] ;[o] s r s r (2 E - --) Cos[2 (E + r) + -] 2 s s :[font = input; preserveAspect; startGroup] % /. {r->1, s->2} :[font = output; output; inactive; preserveAspect; endGroup] (-1/4 + 2*E^2)*Cos[1/2 + 2*(1 + E^2)] ;[o] 1 2 1 2 (-(-) + 2 E ) Cos[- + 2 (1 + E )] 4 2 :[font = input; preserveAspect; startGroup] N[%] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] -0.00940545388944156 ;[o] -0.00940545 :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] Problem 2 :[font = text; inactive; preserveAspect] Let f[x_,y_] = x^3 - x^2 y + x y^2 - y^3. a) Find the gradient of f[x,y], Grad[f[x,y],{x,y}]. b) Find the direction in which f[x_,y_] increases most rapidly at the point {2, 1} and find the derivative of f[x,y] in that direction. c) Plot the level curve of f[x,y] through the point {1,2}. ;[s] 23:0,0;4,1;41,0;42,2;45,0;66,1;72,0;75,1;93,0;95,2;97,0;126,1;134,0;171,1;177,0;205,1;211,0;231,2;233,0;258,1;264,0;283,1;288,0;290,-1; 3:12,13,9,Times,0,12,0,0,0;8,13,10,Courier,1,12,0,0,0;3,13,9,Times,2,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect] Solution :[font = input; preserveAspect; startGroup] f[x_,y_] = x^3 - x^2 y + x y^2 - y^3 :[font = output; output; inactive; preserveAspect; endGroup] x^3 - x^2*y + x*y^2 - y^3 ;[o] 3 2 2 3 x - x y + x y - y :[font = text; inactive; preserveAspect] a) ;[s] 2:0,1;2,0;3,-1; 2:1,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect; startGroup] Grad[f[x,y],{x,y}] :[font = output; output; inactive; preserveAspect; endGroup] {3*x^2 - 2*x*y + y^2, -x^2 + 2*x*y - 3*y^2} ;[o] 2 2 2 2 {3 x - 2 x y + y , -x + 2 x y - 3 y } :[font = text; inactive; preserveAspect] b) The direction in which f[x_,y_] increases most rapidly at the point {2, 1} is given by the gradient and find the derivative in that direction by the norm of the gradient. ;[s] 6:0,1;2,0;27,2;35,0;72,2;78,0;175,-1; 3:3,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] Grad[f[x,y],{x,y}] /. {x->2, y->1} :[font = output; output; inactive; preserveAspect; endGroup] {9, -3} ;[o] {9, -3} :[font = input; preserveAspect; startGroup] Norm[%] :[font = output; output; inactive; preserveAspect; endGroup] 3*10^(1/2) ;[o] 3 Sqrt[10] :[font = input; preserveAspect; startGroup] N[%] :[font = output; output; inactive; preserveAspect; endGroup] 9.48683298050514 ;[o] 9.48683 :[font = text; inactive; preserveAspect] c) The level curve through {1,2} is given by f[x,y] == f[1,2]. ;[s] 6:0,1;2,0;27,2;32,0;45,2;61,0;63,-1; 3:3,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] f[1,2] :[font = output; output; inactive; preserveAspect; endGroup] -5 ;[o] -5 :[font = input; preserveAspect; endGroup] ContourPlot[f[x,y], {x,-1,3}, {y,0,4}, Contours->{-5}, ContourShading->False] :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] Problem 3 :[font = text; inactive; preserveAspect] Let f[x_,y_,z_] = y + z^2 - x^2. a) Plot the level surface f[x,y,z] == 1 b) Find Grad[f[x,y,z]] at the point {1,2,0} and plot this vector with the level surface from part a). ({1,2,0} is on this level surface). ;[s] 17:0,0;4,1;31,0;34,2;36,0;60,1;73,0;74,2;76,0;82,1;96,0;110,1;117,0;172,2;174,0;177,1;184,0;212,-1; 3:9,13,9,Times,0,12,0,0,0;5,13,10,Courier,1,12,0,0,0;3,13,9,Times,2,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect] Solution :[font = input; preserveAspect; startGroup] f[x_,y_,z_] = y + z^2 - x^2 :[font = output; output; inactive; preserveAspect; endGroup] -x^2 + y + z^2 ;[o] 2 2 -x + y + z :[font = text; inactive; preserveAspect] a) ;[s] 2:0,1;3,0;4,-1; 2:1,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect] cp = ContourPlot3D[f[x,y,z],{x,0,2},{y,1,3},{z,-1,1}, Contours->{1}] :[font = text; inactive; preserveAspect] b) ;[s] 2:0,1;2,0;3,-1; 2:1,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect; startGroup] Grad[f[x,y,z]] :[font = output; output; inactive; preserveAspect; endGroup] {-2*x, 1, 2*z} ;[o] {-2 x, 1, 2 z} :[font = input; preserveAspect; startGroup] v = Grad[f[x,y,z]] /. {x->1, y->2, z->0} :[font = output; output; inactive; preserveAspect; endGroup] {-2, 1, 0} ;[o] {-2, 1, 0} :[font = input; preserveAspect; startGroup] w = Vector[{1,2,0}, {1,2,0} + v] :[font = output; output; inactive; preserveAspect; endGroup] Graphics3D["<<>>"] ;[o] -Graphics3D- :[font = input; preserveAspect; endGroup] Show[cp, w,ViewPoint->{-0.804, -2.566, 2.054}] :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] Problem 4 :[font = text; inactive; preserveAspect] a) Find the equation for the plane tangent to the level surface z - x^2 + y^2 == 2 at the point {1, 1, 2}. b) Graph the surface together with this tangent plane. (Suggestion: choose a domain for the tangent plane smaller than the domain for the surface so that the point of tangency is easy to see.) ;[s] 8:0,2;2,0;64,1;82,0;96,1;105,0;107,2;109,0;300,-1; 3:4,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0;2,13,9,Times,2,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect] Solution :[font = text; inactive; preserveAspect] a) The normal vector of the tangent plane is given by the gradient of z - x^2 + y^2 at {1,1,2}. ;[s] 5:0,1;2,0;70,2;83,0;87,2;96,-1; 3:2,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] n = Grad[z - x^2 + y^2] /. {x->1, y->1, z->2} :[font = output; output; inactive; preserveAspect; endGroup] {-2, 2, 1} ;[o] {-2, 2, 1} :[font = text; inactive; preserveAspect] The equation of the tangent plane is then :[font = input; preserveAspect; startGroup] n.{x,y,z} == n.{1,1,2} :[font = output; output; inactive; preserveAspect; endGroup] -2*x + 2*y + z == 2 ;[o] -2 x + 2 y + z == 2 :[font = text; inactive; preserveAspect] b) ;[s] 1:0,1;3,-1; 2:0,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect] tp = Plot3D[2+2x-2y, {x,0,2}, {y,0,2}] :[font = input; preserveAspect] s = Plot3D[2+x^2-y^2, {x,-1,3}, {y,-1,3}] :[font = input; preserveAspect; endGroup] Show[s, tp, ViewPoint->{-2.464, -1.786, 1.479}] :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] Problem 5 :[font = text; inactive; preserveAspect] Let f[x_,y_] = 3 y - y^3 - x^2 a) Test the function f[x,y] for local maxima, local minima, and saddle points. b) Find the value of f[x,y] at the points found in a). c) Produce a graph of f[x,y]that includes the points found in a). ;[s] 19:0,0;4,2;31,0;32,1;34,0;53,2;59,0;111,1;114,0;132,2;138,0;162,1;164,0;166,1;169,0;188,2;194,0;228,1;230,0;232,-1; 3:10,13,9,Times,0,12,0,0,0;5,13,9,Times,2,12,0,0,0;4,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect] Solution :[font = text; inactive; preserveAspect] a) First find critical points. ;[s] 2:0,1;2,0;31,-1; 2:1,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect; startGroup] f[x_,y_] = 3 y - y^3 - x^2 :[font = output; output; inactive; preserveAspect; endGroup] -x^2 + 3*y - y^3 ;[o] 2 3 -x + 3 y - y :[font = input; preserveAspect; startGroup] cp = CriticalPoints[f[x,y]] :[font = output; output; inactive; preserveAspect; endGroup] {{x -> 0, y -> -1}, {x -> 0, y -> 1}} ;[o] {{x -> 0, y -> -1}, {x -> 0, y -> 1}} :[font = text; inactive; preserveAspect] Now the second derivative test. :[font = input; preserveAspect; startGroup] d = Discriminant[f[x,y]] :[font = output; output; inactive; preserveAspect; endGroup] 12*y ;[o] 12 y :[font = input; preserveAspect; startGroup] d /. cp :[font = output; output; inactive; preserveAspect; endGroup] {-12, 12} ;[o] {-12, 12} :[font = text; inactive; preserveAspect] The point {0,-1} is a saddle point. The point {0,1} is either a local max or min. Test D[f[x,y],x,x] to find out. ;[s] 7:0,0;10,1;16,0;46,1;51,0;87,1;100,0;114,-1; 2:4,13,9,Times,0,12,0,0,0;3,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] D[f[x,y],x,x] /. {x->0, y->1} ;[s] 2:0,0;29,1;30,-1; 2:1,12,10,Courier,1,12,0,0,0;1,12,9,Times,0,12,0,0,0; :[font = output; output; inactive; preserveAspect; endGroup] -2 ;[o] -2 :[font = text; inactive; preserveAspect] This means there is a local max at {0,1}. ;[s] 3:0,0;35,1;40,0;42,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = text; inactive; preserveAspect] b) ;[s] 2:0,1;2,0;3,-1; 2:1,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect; startGroup] f[x,y] /. cp :[font = output; output; inactive; preserveAspect; endGroup] {-2, 2} ;[o] {-2, 2} :[font = text; inactive; preserveAspect] c) ;[s] 2:0,1;2,0;3,-1; 2:1,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect; endGroup] Plot3D[f[x,y], {x,-2,2}, {y,-2,2}, ViewPoint->{2.539, -1.868, 1.231}] ^*)