document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=702280>Question 702280</A>: <I><SPAN STYLE=\"word-break: break-all;\"> slope of, y=x^2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #432883 by </B><A HREF=/tutors/aboutme.mpl?userid=neatmath><B>neatmath(302)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=neatmath><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=neatmath><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=432883\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><font face=\"Tahoma\">The equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2 ALIGN=MIDDLE  ALT=\"y=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a quadratic equation.<br>\r<BR>\n" );
document.write( "\n" );
document.write( "We can NOT readily find the slope (or rate of change) of this function like we can with a linear equation simply using Algebra.<br>\r<BR>\n" );
document.write( "\n" );
document.write( "In fact, there are an infinite amount of slopes at an infinite amount of points for this function.<br>\r<BR>\n" );
document.write( "\n" );
document.write( "These slopes are based on the slope of the tangent line to the curve at any given point.<br>\r<BR>\n" );
document.write( "\n" );
document.write( "And to find the slope of that tangent line at any given point, we would need to use Calculus.<br>\r<BR>\n" );
document.write( "\n" );
document.write( "We would need to calculate the derivative of this function to do so.<br>\r<BR>\n" );
document.write( "\n" );
document.write( "The derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is simply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x ALIGN=MIDDLE  ALT=\"2x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
document.write( "\n" );
document.write( "So at any give point x, the slope of the curve at that point is 2x.<br>\r<BR>\n" );
document.write( "\n" );
document.write( "For example, at the point (5,25)<br>\r<BR>\n" );
document.write( "\n" );
document.write( "the slope of the tangent line to the curve<br>\r<BR>\n" );
document.write( "\n" );
document.write( "is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A5 ALIGN=MIDDLE  ALT=\"2%2A5\"   DISABLED_style= \"max-width:90%; height:auto;\" > or 10.<br>\r<BR>\n" );
document.write( "\n" );
document.write( "But at the point (0,0)<br>\r<BR>\n" );
document.write( "\n" );
document.write( "the slope of the tangent line to the curve<br>\r<BR>\n" );
document.write( "\n" );
document.write( "is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%2A5 ALIGN=MIDDLE  ALT=\"0%2A5\"   DISABLED_style= \"max-width:90%; height:auto;\" > or 0, because the tangent line to the curve at (0,0) is horizontal.<br>\r<BR>\n" );
document.write( "\n" );
document.write( "Short answer:<br>\r<BR>\n" );
document.write( "\n" );
document.write( "The slope of the tangent line at any point on the curve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2 ALIGN=MIDDLE  ALT=\"y=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
document.write( "\n" );
document.write( "is given by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x ALIGN=MIDDLE  ALT=\"2x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
document.write( "\n" );
document.write( "which is known as the derivative of y wth respect to x.<br>\r<BR>\n" );
document.write( "\n" );
document.write( "I hope this helps!  Keep practicing!  :)<br>\r<BR>\n" );
document.write( "\n" );
document.write( "*******************************************************<br>\r<BR>\n" );
document.write( "\n" );
document.write( "Email Scott King: neatmath@yahoo.com for help with specific problems,<br>\r<BR>\n" );
document.write( "\n" );
document.write( "or to inquire about income sensitive, low-cost or no-cost mathematics tutoring via email or other methods.<br>\r<BR>\n" );
document.write( "\n" );
document.write( "Paypal is always accepted for detailed assistance with single problems, send contributions to: neatmath@yahoo.com<br>\r<BR>\n" );
document.write( "\n" );
document.write( "Single problems would range from 50 cents to 5 dollars each, depending on their complexity.<br>\r<BR>\n" );
document.write( "\n" );
document.write( "Assistance for single problems can be done via email with scans or photos to help illustrate concepts.<br>\r<BR>\n" );
document.write( "\n" );
document.write( "Thank you and good luck with all your future studies!<br><br> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );