document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1023433>Question 1023433</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Graph the function f(x) = x+sqrt(abs(x)). <BR>\n" );
document.write( "Consider the behavior of the function at the point (-1,0) and at the origin. Find the limit as x approaches -1 and as x approaches 0. <BR>\n" );
document.write( "What is different about the behavior of f(x) near those points? Explain. <BR>\n" );
document.write( "Next, graph its derivative. Discuss differentiability at -1 and 0. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #638943 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=638943\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The roots of the function are at x = -1 and x = 0.<BR>\n" );
document.write( "As x approaches -1 from both sides, y approaches 0.  For negative x, <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28x-%3E-1%2C+%28x%2Bsqrt%28-x%29%29%29+=+0 ALIGN=MIDDLE  ALT=\"lim%28x-%3E-1%2C+%28x%2Bsqrt%28-x%29%29%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "To find the limit at x = 0, the left-hand limit (through negative values) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28x-%3E0%2C+%28x%2Bsqrt%28-x%29%29%29+=+0 ALIGN=MIDDLE  ALT=\"lim%28x-%3E0%2C+%28x%2Bsqrt%28-x%29%29%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, while the right-hand limit (through positive values) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28x-%3E0%2C+%28x%2Bsqrt%28x%29%29%29+=+0 ALIGN=MIDDLE  ALT=\"lim%28x-%3E0%2C+%28x%2Bsqrt%28x%29%29%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Thus <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28x-%3E0%2C+%28x%2Bsqrt%28abs%28x%29%29%29%29+=+0 ALIGN=MIDDLE  ALT=\"lim%28x-%3E0%2C+%28x%2Bsqrt%28abs%28x%29%29%29%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "For x < 0, f'(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-1%2F%282sqrt%28-x%29%29 ALIGN=MIDDLE  ALT=\"1-1%2F%282sqrt%28-x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, hence the derivative exists for all such x values, and the graph of f(x) is smooth there. Incidentally, at x = -1, f'(-1) = 1/2.<BR>\n" );
document.write( "For x >0, f'(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%2B+1%2F%282sqrt%28x%29%29 ALIGN=MIDDLE  ALT=\"1+%2B+1%2F%282sqrt%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, hence the the derivative exists for all such x values, and the graph of f(x) is smooth there.<BR>\n" );
document.write( "For x = 0:<BR>\n" );
document.write( "The left hand derivative at x = 0 is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28h-%3E0%2C+%28f%28h%29+-+f%280%29%29%2Fh%29+ ALIGN=MIDDLE  ALT=\"lim%28h-%3E0%2C+%28f%28h%29+-+f%280%29%29%2Fh%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > through negative values of h..<BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28h-%3E0%2C+f%28h%29%2Fh%29+=+lim%28h-%3E0%2C+%28h%2Bsqrt%28-h%29%29%2Fh%29+ ALIGN=MIDDLE  ALT=\"lim%28h-%3E0%2C+f%28h%29%2Fh%29+=+lim%28h-%3E0%2C+%28h%2Bsqrt%28-h%29%29%2Fh%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28h-%3E0%2C+%281%2Bsqrt%28-h%29%29%2Fh%29+ ALIGN=MIDDLE  ALT=\"lim%28h-%3E0%2C+%281%2Bsqrt%28-h%29%29%2Fh%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28h-%3E0%2C+1-1%2Fsqrt%28-h%29%29+=+-infinity ALIGN=MIDDLE  ALT=\"lim%28h-%3E0%2C+1-1%2Fsqrt%28-h%29%29+=+-infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The right hand derivative at x = 0 is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28h-%3E0%2C+%28f%28h%29+-+f%280%29%29%2Fh%29+ ALIGN=MIDDLE  ALT=\"lim%28h-%3E0%2C+%28f%28h%29+-+f%280%29%29%2Fh%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > through positive values of h..<BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28h-%3E0%2C+f%28h%29%2Fh%29+=+lim%28h-%3E0%2C+%28h%2Bsqrt%28h%29%29%2Fh%29+ ALIGN=MIDDLE  ALT=\"lim%28h-%3E0%2C+f%28h%29%2Fh%29+=+lim%28h-%3E0%2C+%28h%2Bsqrt%28h%29%29%2Fh%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28h-%3E0%2C+%281%2Bsqrt%28h%29%29%2Fh%29+ ALIGN=MIDDLE  ALT=\"lim%28h-%3E0%2C+%281%2Bsqrt%28h%29%29%2Fh%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28h-%3E0%2C+1%2B1%2Fsqrt%28h%29%29+=+%2Binfinity ALIGN=MIDDLE  ALT=\"lim%28h-%3E0%2C+1%2B1%2Fsqrt%28h%29%29+=+%2Binfinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "hence the left-hand and right-hand derivatives are not equal (and opposite infinites) and so there is a cusp at x = 0.\r<BR>\n" );
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document.write( "The graph of f(x) is as follows:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+600%2C+400%2C+-5%2C+5%2C+-5%2C+5%2C+x+%2B+sqrt%28-x%29%2C+x%2Bsqrt%28x%29%29 ALIGN=MIDDLE  ALT=\"graph%28+600%2C+400%2C+-5%2C+5%2C+-5%2C+5%2C+x+%2B+sqrt%28-x%29%2C+x%2Bsqrt%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <BR>\n" );
document.write( "its derivative f'(x):\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+600%2C+400%2C+-5%2C+5%2C+-5%2C+5%2C+1+-+1%2F%282sqrt%28-x%29%29%2C+1+%2B+1%2F%282sqrt%28x%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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