document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192494>Question 1192494</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find where the tangent to the curve y=x^3 at the point where x=2, meets the curve again </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #824378 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=824378\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "f(x) = x^3<br><BR>\n" );
document.write( "f(2) = 8<br><BR>\n" );
document.write( "f'(x) = 3x^2<br><BR>\n" );
document.write( "f'(2) = 12<br><BR>\n" );
document.write( "The tangent has slope 12 and passes through the point (2,8); its equation is y=12x-16.<br><BR>\n" );
document.write( "We are to find the other value of x where the graphs of x^3 and 12x-16 intersect.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+=+12x-16 ALIGN=MIDDLE  ALT=\"x%5E3+=+12x-16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-12x%2B16=0 ALIGN=MIDDLE  ALT=\"x%5E3-12x%2B16=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "We could use standard methods to try to find how that cubic expression factors.  However, we know something about that expression that makes it easy to factor.<br><BR>\n" );
document.write( "We know that the graph of y=12x-16 is tangent to the graph of y=x^3 at x=2.  That means that cubic equation has a double root at x=2.  Removing that double root shows us that<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-12x%2B16=%28x%2B4%29%28x-2%29%5E2 ALIGN=MIDDLE  ALT=\"x%5E3-12x%2B16=%28x%2B4%29%28x-2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "so the other point where the tangent intersects the graph of y=x^3 is when x=-4; that point of intersection is (-4,-64).<br><BR>\n" );
document.write( "A graph....<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-5%2C5%2C-100%2C60%2Cx%5E3%2C12x-16%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-5%2C5%2C-100%2C60%2Cx%5E3%2C12x-16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );