document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168270>Question 1168270</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let  f(x)=4x^2+5x+4 and let g(h)= f(1+h)−f(1)/h\r<BR>\n" );
document.write( "\n" );
document.write( "Determine each of the following:<BR>\n" );
document.write( "(a)  g(1)=    <BR>\n" );
document.write( "(b)  g(0.1)=   <BR>\n" );
document.write( "(c)  g(0.01)=  <BR>\n" );
document.write( " <BR>\n" );
document.write( "You will notice that the values that you entered are getting closer and closer to a number L. This number is called the limit of g(h)as h approaches  0 and is also called the derivative of f(x) at the point when x=1.\r<BR>\n" );
document.write( "\n" );
document.write( "Enter the value of L:  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851601 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851601\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=4x%5E2%2B5x%2B4 ALIGN=MIDDLE  ALT=\"f%28x%29=4x%5E2%2B5x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and let<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+g%28h%29=+%28f%281%2Bh%29-f%281%29%29%2Fh ALIGN=MIDDLE  ALT=\"+g%28h%29=+%28f%281%2Bh%29-f%281%29%29%2Fh\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Determine each of the following:\r<BR>\n" );
document.write( "\n" );
document.write( "(a) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%281%29 ALIGN=MIDDLE  ALT=\"g%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%281%29=+%28f%281%2B1%29-f%281%29%29%2F1=f%282%29-f%281%29 ALIGN=MIDDLE  ALT=\"g%281%29=+%28f%281%2B1%29-f%281%29%29%2F1=f%282%29-f%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%282%29=4%2A2%5E2%2B5%2A2%2B4=16%2B10%2B4=30 ALIGN=MIDDLE  ALT=\"f%282%29=4%2A2%5E2%2B5%2A2%2B4=16%2B10%2B4=30\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%281%29=4%2A1%5E2%2B5%2A1%2B4=4%2B5%2B4=13 ALIGN=MIDDLE  ALT=\"f%281%29=4%2A1%5E2%2B5%2A1%2B4=4%2B5%2B4=13\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "=> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%281%29=30-13=17 ALIGN=MIDDLE  ALT=\"g%281%29=30-13=17\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(b) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%280.1%29=%28f%281%2B0.1%29-f%281%29%29%2F0.1=10f%281.1%29+-+10f%281%29 ALIGN=MIDDLE  ALT=\"g%280.1%29=%28f%281%2B0.1%29-f%281%29%29%2F0.1=10f%281.1%29+-+10f%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10f%281.1%29=10%2A%284%281.1%29%5E2%2B5%281.1%29%2B4%29=143.4 ALIGN=MIDDLE  ALT=\"10f%281.1%29=10%2A%284%281.1%29%5E2%2B5%281.1%29%2B4%29=143.4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10f%281%29=10%2A%284%281%29%5E2%2B5%281%29%2B4%29=130 ALIGN=MIDDLE  ALT=\"10f%281%29=10%2A%284%281%29%5E2%2B5%281%29%2B4%29=130\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%280.1%29=143.4-130=13.4+ ALIGN=MIDDLE  ALT=\"g%280.1%29=143.4-130=13.4+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(c) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%280.01%29 ALIGN=MIDDLE  ALT=\"g%280.01%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=\r<BR>\n" );
document.write( "\n" );
document.write( "similarly\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%280.01%29+=+100+f%281.01%29+-+100+f%281%29 ALIGN=MIDDLE  ALT=\"g%280.01%29+=+100+f%281.01%29+-+100+f%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100+f%281.01%29=100+%284%281.01%29%5E2%2B5%281.01%29%2B4%29=1313.04 ALIGN=MIDDLE  ALT=\"100+f%281.01%29=100+%284%281.01%29%5E2%2B5%281.01%29%2B4%29=1313.04\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100+f%281%29=100+%284%281%29%5E2%2B5%281%29%2B4%29=1300 ALIGN=MIDDLE  ALT=\"100+f%281%29=100+%284%281%29%5E2%2B5%281%29%2B4%29=1300\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%280.01%29=1313.04-1300=13.04 ALIGN=MIDDLE  ALT=\"g%280.01%29=1313.04-1300=13.04\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "You will notice that the values that you entered are getting closer and closer to a number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L ALIGN=MIDDLE  ALT=\"L\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This number is called the limit of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28h%29 ALIGN=MIDDLE  ALT=\"g%28h%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" > approaches <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and is also called the derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > at the point when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28h-%3E0%2C%28f%281+%2B+h%29+-+f%281%29%29%2Fh+%29 ALIGN=MIDDLE  ALT=\"lim%28h-%3E0%2C%28f%281+%2B+h%29+-+f%281%29%29%2Fh+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" >'<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%29 ALIGN=MIDDLE  ALT=\"%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=4x%5E2%2B5x%2B4+ ALIGN=MIDDLE  ALT=\"f%28x%29=4x%5E2%2B5x%2B4+\"   DISABLED_style= \"max-width:90%; height:auto;\" > is\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" >'<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%29=8x%2B5 ALIGN=MIDDLE  ALT=\"%28x%29=8x%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" >..........at the point when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" >'<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%29=8%2A1%2B5=13 ALIGN=MIDDLE  ALT=\"%281%29=8%2A1%2B5=13\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "so, the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L ALIGN=MIDDLE  ALT=\"L\"   DISABLED_style= \"max-width:90%; height:auto;\" > is: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13 ALIGN=MIDDLE  ALT=\"13\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "as you can see, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=17 ALIGN=MIDDLE  ALT=\"17\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13.4 ALIGN=MIDDLE  ALT=\"13.4\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13.04 ALIGN=MIDDLE  ALT=\"13.04\"   DISABLED_style= \"max-width:90%; height:auto;\" > are getting closer and closer to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13 ALIGN=MIDDLE  ALT=\"13\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );