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2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 2 " -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 258 1 {CSTYLE " " -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Norma l" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 260 "" 0 "" {TEXT -1 21 " " }{TEXT 264 47 "Directional Derivatives and the Gradient Vector" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }}{PARA 0 "" 0 "" {TEXT 257 20 "Cal culus III Project" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 281 11 " Objective:" }}{PARA 0 "" 0 "" {TEXT -1 54 " To study directional derivatives and gradients using " } {TEXT 365 5 "Maple" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 259 22 " Required Inform ation:" }}{PARA 0 "" 0 "" {TEXT -1 133 "Vectors are described by bold \+ letters or by using arrows on the letter. We assume the function is d ifferentiable at the given point." }}{PARA 0 "" 0 "" {TEXT -1 2 "A " } {TEXT 366 22 "directional derivative" }{TEXT -1 17 " for a function \+ " }{TEXT 367 1 "f" }{TEXT -1 1 "(" }{TEXT 266 4 "x, y" }{TEXT -1 8 ") \+ at (" }{TEXT 267 4 "a, b" }{TEXT -1 1 ")" }{TEXT 268 2 " " }{TEXT -1 19 " in the direction " }{TEXT 261 1 "u" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "u[1];" "6#&%\"uG6#\"\"\"" }{TEXT -1 0 "" }{TEXT 260 4 " i + " }{XPPEDIT 18 0 "u[2];" "6#&%\"uG6#\"\"#" }{TEXT -1 0 "" }{TEXT 262 4 "j ," }}{PARA 0 "" 0 "" {TEXT -1 7 "where " }{TEXT 263 1 "u" } {TEXT -1 35 " is a unit vector, is given by " }{XPPEDIT 368 1 "lim it((f(a+hu[1],b+hu[2])-f(a,b))/h,h = 0);" "6#-%&limitG6$*&,&-%\"fG6$,& %\"aG\"\"\"&%#huG6#F-F-,&%\"bGF-&F/6#\"\"#F-F--F)6$F,F2!\"\"F-%\"hGF8/ F9\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 47 " By local linearity, this amou nts to finding " }{XPPEDIT 18 0 "f[u](a,b);" "6#-&%\"fG6#%\"uG6$%\"a G%\"bG" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "f[x](a,b);" "6#-&%\"fG6#%\" xG6$%\"aG%\"bG" }{XPPEDIT 18 0 "u[1];" "6#&%\"uG6#\"\"\"" }{TEXT -1 6 " + " }{XPPEDIT 18 0 "f[y](a,b);" "6#-&%\"fG6#%\"yG6$%\"aG%\"bG" } {XPPEDIT 18 0 "u[2];" "6#&%\"uG6#\"\"#" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 369 8 "gradient" }{TEXT -1 19 " of a surface z = " }{TEXT 271 1 "f" }{TEXT -1 1 "(" }{TEXT 272 4 "x, y" }{TEXT -1 8 ") at (" }{TEXT 273 4 "a, b" }{TEXT -1 17 " ), written grad " }{TEXT 275 1 "f" }{TEXT -1 1 "(" }{TEXT 274 4 "a, b " }{TEXT -1 15 "), is a vector " }}{PARA 0 "" 0 "" {TEXT -1 22 " \+ grad " }{TEXT 305 1 "f" }{TEXT -1 2 " (" }{TEXT 306 4 "a, b " }{TEXT -1 6 ") = " }{XPPEDIT 18 0 "f[x](a,b);" "6#-&%\"fG6#%\"xG6$ %\"aG%\"bG" }{TEXT -1 1 " " }{TEXT 307 3 "i " }{TEXT -1 3 "+ " } {XPPEDIT 18 0 "f[y](a,b);" "6#-&%\"fG6#%\"yG6$%\"aG%\"bG" }{TEXT -1 0 "" }{TEXT 308 1 "j" }}{PARA 0 "" 0 "" {TEXT -1 89 "pointing in the dir ection where the directional derivative is a maximum-- i.e., ||grad \+ " }{TEXT 270 1 "f" }{TEXT -1 2 " (" }{TEXT 276 4 "a, b" }{TEXT -1 9 ") || >= " }{XPPEDIT 18 0 "f[v];" "6#&%\"fG6#%\"vG" }{TEXT -1 1 "(" } {TEXT 269 4 "a, b" }{TEXT -1 10 ") for any " }}{PARA 0 "" 0 "" {TEXT -1 7 "vector " }{TEXT 265 1 "v" }{TEXT -1 25 ". Its magnitude, ||grad " }{TEXT 277 1 "f" }{TEXT -1 1 "(" }{TEXT 278 4 "a, b" }{TEXT -1 40 " )||, is the directional derivative of " }{TEXT 279 1 "f" }{TEXT -1 7 " at (" }{TEXT 280 4 "a, b" }{TEXT -1 21 ") in that direction." } }{PARA 0 "" 0 "" {TEXT -1 62 "The gradient vector is perpendicular to \+ the contour curve at (" }{TEXT 298 4 "a, b" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 370 23 "directional derivative " }{TEXT -1 16 "for a function " }{TEXT 371 1 "f" }{TEXT -1 1 "(" }{TEXT 299 7 "x, y, z" }{TEXT -1 22 ") in t he direction " }{TEXT 300 6 "u = " }{XPPEDIT 18 0 "u[1];" "6#&%\"u G6#\"\"\"" }{TEXT 301 1 "i" }{TEXT -1 4 " + " }{XPPEDIT 18 0 "u[2];" "6#&%\"uG6#\"\"#" }{TEXT 302 5 "j + " }{XPPEDIT 18 0 "u[3];" "6#&%\"u G6#\"\"$" }{TEXT -1 1 " " }{TEXT 303 1 "k" }{TEXT -1 8 " at (" } {TEXT 304 7 "a, b, c" }{TEXT -1 15 ") is given by " }}{PARA 0 "" 0 " " {TEXT -1 7 " " }{XPPEDIT 18 0 "f[u](a,b,c);" "6#-&%\"fG6#%\"uG 6%%\"aG%\"bG%\"cG" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "f[x](a,b,c);" " 6#-&%\"fG6#%\"xG6%%\"aG%\"bG%\"cG" }{XPPEDIT 18 0 "u[1];" "6#&%\"uG6# \"\"\"" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "f[y](a,b,c);" "6#-&%\"fG6#% \"yG6%%\"aG%\"bG%\"cG" }{XPPEDIT 18 0 "u[2];" "6#&%\"uG6#\"\"#" } {TEXT -1 3 " + " }{XPPEDIT 18 0 "f[z](a,b,c);" "6#-&%\"fG6#%\"zG6%%\"a G%\"bG%\"cG" }{XPPEDIT 18 0 "u[3];" "6#&%\"uG6#\"\"$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 372 15 "gradient vector" } {TEXT -1 20 " is given by grad " }{TEXT 309 1 "f" }{TEXT -1 2 " (" } {TEXT 310 7 "a, b, c" }{TEXT -1 6 ") = " }{XPPEDIT 18 0 "f[x](a,b,c) ;" "6#-&%\"fG6#%\"xG6%%\"aG%\"bG%\"cG" }{TEXT -1 1 " " }{TEXT 311 2 "i " }{TEXT -1 4 " + " }{XPPEDIT 18 0 "f[y](a,b,c);" "6#-&%\"fG6#%\"yG6 %%\"aG%\"bG%\"cG" }{TEXT -1 0 "" }{TEXT 312 6 "j + " }{XPPEDIT 18 0 "f[z](a,b,c);" "6#-&%\"fG6#%\"zG6%%\"aG%\"bG%\"cG" }{TEXT 313 2 "k," } }{PARA 0 "" 0 "" {TEXT -1 63 "where the directional derivative is a ma ximum-- i.e., ||grad " }{TEXT 342 1 "f" }{TEXT -1 2 " (" }{TEXT 343 7 "a, b, c" }{TEXT -1 9 ")|| >= " }{XPPEDIT 18 0 "f[v];" "6#&%\"fG6# %\"vG" }{TEXT -1 1 "(" }{TEXT 341 7 "a, b, c" }{TEXT -1 10 ") for any \+ " }}{PARA 0 "" 0 "" {TEXT -1 7 "vector " }{TEXT 340 1 "v" }{TEXT -1 25 ". Its magnitude, ||grad " }{TEXT 344 1 "f" }{TEXT -1 1 "(" } {TEXT 345 7 "a, b, c" }{TEXT -1 40 ")||, is the directional derivative of " }{TEXT 346 1 "f" }{TEXT -1 7 " at (" }{TEXT 347 7 "a, b, c" }{TEXT -1 21 ") in that direction." }}{PARA 0 "" 0 "" {TEXT -1 94 " T he gradient vector of a function of 3 variables is perpendicular to \+ the level surfaces. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 258 17 " Solved Example 1" }} {PARA 0 "" 0 "" {TEXT -1 6 " Let " }{TEXT 374 1 "z" }{TEXT -1 5 " = \+ " }{TEXT 284 1 "f" }{TEXT -1 1 "(" }{TEXT 283 4 "x, y" }{TEXT -1 6 ") = " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT 282 1 "y" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 13 " (a) Graph " }{TEXT 373 2 " z " }{TEXT -1 2 "= " }{TEXT 285 1 "f" }{TEXT -1 1 "(" }{TEXT 286 4 "x, y" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 40 " (b) Find the dir ectional derivative " }{XPPEDIT 18 0 "f[v];" "6#&%\"fG6#%\"vG" } {TEXT -1 1 "(" }{TEXT 380 1 "a" }{TEXT -1 2 ", " }{TEXT 381 1 "b" } {TEXT -1 10 "), where " }{TEXT 375 1 "v" }{TEXT -1 28 " is in the di rection of 4" }{TEXT 376 1 "i" }{TEXT -1 6 " - 3" }{TEXT 377 0 "" }{TEXT 378 0 "" }{TEXT 379 3 "j " }{TEXT -1 16 "by using limits." }} {PARA 0 "" 0 "" {TEXT -1 40 " (c) Find the directional derivative \+ " }{XPPEDIT 18 0 "f[v];" "6#&%\"fG6#%\"vG" }{TEXT -1 58 "(2, 6) usin g the formula for the directional derivative." }}{PARA 0 "" 0 "" {TEXT -1 29 " (d) Find the gradient of " }{TEXT 382 3 "f " }{TEXT -1 69 "at (2, 6) using the formula and verify the definition using l imits." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 287 0 "" }{TEXT 288 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 13 "Activate the " }{TEXT 384 5 "plots" }{TEXT -1 5 " and \+ " }{TEXT 383 7 "student" }{TEXT -1 10 " packages." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "with(plots): with(student): " }}}{PARA 0 "" 0 "" {TEXT -1 20 "Define the function." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f := (x, y) -> x^2*y; \+ " }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 7 "Grap h " }{TEXT 385 1 "z" }{TEXT -1 4 " = " }{TEXT 386 2 " f" }{TEXT -1 1 "(" }{TEXT 387 1 "x" }{TEXT -1 2 ", " }{TEXT 388 1 "y" }{TEXT -1 15 ") and use the " }{TEXT 389 6 "plot3d" }{TEXT -1 30 " command, givin g ranges for " }{TEXT 390 1 "x" }{TEXT -1 7 " and " }{TEXT 391 1 "y " }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "plot3d( f(x, y), x = 1..3, y = 5..7, axes = boxed, grid = [5, 5], labels = [x, y, z],title=`z = x^2 * y `);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "G" 0 }}}}{PARA 0 "" 0 "" {TEXT -1 28 "(b) The directi on given is " }{TEXT 289 1 "v" }{TEXT -1 4 " = 4" }{TEXT 290 1 "i" } {TEXT -1 4 " - 3" }{TEXT 291 2 "j." }{TEXT -1 57 " Since ||v|| = 5, \+ the unit vector in the direction of " }{TEXT 392 1 "v" }{TEXT -1 5 " \+ is " }}{PARA 0 "" 0 "" {TEXT -1 14 " 4/5 " }{TEXT 292 1 "i" }{TEXT -1 7 " - 3/5 " }{TEXT 293 1 "j" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "fv (2, 6) := limit ((f(2 + 4/5*h, 6 - 3/5 *h) - f(2, 6))/h, h = 0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#fvG6$ \"\"#\"\"'#\"#%)\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 38 " Note ho w the limit is defined." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "(c) The formula for the directional deri vative is " }{XPPEDIT 18 0 "f[v](a,b);" "6#-&%\"fG6#%\"vG6$%\"aG%\"b G" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "f[x](a,b);" "6#-&%\"fG6#%\"xG6$% \"aG%\"bG" }{XPPEDIT 18 0 "v[1];" "6#&%\"vG6#\"\"\"" }{TEXT -1 6 " + \+ " }{XPPEDIT 18 0 "f[y](a,b);" "6#-&%\"fG6#%\"yG6$%\"aG%\"bG" } {XPPEDIT 18 0 "v[2];" "6#&%\"vG6#\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 22 " For the given " }{TEXT 295 1 "v" }{TEXT -1 25 ", v1 = 4/5, v2 = -3/5. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "fx(x, y) := diff(f(x, y), x); fy(x, y) := diff(f(x, y), y); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#fxG6$ %\"xG%\"yG,$*&F'\"\"\"F(F+\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-% #fyG6$%\"xG%\"yG*$)F'\"\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "fx(2, 6) := subs(x = 2, y = 6, fx(x, y)); fy(2, 6) : = subs(x = 2, y = 6, fy(x, y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-% #fxG6$\"\"#\"\"'\"#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#fyG6$\"\"# \"\"'\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "fv(2, 6) := f x(2, 6)*4/5 + fy(2, 6)*(-3/5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%# fvG6$\"\"#\"\"'#\"#%)\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 22 "(d) The g radient of " }{TEXT 393 1 "f" }{TEXT -1 29 " is the direction in whi ch " }{TEXT 394 3 "f " }{TEXT -1 32 "increases at the greatest rate. " }}{PARA 0 "" 0 "" {TEXT -1 40 " By definition, it is the vect or " }{XPPEDIT 18 0 "f[x](2,6);" "6#-&%\"fG6#%\"xG6$\"\"#\"\"'" } {TEXT -1 1 " " }{TEXT 395 3 "i " }{TEXT -1 3 "+ " }{XPPEDIT 18 0 "f[ y](2,6);" "6#-&%\"fG6#%\"yG6$\"\"#\"\"'" }{TEXT -1 0 "" }{TEXT 396 2 " j" }{TEXT -1 7 " and ." }}{PARA 0 "" 0 "" {TEXT -1 12 " grad \+ " }{TEXT 397 1 "f" }{TEXT -1 11 "(2, 6) = 24" }{TEXT 294 1 "i" }{TEXT -1 4 " + 4" }{TEXT 296 1 "j" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 " Its magnitude, ||grad " }{TEXT 398 1 "f" }{TEXT -1 60 "(2, 6)||, is the directional derivat ive in the direction 24" }{TEXT 399 1 "i" }{TEXT -1 4 " + 4" }{TEXT 400 1 "j" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 " We verif y this in the following steps." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "magnitude := sqrt(24^2 + 4^2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*magnitudeG,$*$-%%sqrtG6#\"#P \"\"\"\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "fv (2, 6) := limit ((f(2 + 24/(magnitude)*h, 6 + 4/(magnitude)*h) - f(2, 6))/h, h \+ = 0); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#fvG6$\"\"#\"\"',$*$-%%sq rtG6#\"#P\"\"\"\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 297 18 " Solved Example 2:" }} {PARA 0 "" 0 "" {TEXT -1 6 " Let " }{TEXT 401 1 "f" }{TEXT -1 1 "(" } {TEXT 314 7 "x, y, z" }{TEXT -1 6 ") = " }{XPPEDIT 18 0 "x^2;" "6#*$ %\"xG\"\"#" }{TEXT -1 1 " " }{TEXT 315 8 "y - z." }}{PARA 0 "" 0 "" {TEXT -1 13 "(a) Graph " }{TEXT 316 1 "f" }{TEXT -1 1 "(" }{TEXT 318 7 "x, y, z" }{TEXT -1 8 ") = 0, " }{TEXT 319 1 "f" }{TEXT -1 1 "( " }{TEXT 320 7 "x, y, z" }{TEXT -1 35 ") = 10 on the same reference a xes." }}{PARA 0 "" 0 "" {TEXT -1 39 "(b) Find the directional derivat ive " }{XPPEDIT 18 0 "f[v];" "6#&%\"fG6#%\"vG" }{TEXT -1 1 "(" } {TEXT 402 1 "a" }{TEXT -1 2 ", " }{TEXT 403 1 "b" }{TEXT -1 2 ", " } {TEXT 404 1 "c" }{TEXT -1 10 "), where " }{TEXT 405 1 "v" }{TEXT -1 28 " is in the direction of 4" }{TEXT 321 1 "i" }{TEXT -1 4 " - 3" }{TEXT 322 1 "j" }{TEXT -1 3 " - " }{TEXT 323 1 "k" }{TEXT -1 19 " b y using limits." }}{PARA 0 "" 0 "" {TEXT -1 38 "(c) Find the directio nal derivative " }{XPPEDIT 18 0 "f[v];" "6#&%\"fG6#%\"vG" }{TEXT -1 61 "(2, 6, 24) using the formula for the directional derivative." }} {PARA 0 "" 0 "" {TEXT -1 28 "(d) Find the gradient of " }{TEXT 406 2 "f " }{TEXT -1 74 " at (2, 6, 24) using the formula and verify the definition using limits." }}{PARA 0 "" 0 "" {TEXT 317 2 " " }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 324 0 "" }{TEXT 325 9 "So lution:" }}{PARA 0 "" 0 "" {TEXT -1 20 "Define the function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f \+ := (x, y, z) -> x^2*y - z; " }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 10 "To graph " }{TEXT 407 2 " f" }{TEXT -1 1 "(" }{TEXT 408 1 "x" }{TEXT -1 2 ", " }{TEXT 409 1 "y" }{TEXT -1 2 " , " }{TEXT 410 1 "z" }{TEXT -1 13 ") = 0 and " }{TEXT 411 1 "f" } {TEXT -1 1 "(" }{TEXT 412 1 "x" }{TEXT -1 2 ", " }{TEXT 413 1 "y" } {TEXT -1 2 ", " }{TEXT 414 1 "z" }{TEXT -1 44 ") = 10 , we rewrite th e functions as z = " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 12 "y and z = " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 23 "y - 1 and we use the " }{TEXT 415 6 "plot3d" }{TEXT -1 28 " comm and, giving ranges for " }{TEXT 416 1 "x" }{TEXT -1 5 " and " }{TEXT 417 1 "y" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "plot3d(\{x^2*y, x^2*y - 10\}, x = 1 ..3, y = 5..7, grid=[10,10],title=`Level Surfaces `);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "G" 0 }}}}{PARA 0 "" 0 "" {TEXT -1 27 "(b) The direction given is " }{TEXT 326 1 "v" }{TEXT -1 4 " = 4" }{TEXT 327 1 "i" }{TEXT -1 4 " - 3" }{TEXT 328 6 "j - k." } {TEXT -1 17 " Since ||v|| = " }{XPPEDIT 18 0 "sqrt(3^2+4^2+1^2);" "6 #-%%sqrtG6#,(*$\"\"$\"\"#\"\"\"*$\"\"%F)F**$F*F)F*" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sqrt(26);" "6#-%%sqrtG6#\"#E" }{TEXT -1 39 ", the unit vector in the direction of " }{TEXT 418 1 "v" }{TEXT -1 5 " is " }} {PARA 0 "" 0 "" {TEXT -1 13 " (4/" }{XPPEDIT 18 0 "sqrt(26)" "6#-%%sqrtG6#\"#E" }{TEXT -1 2 ") " }{TEXT 329 1 "i" }{TEXT -1 6 " - ( 3/" }{XPPEDIT 18 0 "sqrt(26)" "6#-%%sqrtG6#\"#E" }{TEXT -1 2 ") " } {TEXT 330 1 "j" }{TEXT -1 6 " - (1/" }{XPPEDIT 18 0 "sqrt(26)" "6#-%%s qrtG6#\"#E" }{TEXT 334 6 " ) k ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "fv (2, 6,24) := limit ((f(2 + 4/sqrt(26)*h, 6 - 3/sqrt(26)*h,24-1/sqrt(26)*h) - f(2, 6,24))/h, h \+ = 0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#fvG6%\"\"#\"\"'\"#C,$*$-% %sqrtG6#\"#E\"\"\"#\"#&)F/" }}}{PARA 0 "" 0 "" {TEXT -1 36 " Note how the limit is defined." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "(c) The formula for the directional deri vative is " }{XPPEDIT 18 0 "f[v](a,b,c);" "6#-&%\"fG6#%\"vG6%%\"aG% \"bG%\"cG" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "f[x](a,b,c);" "6#-&%\"f G6#%\"xG6%%\"aG%\"bG%\"cG" }{XPPEDIT 18 0 "v[1];" "6#&%\"vG6#\"\"\"" } {TEXT -1 3 " + " }{XPPEDIT 18 0 "f[y](a,b,c);" "6#-&%\"fG6#%\"yG6%%\"a G%\"bG%\"cG" }{XPPEDIT 18 0 "v[2];" "6#&%\"vG6#\"\"#" }{TEXT -1 3 " + \+ " }{XPPEDIT 18 0 "f[z](a,b,c);" "6#-&%\"fG6#%\"zG6%%\"aG%\"bG%\"cG" } {XPPEDIT 18 0 "v[3];" "6#&%\"vG6#\"\"$" }}{PARA 0 "" 0 "" {TEXT -1 21 " For the given " }{TEXT 332 1 "v" }{TEXT -1 9 ", v1 = 4/" } {XPPEDIT 18 0 "sqrt(26);" "6#-%%sqrtG6#\"#E" }{TEXT -1 12 " , v2 = -3/ " }{XPPEDIT 18 0 "sqrt(26)" "6#-%%sqrtG6#\"#E" }{TEXT -1 10 ", v3 = - 1/" }{XPPEDIT 18 0 "sqrt(26)" "6#-%%sqrtG6#\"#E" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "fx (x, y, z) := diff(f(x, y,z), x); fy(x, y, z) := diff(f(x, y, z), y); f z(x, y, z) := diff(f(x, y, z), z); " }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>-%#fxG6%%\"xG%\"yG%\"zG,$*&F'\"\"\"F(F,\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#fyG6%%\"xG%\"yG%\"zG*$)F'\"\"#\"\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>-%#fzG6%%\"xG%\"yG%\"zG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "fx(2, 6, 24) := subs(x = 2, y = 6,z=24, \+ fx(x, y, z)); fy(2, 6, 24) := subs(x = 2, y = 6, z=24, fy(x, y,z));fz (2, 6,24) := subs(x = 2, y = 6, z=24, fz(x, y, z));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>-%#fxG6%\"\"#\"\"'\"#CF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#fyG6%\"\"#\"\"'\"#C\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#fzG6%\"\"#\"\"'\"#C!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "fv(2, 6, 24) := fx(2, 6, 24)*4/sqrt(26) + fy(2, 6, 24)*(-3/ sqrt(26)) +fz(2, 6, 24)*(-1/ sqrt(26));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#fvG6%\"\"#\"\"'\"#C,$*$-%%sqrtG6#\"#E\"\"\"#\"#&)F/ " }}}{PARA 0 "" 0 "" {TEXT -1 53 " This shows that the answers in b) and c) agree." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "(d) The gradient of " }{TEXT 419 2 " f" }{TEXT -1 30 " \+ is the direction in which " }{TEXT 420 1 "f" }{TEXT -1 35 " incre ases at the greatest rate. " }}{PARA 0 "" 0 "" {TEXT -1 40 " By d efinition, it is the vector " }{XPPEDIT 18 0 "f[x](2,6,24);" "6#-&% \"fG6#%\"xG6%\"\"#\"\"'\"#C" }{TEXT -1 1 " " }{TEXT 421 2 "i " }{TEXT -1 4 " + " }{XPPEDIT 18 0 "f[y](2,6,24);" "6#-&%\"fG6#%\"yG6%\"\"#\" \"'\"#C" }{TEXT -1 0 "" }{TEXT 422 6 "j + " }{XPPEDIT 18 0 "f[z](2,6 ,24);" "6#-&%\"fG6#%\"zG6%\"\"#\"\"'\"#C" }{TEXT 423 3 "k " }{TEXT 424 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 12 " grad " }{TEXT 425 1 "f" }{TEXT -1 16 "(2, 6,24) = 24" }{TEXT 331 1 "i" }{TEXT -1 6 " + \+ 4" }{TEXT 333 1 "j" }{TEXT -1 5 " - " }{TEXT 335 1 "k" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 89 " Its magnitude, ||grad f(2, 6, 24)||, is the directional derivative in the direction " }{TEXT 339 1 "v" }{TEXT -1 6 " = 24" }{TEXT 336 1 "i" }{TEXT -1 6 " + 4" } {TEXT 337 1 "j" }{TEXT -1 5 " - " }{TEXT 338 1 "k" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 " We verify this in the following ste ps." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "magnitude := sqrt(24^2 + 4^2 + 1);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%*magnitudeG*$-%%sqrtG6#\"$$f\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "fv (2, 6, 24) := limit ((f(2 + 24/ (magnitude)*h, 6 + 4/(magnitude)*h,24-1/(magnitude)*h) - f(2, 6, 24))/ h, h = 0); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#fvG6%\"\"#\"\"'\"#C *$-%%sqrtG6#\"$$f\"\"\"" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 63 "__________________________________________________ _____________" }}{PARA 258 "" 0 "" {TEXT -1 10 "ASSIGNMENT" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 348 13 " Problem 1: " }}{PARA 0 "" 0 "" {TEXT -1 9 " Let " }{TEXT 426 1 "f" }{TEXT -1 1 "(" } {TEXT 434 1 "x" }{TEXT -1 2 ", " }{TEXT 435 1 "y" }{TEXT -1 6 ") = \+ " }{XPPEDIT 18 0 "(x+y)/(1+x^2);" "6#*&,&%\"xG\"\"\"%\"yGF&F&,&F&F&*$F %\"\"#F&!