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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 31 "Test 1 solutions (pink v
ersion)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
49 "1 (10 points).  Calculate the following:\n\n\n\na)  \n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "vec := t -> crossprod([0, 3*t ,6*t \+
], [exp(t), t ,2*t ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$vecGf*6#%
\"tG6\"6$%)operatorG%&arrowGF(-%*crossprodG6$7%\"\"!,$9$\"\"$,$F2\"\"'
7%-%$expG6#F2F2,$F2\"\"#F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 20 "map(diff,vec(t), t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vec
torG6#7%\"\"!,&-%$expG6#%\"tG\"\"'*(F-\"\"\"F,F/F)F/F/,&F)!\"$*(\"\"$F
/F,F/F)F/!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "b)  " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "map(int,[cos(t/2),t/(t^2-1)], t=2..
5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,&-%$sinG6##\"\"&\"\"#F**&F*
\"\"\"-F&6#F,F,!\"\",$-%#lnG6#F*#\"\"$F*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 84 "This is the best frm of the answer, but in case you prefe
r numerical approximations:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$!*#o(*f[!\"*$\"+r2s
R5F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 252 "2 (10 points).  On the c
urve shown below, clearly  indicate\n\na)  the unit tangent vector at \+
the point P.  Label it T\n\nb) the principal unit normal vector at the
 point P.  Label it N\n\nc) at least one point where the curvature is \+
maximized.  Label it M.\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 97 "For part a), the unit tangent vector can point in \+
either direction.  For part b), the unit normal" }}{PARA 0 "" 0 "" 
{TEXT -1 99 "vector should point to the inside of the curve and have t
he same length as T.  For part c, M should" }}{PARA 0 "" 0 "" {TEXT 
-1 31 "be where the curve is sharpest." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 112 "In problems 3 and 4, an object of ma
ss 2 moves along the helix\n\n\011r(t) = 2 cos(3 t^2) i + t^2 j + 2 si
n(3 t^2) k\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "r := t -> v
ector([2*cos(3*t^2),3*t^2,2*sin(3*t^2)]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"rGf*6#%\"tG6\"6$%)operatorG%&arrowGF(-%'vectorG6#7%
,$-%$cosG6#,$*$)9$\"\"#\"\"\"\"\"$F8F4,$-%$sinGF3F8F(F(F(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 24 "First, let's look at it:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "with(plots): spacecurve([2*cos(3*t^
2),3*t^2,2*sin(3*t^2)], t=0..2, axes=NORMAL);" }}{PARA 7 "" 1 "" 
{TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}
{PARA 13 "" 1 "" {GLPLOT3D 349 262 262 {PLOTDATA 3 "6$-%'CURVESG6#7T7%
$\"\"#\"\"!$F*F*F+7%$\"+@](***>!\"*$\"+Mv\"z*\\!#7$\"+aMz&***F27%$\"+Y
.g**>F/$\"+9q;**>!#6$\"+*pn!)*RF:7%$\"+Gq(z*>F/$\"+#yD\")\\%F:$\"+k\"=
K**)F:7%$\"+P(3O*>F/$\"+`!om*zF:$\"+?'Hwf\"!#57%$\"+8$3W)>F/$\"+$Qz%\\
7FK$\"+a:Y#\\#FK7%$\"+BUrn>F/$\"+6.D*z\"FK$\"+0j6zNFK7%$\"+TTKS>F/$\"+
!fz*[CFK$\"+&fY\"\\[FK7%$\"+pYb)*=F/$\"+>sm)>$FK$\"+64!))G'FK7%$\"+%pP
$Q=F/$\"+*>8$[SFK$\"+5WFxyFK7%$\"+kZOb<F/$\"+Hv\"z*\\FK$\"+%[b[e*FK7%$
\"+v2HX;F/$\"+5-[ZgFK$\"+5'4r8\"F/7%$\"+eo+/:F/$\"+U7+(>(FK$\"+)Q=$=8F
/7%$\"+h*))zK\"F/$\"+C1[Y%)FK$\"+[\\Z&\\\"F/7%$\"+gHs96F/$\"+c$=fz*FK$
\"+*))R0m\"F/7%$\"+\\Y)>j)FK$\"+W9`C6F/$\"+H48/=F/7%$\"+')z^WdFK$\"+()
oYz7F/$\"+Xds:>F/7%$\"+lsB@DFK$\"+m\")RW9F/$\"+kZ/%)>F/7%$!++8)>q*F:$
\"+z_K>;F/$\"+.ak(*>F/7%$!+qJuEYFK$\"+G#[U!=F/$\"+!GZd%>F/7%$!+a^y2$)F
K$\"+6q;**>F/$\"+9vG>=F/7%$!+yBf$=\"F/$\"+I;3/AF/$\"+k_<7;F/7%$!+Eo<+:
F/$\"+%3#**=CF/$\"+F^nA8F/7%$!+(Hrtv\"F/$\"+s$)*Qk#F/$\"+]h,[&*FK7%$!+
0oLJ>F/$\"+'\\+)yGF/$\"+'=db>&FK7%$!+1)z'**>F/$\"+d%)pBJF/$\"+f#['yNF:
7%$!+4O4W>F/$\"+bAfyLF/$!+'=Vdp%FK7%$!+*\\]Lv\"F/$\"+()=[VOF/$!+2Gl@'*
FK7%$!+XiNE9F/$\"+btO=RF/$!+8d'>S\"F/7%$!+cd`[(*FK$\"+e'[K?%F/$!+?xKY<
F/7%$!+w.b_UFK$\"+'zD\")\\%F/$!+amEa>F/7%$\"+zqr4=FK$\"+q()*H![F/$!+#[
&z\"*>F/7%$\"+D><*)yFK$\"+zv'y6&F/$!+Iu#y$=F/7%$\"+V2DM8F/$\"+BAtUaF/$
!+@4*)*[\"F/7%$\"+H_x\\<F/$\"+.FfxdF/$!+`(>lo*FK7%$\"+B%>U(>F/$\"+=!\\
C7'F/$!+r4!4?$FK7%$\"+0sVi>F/$\"+o6IxkF/$\"+G:)z&QFK7%$\"+%45cp\"F/$\"
+`\"\\@%oF/$\"+_ghg5F/7%$\"+4Zb*=\"F/$\"+uH*p@(F/$\"+a4y2;F/7%$\"+g;r*
)\\FK$\"+IE$=g(F/$\"+DqvO>F/7%$!+8i0WGFK$\"+@\"om*zF/$\"+&3v'z>F/7%$!+
$**Q6/\"F/$\"+[%*\\,%)F/$\"+.'Qwq\"F/7%$!+6%o5k\"F/$\"+5mK;))F/$\"+`[>
V6F/7%$!+r_Pm>F/$\"+2'\\6C*F/$\"+)Ri>l$FK7%$!+7WBP>F/$\"+S%ofn*F/$!+r]
9r\\FK7%$!+l#>[`\"F/$\"+6$y?,\"!\")$!+XUJ#G\"F/7%$!+%=c[:)FK$\"+h$fv0
\"Fc_l$!+RO>E=F/7%$\"+)HI@'*)F:$\"+&**RS5\"Fc_l$!+'*4*z*>F/7%$\"+PE9J*
*FK$\"+7-_^6Fc_l$!+#33gt\"F/7%$\"+f!3xo\"F/$\"+8+++7Fc_l$!+8c9t5F/-%*A
XESSTYLEG6#%'NORMALG" 1 2 0 1 10 0 2 1 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "3 (10 \+
points).  \n\na)   The speed of the object at t= 3 is" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "with(linalg): map(diff,r(t),t): nor
m(%, 2);" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names \+
norm and trace have been redefined and unprotected\n" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#,$*$-%%sqrtG6#,(*$)-%$absG6#*&-%$sinG6#,$*$)%\"tG\"
\"#\"\"\"\"\"$F7F5F7F6F7\"\"%*$)-F,6#F5F6F7F7*&F9F7)-F,6#*&-%$cosGF1F7
F5F7F6F7F7F7\"\"'" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 19 "which simplifies to" }{TEXT 256 1 " " }
{TEXT -1 28 "6 t sqrt(5).  At t=3, it is " }{TEXT 257 9 "18sqrt(5)" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 118 "b)  Tthe length of the curve between the points (2,0,0) \+
and (-2,3 \271,0) is the same as the curve from t=0 to t=sqrt(\271):" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "int(6*t*sqrt(5), t=0..sqr
t(Pi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%%sqrtG6#\"\"&\"\"\"%#
PiGF)\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "c)  As for the tang
ent line at t=sqrt(\271)/3, it passes through " }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 14 "r(sqrt(Pi)/3);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'vectorG6#7%\"\"\",$%#PiG#F'\"\"$*$-%%sqrtG6#F+F'" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 39 "and has direction given by the velocity" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "map(diff,r(t),t);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,$*&-%$sinG6#,$*$)%\"tG
\"\"#\"\"\"\"\"$F1F/F1!#7,$F/\"\"',$*&-%$cosGF+F1F/F1\"#7" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 16 "at t= sqrt(\271)/3:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 22 "subs(t=sqrt(Pi)/3, %);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'vectorG6#7%,$*&-%$sinG6#,$%#PiG#\"\"\"\"\"$F/-%%sqrt
G6#F-F/!\"%,$*$F1F/\"\"#,$*&-%$cosGF+F/F1F/\"\"%" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#-%'vectorG6#7%,$*&-%%sqrtG6#\"\"$\"\"\"-F*6#%#PiGF-!\"#,$*$F.F-\"\"
#F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 17 "Hence the line is" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 257 "" 0 "" {TEXT -1 83 "R(u) = (1 - 2 sqrt(3 \271) u) i + (
\271/3 + 2 sqrt(\271) u) j + (sqrt(3) + 2 sqrt(pi) u) k. " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 1 "4" 
}{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 65 "a)  Find the force on \+
the object in the moving coordinate system." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "There is no " }{TEXT 259 1 "B" 
}{TEXT -1 63 " component, because the force is just 2* acceleration, w
hich is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
10 "     s''  " }{TEXT 260 1 "T" }{TEXT -1 12 " + (s')^2 k " }{TEXT 
261 1 "N" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 72 "hence we need to calculate the speed, its derivati
ve, and the curvature." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 27 "We already know the speed, " }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 26 "speed := t -> 6*sqrt(5)*t;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&speedGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,$*&-%%sqrt
G6#\"\"&\"\"\"9$F2\"\"'F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 
"so the acceleration is  6 sqrt(5).   The only tricky part is the curv
ature, but since the curve is a helix, it won't be too bad." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "crossprod(map(diff,r(t),t), \+
map(diff,r(t),t$2));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%
,&*&%\"tG\"\"\",&*&-%$sinG6#,$*$)F)\"\"#F*\"\"$F*F2F*!#s*&\"#7F*-%$cos
GF/F*F*F*\"\"'*(\"#sF*F8F*F)F*!\"\",&*(F8F*F)F*,&*&F8F*F2F*F5*&F7F*F-F
*F=F*F7**F7F*F-F*F)F*F+F*F*,&*&F-F*F)F*F5*(F:F*F)F*F@F*F=" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "simplify(norm(%,2)/(6 * sqrt(5)* t)
^3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,(-%$absG6#*&,&!\"\"\"\"\"
*$)-%$cosG6#,$*$)%\"tG\"\"#F,\"\"$F6F,F,F,)F5\"\"'F,F,*&\"\"%F,)-F'6#F
5F9F,F,-F'6#*&F.F,F8F,F,#F,F6F5!\"$\"\"&FB#F6\"#D" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 146 "Come on, Maple, can't you see the common factor o
f t^3 in the numerator and denominator?  And what about the cancellati
on of the cosines?  Sheesh!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "curvature := 2/5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*curvatureG
#\"\"#\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 262 1 
"T" }{TEXT -1 19 " component is      " }{TEXT 264 10 "12 sqrt(5)" }
{TEXT -1 12 "    and the " }{TEXT 263 1 "N" }{TEXT -1 13 " component i
s" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "2* speed(t)^2 * curvat
ure;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%\"tG\"\"#\"\"\"\"$W\"" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "33 0 0" 0 }
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