document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=894150>Question 894150</A>: <I><SPAN STYLE=\"word-break: break-all;\"> if a graph is created showing ln z against t for this :  z= 2e^(4t) - what is the gradient, and how do I work it out - I am struggling with understanding a text book. Please note this is not the actual problem I need to solve - it is similar - I need to understand how to do this type of question.<BR>\n" );
document.write( "I have tried to follow the text and thought the gradient was equal to the power when I multiply by base e and reverse the order etc then use the line equation y=mx+c. Please can you help. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #541791 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=541791\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The gradient is the derivative of the function.<BR>\n" );
document.write( "In this case,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=2e%5E%284t%29 ALIGN=MIDDLE  ALT=\"z=2e%5E%284t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dz%2Fdt=2e%5E%284t%29%2A4 ALIGN=MIDDLE  ALT=\"dz%2Fdt=2e%5E%284t%29%2A4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dz%2Fdt=8e%5E4t ALIGN=MIDDLE  ALT=\"dz%2Fdt=8e%5E4t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The value of the tangent line to the function at any point is the value of the derivative at that point. <BR>\n" );
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document.write( "As an example, when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=2 ALIGN=MIDDLE  ALT=\"z=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dz%2Fdt=8 ALIGN=MIDDLE  ALT=\"dz%2Fdt=8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the equation of the tangent line is,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z-2=8%28t-0%29 ALIGN=MIDDLE  ALT=\"z-2=8%28t-0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=8t%2B2 ALIGN=MIDDLE  ALT=\"z=8t%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "On the graph, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=t ALIGN=MIDDLE  ALT=\"x=t\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=z ALIGN=MIDDLE  ALT=\"y=z\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-2%2C2%2C-1%2C5%2C2e%5E%284x%29%2C8x%2B2%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-2%2C2%2C-1%2C5%2C2e%5E%284x%29%2C8x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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