document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1099289>Question 1099289</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find f(x) given that<BR>\n" );
document.write( "f'(x) = x^3 &#8722; 2&#8730;x, f(1) = 4 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #713721 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=713721\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The derivative of a sum of functions is the sum of the derivatives,<BR>\n" );
document.write( "so the same applies to antiderivatives,<BR>\n" );
document.write( "So to find a function with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-2sqrt%28x%29 ALIGN=MIDDLE  ALT=\"x%5E3-2sqrt%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for a derivative,<BR>\n" );
document.write( "we need to find what function has for derivative <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3 ALIGN=MIDDLE  ALT=\"x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and what function has for derivative <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%29=x%5E%221+%2F+2%22 ALIGN=MIDDLE  ALT=\"sqrt%28x%29=x%5E%221+%2F+2%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We also know that if we find a function with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-2sqrt%28x%29 ALIGN=MIDDLE  ALT=\"x%5E3-2sqrt%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for a derivative,<BR>\n" );
document.write( "adding any constant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C ALIGN=MIDDLE  ALT=\"C\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we get another function with the same derivative.<BR>\n" );
document.write( "We will use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%281%29=4 ALIGN=MIDDLE  ALT=\"f%281%29=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to figure out what constant to add.<BR>\n" );
document.write( " <BR>\n" );
document.write( "We know about derivatives of polynomials.<BR>\n" );
document.write( "For any exponent <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > (integer or not, as long as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%3C%3E-1 ALIGN=MIDDLE  ALT=\"r%3C%3E-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ), <BR>\n" );
document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=x%5E%28r%2B1%29%2F%28r%2B1%29 ALIGN=MIDDLE  ALT=\"g%28x%29=x%5E%28r%2B1%29%2F%28r%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22g+%27+%28+x+%29%22=x%5Er ALIGN=MIDDLE  ALT=\"%22g+%27+%28+x+%29%22=x%5Er\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and that lets us figure out the antiderivative of any <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5Er ALIGN=MIDDLE  ALT=\"x%5Er\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "except <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%28-1%29=1%2Fx ALIGN=MIDDLE  ALT=\"x%5E%28-1%29=1%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "So, the antiderivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3 ALIGN=MIDDLE  ALT=\"x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2F4 ALIGN=MIDDLE  ALT=\"x%5E4%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > ;<BR>\n" );
document.write( "and the antiderivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%29=x%5E%221+%2F+2%22 ALIGN=MIDDLE  ALT=\"sqrt%28x%29=x%5E%221+%2F+2%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%223+%2F+2%22%2F%223+%2F+2%22=%282%2F3%29x%5E%223+%2F+2%22=2x%5E%223+%2F+2%22%2F3 ALIGN=MIDDLE  ALT=\"x%5E%223+%2F+2%22%2F%223+%2F+2%22=%282%2F3%29x%5E%223+%2F+2%22=2x%5E%223+%2F+2%22%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "So, we can see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E4%2F4-2%282x%5E%223+%2F+2%22%2F3%29%2BC=x%5E4%2F4-4x%5E%223+%2F+2%22%2F3%2BC ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E4%2F4-2%282x%5E%223+%2F+2%22%2F3%29%2BC=x%5E4%2F4-4x%5E%223+%2F+2%22%2F3%2BC\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "for some constant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C ALIGN=MIDDLE  ALT=\"C\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Then, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%281%29=1%2F4-4%2F3%2BC ALIGN=MIDDLE  ALT=\"f%281%29=1%2F4-4%2F3%2BC\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%281%29=4 ALIGN=MIDDLE  ALT=\"f%281%29=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4-4%2F3%2BC=4 ALIGN=MIDDLE  ALT=\"1%2F4-4%2F3%2BC=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=4%2B4%2F3-1%2F4 ALIGN=MIDDLE  ALT=\"C=4%2B4%2F3-1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=61%2F12 ALIGN=MIDDLE  ALT=\"C=61%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28f%28x%29=x%5E4%2F4-4x%5E%223%2F2%22%2F3%2B61%2F12%29 ALIGN=MIDDLE  ALT=\"highlight%28f%28x%29=x%5E4%2F4-4x%5E%223%2F2%22%2F3%2B61%2F12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );