document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=661150>Question 661150</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the average rate of change of the function f(x)=sqrt(1-2x^2) between x=-a and x=-a+h. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #411541 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=411541\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "The average rate of change of a function between two points is given by the formula for the slope of the secant line through the two points.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ m\ =\ \frac{f(x_2)\ -\ f(x_1)}{x_2\ -\ x_1} \">\r<BR>\n" );
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document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(x_1,f(x_1)\right)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(x_2,f(x_2)\right)\"> are the coordinates of the given points.\r<BR>\n" );
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document.write( "Where the two abscissas differ only by a constant, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE h\"> in your problem, the denominator simplifies to just that constant.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ m\ =\ \frac{f(-a\ +\ h)\ -\ f(-a)}{h} \">\r<BR>\n" );
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document.write( "All you need to do is to evaluate the function at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE -a\ +\ h\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE -a\">, simplify the numerator, divide out the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE h\"> because every term in the simplified numerator (given correct algebra) will contain at least one factor of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE h\">.\r<BR>\n" );
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document.write( "In the case of your problem, simplifying the numerator is somewhat more than a trivial exercise because you will have the difference of two expressions containing radicals and you will not be able to reduce the two radicands to be equal expressions.  The trick is to rationalize the numerator.  The process is the same as rationalizing a denominator, except that you will choose the conjugate of the numerator so that when you multiply you get the difference of two squares effectively eliminating the numerator radicals.  You will still have radical expressions in the denominator, but you will be able to eliminate the factor of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE h\"> in the denominator because you have taken all of the factors of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE h\"> out of the radicals in the numerator.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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