document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1086729>Question 1086729</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Evaluate [f(x+h)-f(x)]/h and simplify if f(x)=x^2+x+1<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 #700941 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=700941\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\"><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E2%2Bx%2B1 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2%2Bx%2B1\"   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=%28x%2Bh%29%5E2%2B%28x%2Bh%29%2B1 ALIGN=MIDDLE  ALT=\"f%28x%2Bh%29=%28x%2Bh%29%5E2%2B%28x%2Bh%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > 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=x%5E2%2B2xh%2Bh%5E2%2Bx%2Bh%2B1 ALIGN=MIDDLE  ALT=\"f%28x%2Bh%29=x%5E2%2B2xh%2Bh%5E2%2Bx%2Bh%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Expand out (x+h)^2\r<BR>\n" );
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document.write( "----------------------------\r<BR>\n" );
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document.write( "Now that we know what f(x+h) is equal to, we can find the difference quotient\r<BR>\n" );
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document.write( "We'll be making two substitutions. The first substitution will have us replace the \"f(x+h)\" with the expression we got above\r<BR>\n" );
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document.write( "So we go from this\r<BR>\n" );
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document.write( "to this\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28highlight%28f%28x%2Bh%29%29-f%28x%29%29%2Fh=%28highlight%28x%5E2%2B2xh%2Bh%5E2%2Bx%2Bh%2B1%29-f%28x%29%29%2Fh ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Note the red boxes highlighted to show exactly what is replaced and how it looks after the replacement is done\r<BR>\n" );
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document.write( "Similarly, we will replace f(x) with x^2+x+1 to go from this\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-highlight%28f%28x%29%29%29%2Fh=%28x%5E2%2B2xh%2Bh%5E2%2Bx%2Bh%2B1-%28highlight%28f%28x%29%29%29%29%2Fh ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "to this\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-highlight%28f%28x%29%29%29%2Fh=%28x%5E2%2B2xh%2Bh%5E2%2Bx%2Bh%2B1-%28highlight%28x%5E2%2Bx%2B1%29%29%29%2Fh ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "----------------------------\r<BR>\n" );
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document.write( "After the two substitutions, we now have\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=%28x%5E2%2B2xh%2Bh%5E2%2Bx%2Bh%2B1-%28x%5E2%2Bx%2B1%29%29%2Fh ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's simplify the right hand side (RHS)\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=%28x%5E2%2B2xh%2Bh%5E2%2Bx%2Bh%2B1-x%5E2-x-1%29%2Fh ALIGN=MIDDLE  ALT=\"%28f%28x%2Bh%29-f%28x%29%29%2Fh=%28x%5E2%2B2xh%2Bh%5E2%2Bx%2Bh%2B1-x%5E2-x-1%29%2Fh\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute the negative through\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=%282xh%2Bh%5E2%2Bx%2Bh%2B1-x-1%29%2Fh ALIGN=MIDDLE  ALT=\"%28f%28x%2Bh%29-f%28x%29%29%2Fh=%282xh%2Bh%5E2%2Bx%2Bh%2B1-x-1%29%2Fh\"   DISABLED_style= \"max-width:90%; height:auto;\" > Note how x^2-x^2 = 0x^2 = 0 so the x^2 terms go away\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=%282xh%2Bh%5E2%2Bh%2B1-1%29%2Fh ALIGN=MIDDLE  ALT=\"%28f%28x%2Bh%29-f%28x%29%29%2Fh=%282xh%2Bh%5E2%2Bh%2B1-1%29%2Fh\"   DISABLED_style= \"max-width:90%; height:auto;\" > Similarly, x-x = 0x = 0\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=%282xh%2Bh%5E2%2Bh%29%2Fh ALIGN=MIDDLE  ALT=\"%28f%28x%2Bh%29-f%28x%29%29%2Fh=%282xh%2Bh%5E2%2Bh%29%2Fh\"   DISABLED_style= \"max-width:90%; height:auto;\" > And also 1-1 = 0. Each term now has something with an 'h' as a factor\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%282x%2Bh%2B1%29%29%2Fh ALIGN=MIDDLE  ALT=\"%28f%28x%2Bh%29-f%28x%29%29%2Fh=%28h%282x%2Bh%2B1%29%29%2Fh\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out that common term h\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=%28cross%28h%29%282x%2Bh%2B1%29%29%2F%28cross%28h%29%29 ALIGN=MIDDLE  ALT=\"%28f%28x%2Bh%29-f%28x%29%29%2Fh=%28cross%28h%29%282x%2Bh%2B1%29%29%2F%28cross%28h%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > The 'h's cancel on top and bottom (since h/h = 1)\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=2x%2Bh%2B1 ALIGN=MIDDLE  ALT=\"%28f%28x%2Bh%29-f%28x%29%29%2Fh=2x%2Bh%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "----------------------------<BR>\n" );
document.write( "----------------------------\r<BR>\n" );
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document.write( "The final simplified answer is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2Bh%2B1 ALIGN=MIDDLE  ALT=\"2x%2Bh%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ></font> </SPAN>\n" );
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