document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200529>Question 1200529</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the function f(x) = x^2 + 9\r<BR>\n" );
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document.write( "a. Demonstrate how to find the average rate of change from x= -3 to x= 1.\r<BR>\n" );
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document.write( "b. Demonstrate algebraically how to find the simplification of f(a+h)-f(a)/h for the given f(x).\r<BR>\n" );
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document.write( "c. Let -3 = a, and 1 = a+h, find h. Put that into the simplification in part b. Compare it to the answer for part a. What do you notice? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848066 by </B><A HREF=/tutors/aboutme.mpl?userid=GingerAle><B>GingerAle(43)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=GingerAle><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=848066\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a. Find the Average Rate of Change**\r<BR>\n" );
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document.write( "* **Calculate f(-3):** <BR>\n" );
document.write( "    f(-3) = (-3)² + 9 = 9 + 9 = 18\r<BR>\n" );
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document.write( "* **Calculate f(1):**<BR>\n" );
document.write( "    f(1) = (1)² + 9 = 1 + 9 = 10\r<BR>\n" );
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document.write( "* **Calculate the Average Rate of Change:**<BR>\n" );
document.write( "    Average Rate of Change = (f(1) - f(-3)) / (1 - (-3)) <BR>\n" );
document.write( "                              = (10 - 18) / (1 + 3) <BR>\n" );
document.write( "                              = -8 / 4 <BR>\n" );
document.write( "                              = -2\r<BR>\n" );
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document.write( "**Therefore, the average rate of change of f(x) from x = -3 to x = 1 is -2.**\r<BR>\n" );
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document.write( "**b. Simplify f(a+h) - f(a) / h**\r<BR>\n" );
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document.write( "1. **Find f(a+h):**<BR>\n" );
document.write( "   f(a+h) = (a+h)² + 9 <BR>\n" );
document.write( "          = a² + 2ah + h² + 9\r<BR>\n" );
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document.write( "2. **Find f(a):**<BR>\n" );
document.write( "   f(a) = a² + 9\r<BR>\n" );
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document.write( "3. **Substitute and Simplify:**<BR>\n" );
document.write( "   (f(a+h) - f(a)) / h <BR>\n" );
document.write( "   = [(a² + 2ah + h² + 9) - (a² + 9)] / h<BR>\n" );
document.write( "   = (a² + 2ah + h² + 9 - a² - 9) / h<BR>\n" );
document.write( "   = (2ah + h²) / h<BR>\n" );
document.write( "   = 2a + h\r<BR>\n" );
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document.write( "**Therefore, (f(a+h) - f(a)) / h simplifies to 2a + h.**\r<BR>\n" );
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document.write( "**c. Find h and Substitute**\r<BR>\n" );
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document.write( "* Given: <BR>\n" );
document.write( "    * -3 = a<BR>\n" );
document.write( "    * 1 = a + h\r<BR>\n" );
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document.write( "* Find h:<BR>\n" );
document.write( "    1 = -3 + h<BR>\n" );
document.write( "    h = 4\r<BR>\n" );
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document.write( "* Substitute h = 4 and a = -3 into the simplified expression:<BR>\n" );
document.write( "    2a + h = 2(-3) + 4 = -6 + 4 = -2\r<BR>\n" );
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document.write( "**Observation:**\r<BR>\n" );
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document.write( "The result of the simplification in part (c) (-2) is the same as the average rate of change calculated in part (a). \r<BR>\n" );
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document.write( "**Interpretation:**\r<BR>\n" );
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document.write( "This demonstrates that the simplification of (f(a+h) - f(a)) / h represents the slope of the secant line between the points (a, f(a)) and (a+h, f(a+h)) on the graph of f(x). In this case, it gives the slope of the secant line between the points (-3, f(-3)) and (1, f(1)). <BR>\n" );
document.write( " </SPAN>\n" );
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