(*^ ::[ 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, M18, N18, O431, bold, e8, 24, "Times"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M18, N18, O431, bold, e6, 18, "Times"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M18, N18, O431, italic, e6, 14, "Times"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M18, N18, O431, bold, a20, 18, "Times"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M18, N18, O431, bold, a15, 14, "Times"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, N18, O431, bold, a12, 12, "Times"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, N18, O431, 12, "Times"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, N18, O431, 10, "Times"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M18, N18, O431, bold, L-5, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N18, O431, L-5, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N18, O431, R65535, L-5, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N18, O431, L-5, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N18, O431, B65535, L-5, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M18, N18, O431, l34, w282, h287, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, N18, O431, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M18, N18, O431, 12, "Times"; fontset = leftheader, inactive, M18, N18, O431, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M18, N18, O431, 12, "Times"; fontset = leftfooter, inactive, M18, N18, O431, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, N18, O431, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, N18, O431, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, N18, O431, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, N18, O431, 12, "Times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, N18, O431, 12, "Times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, N18, O431, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, N18, O431, 12, "Times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, N18, O431, 12, "Times"; paletteColors = 128; showRuler; currentKernel; ] :[font = subsection; inactive; preserveAspect; rightWrapOffset = 504] Math 225: Calculus III Quiz 4 2/20&2/22/96 :[font = input; initialization; preserveAspect; rightWrapOffset = 504] *) <3, y[t]->5, x'[t]->0.35, y'[t]->0.4} :[font = output; output; inactive; preserveAspect; rightWrapOffset = 504; endGroup] 2.95 ;[o] 2.95 :[font = subsubsection; inactive; preserveAspect; rightWrapOffset = 504; startGroup] 2. Let f[x_,y_] = E^(x^3 y)Sin[x y]. Compute D[f[x,y], x.y] at x -> 1, y -> Pi. :[font = subsubsection; inactive; preserveAspect; rightWrapOffset = 504] Solution :[font = input; preserveAspect; rightWrapOffset = 504; startGroup] f[x_,y_] = E^(x^3 y)Sin[x y] ;[s] 1:0,1;29,-1; 2:0,12,10,Courier,1,12,0,0,0;1,12,9,Times,1,12,0,0,0; :[font = output; output; inactive; preserveAspect; rightWrapOffset = 504; endGroup] E^(x^3*y)*Sin[x*y] ;[o] 3 x y E Sin[x y] :[font = input; preserveAspect; rightWrapOffset = 504; startGroup] D[f[x,y],x] :[font = output; output; inactive; preserveAspect; endGroup] E^(x^3*y)*y*Cos[x*y] + 3*E^(x^3*y)*x^2*y*Sin[x*y] ;[o] 3 3 x y x y 2 E y Cos[x y] + 3 E x y Sin[x y] :[font = input; preserveAspect; startGroup] D[%,y] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] E^(x^3*y)*Cos[x*y] + 4*E^(x^3*y)*x^3*y*Cos[x*y] + 3*E^(x^3*y)*x^2*Sin[x*y] - E^(x^3*y)*x*y*Sin[x*y] + 3*E^(x^3*y)*x^5*y*Sin[x*y] ;[o] 3 3 3 x y x y 3 x y 2 E Cos[x y] + 4 E x y Cos[x y] + 3 E x Sin[x y] - 3 3 x y x y 5 E x y Sin[x y] + 3 E x y Sin[x y] :[font = input; preserveAspect; startGroup] % /. {x->1, y->Pi} :[font = output; output; inactive; preserveAspect; endGroup] -E^Pi - 4*E^Pi*Pi ;[o] Pi Pi -E - 4 E Pi ^*)