document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1036508>Question 1036508</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am really having a hard time with this one. Been working on it for nearly 3 hours and cant seem to figure out how to make it work.\r<BR>\n" );
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document.write( "Problem: r(t) is the position of a particle in the xy-plane at time t. Find an equation in x and y whose graph is the path of the particle. Then find the particle’s velocity and acceleration vectors at the given value of t.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+r%28t%29+=+e%5Et+i+%2B+%282%2F9%29+e%5E%282t%29+j+ ALIGN=MIDDLE  ALT=\"+r%28t%29+=+e%5Et+i+%2B+%282%2F9%29+e%5E%282t%29+j+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+t+=+ln3+ ALIGN=MIDDLE  ALT=\"+t+=+ln3+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "To find the Equation, I first organize it into a set:<BR>\n" );
document.write( "[ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+e%5Et+ ALIGN=MIDDLE  ALT=\"+e%5Et+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%282%2F9%29+e%5E%282t%29+ ALIGN=MIDDLE  ALT=\"+%282%2F9%29+e%5E%282t%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ]<BR>\n" );
document.write( "Then I just plug in the value of t (ln3)<BR>\n" );
document.write( "[ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+e%5E%28ln3%29+ ALIGN=MIDDLE  ALT=\"+e%5E%28ln3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%282%2F9%29+e%5E%282%28ln3%29%29+ ALIGN=MIDDLE  ALT=\"+%282%2F9%29+e%5E%282%28ln3%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ]<BR>\n" );
document.write( "I then reconstruct the original problem with the new values:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+r%28t%29+=+e%5E%28ln3%29+i+%2B+%282%2F9%29+e%5E%282%28ln3%29%29+j+ ALIGN=MIDDLE  ALT=\"+r%28t%29+=+e%5E%28ln3%29+i+%2B+%282%2F9%29+e%5E%282%28ln3%29%29+j+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+r%28ln3%29+=+e%5E%281.0986%29+i+%2B+%282%2F9%29+e%5E%282.1972%29+j+ ALIGN=MIDDLE  ALT=\"+r%28ln3%29+=+e%5E%281.0986%29+i+%2B+%282%2F9%29+e%5E%282.1972%29+j+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "then I change the i / j coordinates to x/y coordinates<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+r%28ln3%29+=+e%5E%281.0986%29+x+%2B+%282%2F9%29e%5E%282.1972%29+y+ ALIGN=MIDDLE  ALT=\"+r%28ln3%29+=+e%5E%281.0986%29+x+%2B+%282%2F9%29e%5E%282.1972%29+y+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----------------------------\r<BR>\n" );
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document.write( "As for velocity and acceleration, so far I have figured it like this:<BR>\n" );
document.write( "Velocity:<BR>\n" );
document.write( "[ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+e%5Et+ ALIGN=MIDDLE  ALT=\"+e%5Et+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%282%2F9%29+e%5E%282t%29+ ALIGN=MIDDLE  ALT=\"+%282%2F9%29+e%5E%282t%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ]<BR>\n" );
document.write( "[ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+te+ ALIGN=MIDDLE  ALT=\"+te+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%284%2F9%29+e%5Et+ ALIGN=MIDDLE  ALT=\"+%284%2F9%29+e%5Et+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ]<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+Velocity+=+tei+%2B+%284%2F9%29etj+ ALIGN=MIDDLE  ALT=\"+Velocity+=+tei+%2B+%284%2F9%29etj+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Acceleration:<BR>\n" );
document.write( "[ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+e+ ALIGN=MIDDLE  ALT=\"+e+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%284%2F9%29+te+ ALIGN=MIDDLE  ALT=\"+%284%2F9%29+te+\"   DISABLED_style= \"max-width:90%; height:auto;\" >]<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+Acceleration+=+tei+%2B+%284%2F9%29+tej+ ALIGN=MIDDLE  ALT=\"+Acceleration+=+tei+%2B+%284%2F9%29+tej+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Am I taking the derivative correctly? As far as I know, e remains as e, even after the derivative, right?\r<BR>\n" );
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document.write( "thanks in advance  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #651227 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=651227\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I think you meant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+r%28t%29+=+e%5Et%2A+i+%2B+%282%2F9%29+e%5E%282t%29%2A+j+ ALIGN=MIDDLE  ALT=\"+r%28t%29+=+e%5Et%2A+i+%2B+%282%2F9%29+e%5E%282t%29%2A+j+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is a vector-valued function.<BR>\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+e%5Et ALIGN=MIDDLE  ALT=\"x+=+e%5Et\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%282%2F9%29+e%5E%282t%29 ALIGN=MIDDLE  ALT=\"y+=+%282%2F9%29+e%5E%282t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as they were meant to be.\r<BR>\n" );
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document.write( "Then after a simple substitution, you should get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F9%29+%28e%5Et%29%5E2+=+%282%2F9%29x%5E2 ALIGN=MIDDLE  ALT=\"y=%282%2F9%29+%28e%5Et%29%5E2+=+%282%2F9%29x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+Velocity+=+e%5Et%2Ai+%2B+%284%2F9%29e%5E%282t%29%2Aj+ ALIGN=MIDDLE  ALT=\"+Velocity+=+e%5Et%2Ai+%2B+%284%2F9%29e%5E%282t%29%2Aj+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+Acceleration+=+e%5Et%2Ai+%2B+%288%2F9%29e%5E%282t%29%2Aj+ ALIGN=MIDDLE  ALT=\"+Acceleration+=+e%5Et%2Ai+%2B+%288%2F9%29e%5E%282t%29%2Aj+\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );