document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=828243>Question 828243</A>: <I><SPAN STYLE=\"word-break: break-all;\"> hi! can u help me with these questions.? thank u!<BR>\n" );
document.write( "(applications of the derivative)\r<BR>\n" );
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document.write( "1) after x weeks, the number of people(or population) living in a communitty is given by the function f(x)=2x^2-4x+100 provided that x is less than or equal to 1.<BR>\n" );
document.write( "   a.) what is the average rate of change of the population from the second week (x=2) to the fifth      week (x=5)?<BR>\n" );
document.write( "   b.) find the rate of change of the population after the second week.\r<BR>\n" );
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document.write( "2) During a flu epidemic, medical researchers found out that the number of people infected in x days is given by the function p(x)=15x^2-x^3 for 0 less than or equal to x less than or equal to 10.<BR>\n" );
document.write( "   a.) find the average rate of infection for the first seven days (from day 1 to day 7).<BR>\n" );
document.write( "   b,) what was the rate of infection after the fourth day? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #499173 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=499173\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1) after x weeks, the number of people(or population) living in a community is given by the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=2x%5E2-4x%2B100 ALIGN=MIDDLE  ALT=\"f%28x%29=2x%5E2-4x%2B100\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "After that, you typed \"provided that x is less than or equal to 1\".<BR>\n" );
document.write( "That does not make sense I think you meant<BR>\n" );
document.write( "provided that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E=1 ALIGN=MIDDLE  ALT=\"x%3E=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or<BR>\n" );
document.write( "provided that x is greater than or equal to 1.<BR>\n" );
document.write( " <BR>\n" );
document.write( "a.) The average rate of change of the population from the second week (x=2) to the fifth week (x=5) can be found by dividing the difference in population by the number of weeks passed.<BR>\n" );
document.write( "So, in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2-5=3 ALIGN=MIDDLE  ALT=\"2-5=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > weeks the population went from<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%282%29=2%2A2%5E2-4%2A2%2B100=2%2A4-8%2B100=8-8%2A100 ALIGN=MIDDLE  ALT=\"f%282%29=2%2A2%5E2-4%2A2%2B100=2%2A4-8%2B100=8-8%2A100\"   DISABLED_style= \"max-width:90%; height:auto;\" > to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%285%29=2%2A5%5E2-4%2A5%2B100=2%2A25-20%2B100=50-20%2B100=130 ALIGN=MIDDLE  ALT=\"f%285%29=2%2A5%5E2-4%2A5%2B100=2%2A25-20%2B100=50-20%2B100=130\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The average rate of change of the population from the second week (x=2) to the fifth week (x=5) is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28f%285%29-f%282%29%29%2F%285-2%29=%28130-100%29%2F%285-2%29=30%2F3=highlight%2810%29 ALIGN=MIDDLE  ALT=\"%28f%285%29-f%282%29%29%2F%285-2%29=%28130-100%29%2F%285-2%29=30%2F3=highlight%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That average rate is the slope of the line connecting the points for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the graph.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-2%2C8%2C-20%2C180%2C10x%2B80%2C2x%5E2-4x%2B100%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-2%2C8%2C-20%2C180%2C10x%2B80%2C2x%5E2-4x%2B100%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "b.) find the rate of change of the population after the second week.<BR>\n" );
document.write( "At the point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (after the second week, the change of the population is given by the value of the derivative of f(x) at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If you have been taught about derivatives, you would know that 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;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=df%2Fdx ALIGN=MIDDLE  ALT=\"df%2Fdx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%29=2%2A%282x%29-4=4x-4 ALIGN=MIDDLE  ALT=\"%28x%29=2%2A%282x%29-4=4x-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > its value is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=df%2Fdx ALIGN=MIDDLE  ALT=\"df%2Fdx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%29=4%2A2-4=8-4=highlight%284%29 ALIGN=MIDDLE  ALT=\"%282%29=4%2A2-4=8-4=highlight%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That rate at the point with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the slope of the tangent to the graph at the point with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-2%2C8%2C-20%2C180%2C4x%2B92%2C2x%5E2-4x%2B100%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-2%2C8%2C-20%2C180%2C4x%2B92%2C2x%5E2-4x%2B100%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "2) During a flu epidemic, medical researchers found out that the number of people infected in x days is given by the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28x%29=15x%5E2-x%5E3 ALIGN=MIDDLE  ALT=\"p%28x%29=15x%5E2-x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=x%3C=10 ALIGN=MIDDLE  ALT=\"0%3C=x%3C=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > (for x between 0 and 10, inclusive) .<BR>\n" );
document.write( "(Clumsier alternative wordings:<BR>\n" );
document.write( "for x greater than or equal to 0, but lesser than or equal to 10,<BR>\n" );
document.write( "or if you want a more unwieldy mouthful, for x such that 0 is less than or equal to x, and x is less than or equal to 10).<BR>\n" );
document.write( "a.) find the average rate of infection for the first seven days (from day 1 to day 7).<BR>\n" );
document.write( "As done above,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%281%29=15%2A1%5E2-1%5E3=15-1=14 ALIGN=MIDDLE  ALT=\"p%281%29=15%2A1%5E2-1%5E3=15-1=14\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%287%29=15%2A7%5E2-7%5E3=15%2A49-343=735-343=392 ALIGN=MIDDLE  ALT=\"p%287%29=15%2A7%5E2-7%5E3=15%2A49-343=735-343=392\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28f%287%29-f%281%29%29%2F%287-1%29=%28392-14%29%2F%287-1%29=378%2F6=highlight%2863%29 ALIGN=MIDDLE  ALT=\"%28f%287%29-f%281%29%29%2F%287-1%29=%28392-14%29%2F%287-1%29=378%2F6=highlight%2863%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "b) what was the rate of infection after the fourth day?<BR>\n" );
document.write( "The derivative is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dp%2Fdx ALIGN=MIDDLE  ALT=\"dp%2Fdx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%29=15%2A%282x%29-3x%5E2=30x-3x%5E2 ALIGN=MIDDLE  ALT=\"%28x%29=15%2A%282x%29-3x%5E2=30x-3x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > its value is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dp%2Fdx ALIGN=MIDDLE  ALT=\"dp%2Fdx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%29=30%2A4-3%2A4%5E2=120-3%2A16=120-48=highlight%2872%29 ALIGN=MIDDLE  ALT=\"%284%29=30%2A4-3%2A4%5E2=120-3%2A16=120-48=highlight%2872%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That rate at the point with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the slope of the tangent to the graph at the point with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > , with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%284%29=15%2A4%5E2-4%5E3=15%2A16-64=240-64=176 ALIGN=MIDDLE  ALT=\"p%284%29=15%2A4%5E2-4%5E3=15%2A16-64=240-64=176\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The graphs of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28x%29 ALIGN=MIDDLE  ALT=\"p%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > with one or the other line are shown below.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-2%2C12%2C-75%2C675%2C63x-49%2C15x%5E2-x%5E3%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-2%2C12%2C-75%2C675%2C63x-49%2C15x%5E2-x%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-2%2C6%2C-30%2C270%2C72x-112%2C15x%5E2-x%5E3%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-2%2C6%2C-30%2C270%2C72x-112%2C15x%5E2-x%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > It's hard to see that the line at left is the tangent at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > , but a close-up does not help much either. The function is almost a straight line in the vicinity of that point. </SPAN>\n" );
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