(*^ ::[ 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; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Math 225: Calculus III Assignment 4 :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Name: :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] 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; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450; endGroup] Signature: :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Instructions :[font = text; inactive; preserveAspect] The problems in this assignment involve curves, acceleration, functions of 2 variables, their graphs, level curves, and derivatives (Chapter 12-13 in Finney and Thomas, Chapter 2-3 in the Lecture Notes). Before starting, you may want to review the notebooks Derivatives, and Exercises (in the Notebook folder or the back of the Lecture Notes). To make the graphs look smoother you can increasing the number of plotted points with the optional optional argument PlotPoints. For example, Plot3D[ x^2 - y^2, {x,-2,2}, {y,-2,2}, PlotPoints->30] or ContourPlot[ x^2 - y^2, {x,-2,2}, {y,-2,2}, PlotPoints->30] The function Limit[] only finds limits of functions of one variable. In problem 4, you should substitute different expressions for y, then take the limit as x -> 0. For example, z = Sin[x^3y^2]/(x^5 + y^5) z1 = z /. {y -> 2x} Limit[z1, x->0] 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] 25:0,0;258,2;269,0;274,2;284,0;461,1;471,0;486,1;541,0;543,1;605,0;618,1;625,0;736,1;737,0;762,1;768,0;783,1;851,0;852,1;853,0;933,1;950,0;958,1;962,0;1006,-1; 3:13,13,9,Times,0,12,0,0,0;10,13,10,Courier,1,12,0,0,0;2,13,9,Times,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Initialization :[font = input; initialization; preserveAspect] *) < 1}, {t -> 1}} ;[o] {{t -> 1}, {t -> 1}} :[font = text; inactive; preserveAspect] The non-smooth point occurs when t == 1. ;[s] 3:0,0;32,1;39,0;41,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] r[0] :[font = output; output; inactive; preserveAspect; endGroup] {0, 1, 1} ;[o] {0, 1, 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] ParametricPlot3D[r[t], {t,0,2}] :[font = text; inactive; preserveAspect] c) Length in the integral of "speed" with respect to "time". ;[s] 2:0,1;2,0;61,-1; 2:1,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect; startGroup] NIntegrate[Norm[r'[t]], {t,0,2}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 0.905439234748809836 ;[o] 0.905439 :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 2 :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] A particle's position is given by r[t_] = (1 + t - t^2)i + (3 Cos[t])j + (t + t^3)k. a) Find the acceleration, a[0], of the particle at time t == 0. b) Find the tangential and normal components of a[0], i.e., find NUMBERS u and v such that a[0] = u T[0] + v n[0] without calculating T[0] or n[0]! (See Section 12.4 in Finney &Thomas or 2.4 in the Lecture Notes). ;[s] 23:0,0;34,1;83,0;85,2;87,0;111,1;115,0;141,1;147,0;149,2;151,0;197,1;201,0;222,1;223,0;228,1;229,0;241,1;263,0;285,1;289,0;293,1;297,0;365,-1; 3:12,13,9,Times,0,12,0,0,0;9,13,10,Courier,1,12,0,0,0;2,13,9,Times,2,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450] Solution :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] a) Acceleration is given by the second derivative. ;[s] 2:0,1;2,0;51,-1; 2:1,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect; startGroup] r[t_] = (1 + t - t^2)i + (3 Cos[t])j + (t + t^3)k :[font = output; output; inactive; preserveAspect; endGroup] {1 + t - t^2, 3*Cos[t], t + t^3} ;[o] 2 3 {1 + t - t , 3 Cos[t], t + t } :[font = input; preserveAspect; startGroup] r'[t] :[font = output; output; inactive; preserveAspect; endGroup] {1 - 2*t, -3*Sin[t], 1 + 3*t^2} ;[o] 2 {1 - 2 t, -3 Sin[t], 1 + 3 t } :[font = input; preserveAspect; startGroup] a[t_] = r''[t] :[font = output; output; inactive; preserveAspect; endGroup] {-2, -3*Cos[t], 6*t} ;[o] {-2, -3 Cos[t], 6 t} :[font = input; preserveAspect; startGroup] a[0] :[font = output; output; inactive; preserveAspect; endGroup] {-2, -3, 0} ;[o] {-2, -3, 0} :[font = text; inactive; preserveAspect] b) The tangential component of acceleration is the derivtive of speed: :[font = input; preserveAspect; startGroup] s[t_] = Norm[r'[t]] :[font = output; output; inactive; preserveAspect; endGroup] ((1 - 2*t)^2 + (1 + 3*t^2)^2 + 9*Sin[t]^2)^(1/2) ;[o] 2 2 2 2 Sqrt[(1 - 2 t) + (1 + 3 t ) + 9 Sin[t] ] :[font = input; preserveAspect; startGroup] u = s'[0] :[font = output; output; inactive; preserveAspect; endGroup] -2^(1/2) ;[o] -Sqrt[2] :[font = text; inactive; preserveAspect] The normal component of acceleration is given by the formula: :[font = input; preserveAspect; startGroup] v = Sqrt[a[0].a[0] - u^2] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 11^(1/2) ;[o] Sqrt[11] :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 3 :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Plot the graphs of the following functions, z = f[x,y], with Plot3D[] and plot several level curves of each function with ContourPlot[] (See Section 3.1 in the Lecture Notes and the examples in the Notebook directory.) Compare the graphs to the level curves. a) z = x y^2 - x^3 + y^3 b) z = x y(x^2 - y^2)/(x^2 + y^2) c) z = E^(-x^2-y^2)x ;[s] 15:0,0;44,1;54,0;61,1;69,0;122,1;135,0;259,2;261,0;262,1;284,2;286,0;287,1;318,2;321,1;339,-1; 3:6,13,9,Times,0,12,0,0,0;6,13,10,Courier,1,12,0,0,0;3,13,9,Times,2,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; endGroup] Solution :[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] z = x y^2 - x^3 + y^3 :[font = output; output; inactive; preserveAspect; endGroup] -x^3 + x*y^2 + y^3 ;[o] 3 2 3 -x + x y + y :[font = input; preserveAspect] Plot3D[z, {x,-2,2}, {y,-2,2}] :[font = input; preserveAspect] ContourPlot[z, {x,-2,2}, {y,-2,2}] :[font = text; inactive; preserveAspect] b) ;[s] 2:0,1;2,0;4,-1; 2:1,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect; startGroup] z = x y(x^2 - y^2)/(x^2 + y^2) :[font = output; output; inactive; preserveAspect; endGroup] (x*y*(x^2 - y^2))/(x^2 + y^2) ;[o] 2 2 x y (x - y ) ------------- 2 2 x + y :[font = input; preserveAspect] Plot3D[z, {x,-2,2}, {y,-2,2}] :[font = input; preserveAspect] ContourPlot[z, {x,-2,2}, {y,-2,2}] :[font = text; inactive; preserveAspect] c) ;[s] 1:0,1;4,-1; 2:0,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect; startGroup] z = E^(-x^2-y^2)x :[font = output; output; inactive; preserveAspect; endGroup] E^(-x^2 - y^2)*x ;[o] 2 2 -x - y E x :[font = input; preserveAspect] Plot3D[z,{x,-2,2},{y,-2,2}] :[font = input; preserveAspect] ContourPlot[z,{x,-2,2},{y,-2,2}] :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 4 :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] By considering different paths of approach, show that the limit of the function f[x_,y_] = x^2y^2/(x^4 + x y^3 + y^4) as x and y both go to 0 does not exist. Graph the function near {0,0} and explain why this is so based on the appearance of the graph. ;[s] 11:0,0;80,1;117,0;121,1;122,0;127,1;128,0;140,1;141,0;182,1;187,0;253,-1; 2:6,13,9,Times,0,12,0,0,0;5,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450] Solution :[font = input; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] :[font = input; preserveAspect; startGroup] f[x_,y_] = x^2y^2/(x^4 + x y^3 + y^4) :[font = output; output; inactive; preserveAspect; endGroup] (x^2*y^2)/(x^4 + x*y^3 + y^4) ;[o] 2 2 x y -------------- 4 3 4 x + x y + y :[font = text; inactive; preserveAspect] Restrict f[x,y] to the line y == 2x by substituting y->2x into f[x,y]. Then take the limit (f[x,y] is actually constant on this line). ;[s] 11:0,0;9,1;15,0;28,1;35,0;53,1;58,0;64,1;70,0;93,1;99,0;136,-1; 2:6,13,9,Times,0,12,0,0,0;5,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] Limit[f[x,y] /. {y->2x}, x->0] :[font = output; output; inactive; preserveAspect; endGroup] 4/25 ;[o] 4 -- 25 :[font = text; inactive; preserveAspect] Along the line y == 3x: ;[s] 3:0,0;15,1;22,0;24,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] Limit[f[x,y] /. {y->3x}, x->0] :[font = output; output; inactive; preserveAspect; endGroup] 9/109 ;[o] 9 --- 109 :[font = text; inactive; preserveAspect] We don't have to restrict to straight lines. For example, along the curve y == x^2: ;[s] 3:0,0;74,1;82,0;84,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] f[x,y] /. {y->x^2} :[font = output; output; inactive; preserveAspect; endGroup] x^6/(x^4 + x^7 + x^8) ;[o] 6 x ------------ 4 7 8 x + x + x :[font = input; preserveAspect; startGroup] Simplify[%] :[font = output; output; inactive; preserveAspect; endGroup] x^2/(1 + x^3 + x^4) ;[o] 2 x ----------- 3 4 1 + x + x :[font = input; preserveAspect; startGroup] Limit[%,x->0] :[font = output; output; inactive; preserveAspect; endGroup] 0 ;[o] 0 :[font = text; inactive; preserveAspect] Since these limits are diufferent (you only need to find two different!) the limit does not exist. The graph of f[x,y] shows this as well: ;[s] 3:0,0;112,1;118,0;139,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] Plot3D[f[x,y], {x,-1,1}, {y,-1,1}, PlotPoints->50] :[font = text; inactive; preserveAspect; endGroup] You cans see how the graph is "pinched" together over the point {0,0}, indicating that different function values are obtained depending on how one approaches {0,0}. ;[s] 5:0,0;64,1;69,0;158,1;163,0;165,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] Problem 5 :[font = text; inactive; preserveAspect] a) Find the partial derivatives of f[u_,v_] = (u^2 + v^2)E^(u/v) with respect to u and v. ;[s] 8:0,1;2,0;39,2;68,0;85,2;86,0;91,2;92,0;94,-1; 3:4,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0;3,13,10,Courier,1,12,0,0,0; :[font = text; inactive; preserveAspect] b) Find the second order partial derivatives of f[x_,y_] = x^3 y^2 - y/x^2 + Sqrt[x y] ;[s] 3:0,1;2,0;52,2;91,-1; 3:1,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect] Solution :[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] Clear[f,u,v] :[font = input; preserveAspect; startGroup] f[u_,v_] = (u^2 + v^2)E^(u/v) ;[s] 2:0,1;1,0;31,-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] E^(u/v)*(u^2 + v^2) ;[o] u/v 2 2 E (u + v ) :[font = input; preserveAspect; startGroup] D[f[u,v], u] :[font = output; output; inactive; preserveAspect; endGroup] 2*E^(u/v)*u + (E^(u/v)*(u^2 + v^2))/v ;[o] u/v 2 2 u/v E (u + v ) 2 E u + -------------- v :[font = input; preserveAspect; startGroup] Simplify[%] :[font = output; output; inactive; preserveAspect; endGroup] (E^(u/v)*(u + v)^2)/v ;[o] u/v 2 E (u + v) ------------- v :[font = input; preserveAspect; startGroup] D[f[u,v], v] //Simplify :[font = output; output; inactive; preserveAspect; endGroup] (E^(u/v)*(-u^3 - u*v^2 + 2*v^3))/v^2 ;[o] u/v 3 2 3 E (-u - u v + 2 v ) ------------------------ 2 v :[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] Clear[f,x,y] :[font = input; preserveAspect; startGroup] f[x_,y_] = x^3 y^2 - y/x^2 + Sqrt[x y] :[font = output; output; inactive; preserveAspect; endGroup] -(y/x^2) + x^3*y^2 + (x*y)^(1/2) ;[o] y 3 2 -(--) + x y + Sqrt[x y] 2 x :[font = input; preserveAspect; startGroup] D[f[x,y],x,x] :[font = output; output; inactive; preserveAspect; endGroup] (-6*y)/x^4 + 6*x*y^2 - y^2/(4*(x*y)^(3/2)) ;[o] 2 -6 y 2 y ---- + 6 x y - ---------- 4 3/2 x 4 (x y) :[font = input; preserveAspect; startGroup] D[f[x,y],x,y] :[font = output; output; inactive; preserveAspect; endGroup] 2/x^3 + 6*x^2*y - (x*y)/(4*(x*y)^(3/2)) + 1/(2*(x*y)^(1/2)) ;[o] 2 2 x y 1 -- + 6 x y - ---------- + ----------- 3 3/2 2 Sqrt[x y] x 4 (x y) :[font = input; preserveAspect; startGroup] D[f[x,y],y,x] :[font = output; output; inactive; preserveAspect; endGroup] 2/x^3 + 6*x^2*y - (x*y)/(4*(x*y)^(3/2)) + 1/(2*(x*y)^(1/2)) ;[o] 2 2 x y 1 -- + 6 x y - ---------- + ----------- 3 3/2 2 Sqrt[x y] x 4 (x y) :[font = text; inactive; preserveAspect] Notice how D[f[x,y],x,y]] == D[f[x,y],y,x]. This is not a coincindence! ;[s] 3:0,0;11,1;42,0;72,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] D[f[x,y],y,y] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 2*x^3 - x^2/(4*(x*y)^(3/2)) ;[o] 2 3 x 2 x - ---------- 3/2 4 (x y) ^*)