document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=958546>Question 958546</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Of the infinitely many lines that are tangent to the curve of y = -sin x and pass through the origin, there is one that has the largest slope. Use Newton's method to find the slope of that line correct to six decimal places. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #585936 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=585936\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-7%2C7%2C-2%2C2%2C-sin%28x%29%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-7%2C7%2C-2%2C2%2C-sin%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "A line tangent to the curve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-sin%28x%29 ALIGN=MIDDLE  ALT=\"y=-sin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > at point has a slope of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=-cos%28x%29 ALIGN=MIDDLE  ALT=\"m=-cos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Since it goes through point (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and the origin then the slope also equals.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=%28y-0%29%2F%28x-0%29=y%2Fx=-sin%28x%29%2Fx ALIGN=MIDDLE  ALT=\"m=%28y-0%29%2F%28x-0%29=y%2Fx=-sin%28x%29%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So then, <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sin%28x%29%2Fx=-cos%28x%29 ALIGN=MIDDLE  ALT=\"-sin%28x%29%2Fx=-cos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29-xcos%28x%29=0 ALIGN=MIDDLE  ALT=\"sin%28x%29-xcos%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So to find the root, let<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=sin%28x%29-xcos%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29=sin%28x%29-xcos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=df%2Fdx=xsin%28x%29 ALIGN=MIDDLE  ALT=\"df%2Fdx=xsin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now using Newton's method with a starting value of <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;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/display-illustration.mpl?tutor=Fombitz&illustration_name=new.JPG\">.<BR>\n" );
document.write( "So then at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4.493409 ALIGN=MIDDLE  ALT=\"x=4.493409\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the slope is equal to,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=-cos%284.493409%29 ALIGN=MIDDLE  ALT=\"m=-cos%284.493409%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28m=0.217234%29 ALIGN=MIDDLE  ALT=\"highlight%28m=0.217234%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-7%2C7%2C-2%2C2%2C-sin%28x%29%2C0.217234x%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-7%2C7%2C-2%2C2%2C-sin%28x%29%2C0.217234x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );