document.write( "Question 1108914: What is the slope of the tangent line at x=-2 and y=x^3 \n" ); document.write( "
Algebra.Com's Answer #723907 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!

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document.write( "How you go about this problem depends on whether you are in \r\n" );
document.write( "algebra or calculus.  I'll do it both ways.  It's much, much \r\n" );
document.write( "easier by calculus but it can be done by algebra only.  Just \r\n" );
document.write( "for fun, I'll do it both ways, the hard way using algebra \r\n" );
document.write( "only, and the easy way with calculus.  First the hard algebra \r\n" );
document.write( "way:\r\n" );
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document.write( "1.  By algebra only (the hard way).\r\n" );
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document.write( "Let the tangent line have equation y = mx + b\r\n" );
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document.write( "A tangent line must touch the graph at only one point,\r\n" );
document.write( "so the system of equations:\r\n" );
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document.write( "\"system%28y=x%5E3%2Cy=mx%2Bb%29\"\r\n" );
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document.write( "must have a double solution at x = -2, that is,\r\n" );
document.write( "it must have multiplicity 2.\r\n" );
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document.write( "Equating the values of y\r\n" );
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document.write( "\"mx%2Bb=x%5E3\"\r\n" );
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document.write( "Substituting x=-2\r\n" );
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document.write( "\"m%28-2%29%2Bb=%28-2%29%5E3\"\r\n" );
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document.write( "\"-2m%2Bb=-8\"\r\n" );
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document.write( "\"b=2m-8\"\r\n" );
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document.write( " Substitute for b in\r\n" );
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document.write( "\"mx%2Bb=x%5E3\"\r\n" );
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document.write( "\"mx%2B2m-8=x%5E3\"\r\n" );
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document.write( "\"x%5E3-mx-2m%2B8=0\"\r\n" );
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document.write( "Since the graph goes through the point\r\n" );
document.write( "where x=-2. the polynomial must be divisible by\r\n" );
document.write( "x+2.\r\n" );
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document.write( "-2 | 1  0    -m   -2m+8\r\n" );
document.write( "   |   -2    +4    2m-8 \r\n" );
document.write( "     1 -2  -m+4     0\r\n" );
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document.write( "So the polynomial equation factors as\r\n" );
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document.write( "\"%28x%2B2%29%28x%5E2-m%2B4%29=0\"\r\n" );
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document.write( "That has a solution of x=-2.  But since it must \r\n" );
document.write( "have a double solution at x=-2, x+2 must also be\r\n" );
document.write( "a factor of \"x%5E2-m%2B4\"\r\n" );
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document.write( "-2 | 1 -2  -m+4\r\n" );
document.write( "   |   -2     8 \r\n" );
document.write( "     1 -4  -m+12\r\n" );
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document.write( "The remainder must equal to 0, so\r\n" );
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document.write( "-m+12 = 0\r\n" );
document.write( "   -m = -12\r\n" );
document.write( "    m = 12\r\n" );
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document.write( "So the slope of the tangent line at x=-2 is 12.\r\n" );
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document.write( "With calculus, it's a cinch!\r\n" );
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document.write( "\"y=x%5E3\"\r\n" );
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document.write( "\"%22y%27%22=3x%5E2\"\r\n" );
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document.write( "Substitute x=-2\r\n" );
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document.write( "\"%22y%27%22=3%28-2%29%5E2\"\r\n" );
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document.write( "\"%22y%27%22=12\"\r\n" );
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document.write( "That's the desired slope.\r\n" );
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document.write( "Edwin
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