document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1176449>Question 1176449</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find and simplify the difference quotient for the given function.\r<BR>\n" );
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document.write( "f(x) = 6x2\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #803390 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=803390\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "6x^2 is the same as writing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2 ALIGN=MIDDLE  ALT=\"6x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+6x%5E2 ALIGN=MIDDLE  ALT=\"f%28x%29+=+6x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%2Bh%29+=+6%28x%2Bh%29%5E2 ALIGN=MIDDLE  ALT=\"f%28x%2Bh%29+=+6%28x%2Bh%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > We replace every x with x+h\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%2Bh%29+=+6%28x%5E2%2B2xh%2Bh%5E2%29 ALIGN=MIDDLE  ALT=\"f%28x%2Bh%29+=+6%28x%5E2%2B2xh%2Bh%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > FOIL \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%2Bh%29-f%28x%29+=+6%28x%5E2%2B2xh%2Bh%5E2%29-6x%5E2 ALIGN=MIDDLE  ALT=\"f%28x%2Bh%29-f%28x%29+=+6%28x%5E2%2B2xh%2Bh%5E2%29-6x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract f(x+h) and f(x). This is the \"difference\" portion of \"difference quotient\".\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%2Bh%29-f%28x%29+=+6x%5E2%2B12xh%2B6h%5E2-6x%5E2 ALIGN=MIDDLE  ALT=\"f%28x%2Bh%29-f%28x%29+=+6x%5E2%2B12xh%2B6h%5E2-6x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%2Bh%29-f%28x%29+=+12xh%2B6h%5E2 ALIGN=MIDDLE  ALT=\"f%28x%2Bh%29-f%28x%29+=+12xh%2B6h%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > The 6x^2 terms subtract and cancel.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+%2812xh%2B6h%5E2%29%2Fh ALIGN=MIDDLE  ALT=\"%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+%2812xh%2B6h%5E2%29%2Fh\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 'h'. This is where the \"quotient\" part comes in of \"difference quotient\".\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+%28h%2812x%2B6h%29%29%2Fh ALIGN=MIDDLE  ALT=\"%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+%28h%2812x%2B6h%29%29%2Fh\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out h from the numerator\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+12x%2B6h ALIGN=MIDDLE  ALT=\"%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+12x%2B6h\"   DISABLED_style= \"max-width:90%; height:auto;\" > Cancel the pair of 'h' terms from the numerator and denominator\r<BR>\n" );
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document.write( "<font color=red>Answer: </font><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+12x%2B6h ALIGN=MIDDLE  ALT=\"%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+12x%2B6h\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Side note: Later in calculus you'll learn that as h approaches 0, the expression 12x+6h will approach 12x. This is the derivative of f(x) = 6x^2. Keep in mind that h = 0 is not allowed, since division by zero is not allowed, but we can get closer and closer to it. <BR>\n" );
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