\"\"" }{TEXT -1 10 " and let " }{TEXT 427 1 "P" }{TEXT -1 69 " = (1, -2). Graph the surface. Find the directional derivative a t " }{TEXT 428 1 "P" }{TEXT -1 39 " in the direction of the vectors \+ (a) " }{TEXT 349 1 "v" }{TEXT -1 4 " = 3" }{TEXT 350 3 "i " }{TEXT -1 4 "- 2" }{TEXT 351 1 "j" }{TEXT -1 6 " and" }}{PARA 0 "" 0 "" {TEXT -1 6 " (b) " }{TEXT 364 1 "v" }{TEXT -1 5 " = - " }{TEXT 352 1 "i" }{TEXT -1 4 " + 4" }{TEXT 353 1 "j" }{TEXT -1 62 " using the for mula and verify your answers using limits. " }}{PARA 0 "" 0 "" {TEXT -1 53 " (c) What is the direction of greatest increase at " } {TEXT 429 1 "P" }{TEXT -1 31 "? Calculate the gradient of " }{TEXT 430 1 "f" }{TEXT -1 6 " at " }{TEXT 431 1 "P" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 363 10 "Problem 2:" }}{PARA 259 "" 0 "" {TEXT 432 3 "Let" }{TEXT -1 4 " f" }{TEXT 433 1 "(" }{TEXT 354 7 "x, y, z" }{TEXT 436 1 ")" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "(x+y)/(1+x^2);" "6#*&,&%\"xG\"\"\"%\"yGF&F&,&F& F&*$F%\"\"#F&!\"\"" }{TEXT 355 9 " - z." }}{PARA 0 "" 0 "" {TEXT -1 12 "(a) Graph " }{TEXT 356 1 "f" }{TEXT -1 1 "(" }{TEXT 357 7 "x, y, z" }{TEXT -1 9 ") = 0, " }{TEXT 358 1 "f" }{TEXT -1 1 "(" } {TEXT 359 7 "x, y, z" }{TEXT -1 35 ") = 10 on the same reference axes ." }}{PARA 0 "" 0 "" {TEXT -1 39 "(b) Find the directional derivative " }{XPPEDIT 18 0 "f[v];" "6#&%\"fG6#%\"vG" }{TEXT -1 1 "(" }{TEXT 437 1 "a" }{TEXT -1 2 ", " }{TEXT 438 1 "b" }{TEXT -1 2 ", " }{TEXT 439 1 "c" }{TEXT -1 10 "), where " }{TEXT 440 1 "v" }{TEXT -1 28 " i s in the direction of 4" }{TEXT 360 1 "i" }{TEXT -1 4 " - 3" }{TEXT 361 1 "j" }{TEXT -1 3 " - " }{TEXT 362 1 "k" }{TEXT -1 19 " by using limits." }}{PARA 0 "" 0 "" {TEXT -1 39 "(c) Find the directional der ivative " }{XPPEDIT 18 0 "f[v];" "6#&%\"fG6#%\"vG" }{TEXT -1 8 "(1, \+ -2, " }{XPPEDIT 18 0 "-1/2" "6#,$*&\"\"\"F%\"\"#!\"\"F'" }{TEXT -1 53 ") using the formula for the directional derivative." }}{PARA 0 "" 0 "" {TEXT -1 28 "(d) Find the gradient of " }{TEXT 441 1 "f" } {TEXT -1 15 " at (1, -2, " }{XPPEDIT 18 0 "-1/2;" "6#,$*&\"\"\"F%\" \"#!\"\"F'" }{TEXT -1 60 ") using the formula and verify the definiti on using limits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 69 "_________________________________________________________ ____________" }}{PARA 0 "" 0 "" {TEXT -1 172 "MSIP Grant #P120A80089-9 8: \"Three Urban Calculus Reform programs: Adopting the Best\" 1998-2 001, MSEIP Grant #P120A010031: \"Four Colleges: Calculus + Enhancement s\" 2001-04" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "2 0" 12 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }