document.write( "Question 1183948: A line tangent to the curve x = y^(1/2) at the point P intersects the x axis at
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Algebra.Com's Answer #814471 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Suppose the line is tangent to the curve at P(h,h2). \r\n" );
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document.write( "Then its slope is the derivative of \"y=x%5E2\" at P(h,h2).\r\n" );
document.write( "The derivative is \"dy%2Fdx=2x\", so the slope at P(h,h2) is 2h.\r\n" );
document.write( "Since it goes through (h,h2), its equation is:\r\n" );
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document.write( "\"y-h%5E2=2h%28x-h%29\"\r\n" );
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document.write( "Its x-intercept P is the x-value when y=0, so\r\n" );
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document.write( "\"0-h%5E2=2h%28x-h%29\"\r\n" );
document.write( "\"-h%5E2=2hx-2h%5E2%29\"\r\n" );
document.write( "Divide through by h\r\n" );
document.write( "\"-h=2x-2h\"\r\n" );
document.write( "\"h=2x\"\r\n" );
document.write( "\"expr%281%2F2%29h=x\"\r\n" );
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document.write( "So the coordinates of Q are \"Q%28+expr%281%2F2%29h%2C0%29\".\r\n" );
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document.write( "Now let's switch the letter h to x.  [It would have been too confusing\r\n" );
document.write( "if we had started out with x because the equation of a line uses x).\r\n" );
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document.write( "We want to know how fast Q is moving.  Since Q is on the x-axis, Q moves the\r\n" );
document.write( "same speed as its own x-coordinate \"expr%281%2F2%29x\".  So let the variable\r\n" );
document.write( "q = the x-coordinate of Q.  \"q=expr%281%2F2%29x\". We want to know \"%28dq%29%2F%28dt%29\" at P(2,4) which is when x=2.\r\n" );
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document.write( "The point P moves up the curve at the rate of 2 units per second, so\r\n" );
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document.write( "\"ds%2Fdt+=+2\"\r\n" );
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document.write( "\"%28ds%2Fdt%29%5E2=%28dx%2Fdt%29%5E2%2B%28dy%2Fdt%29%5E2\"\r\n" );
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document.write( "\"2%5E2=%28dx%2Fdt%29%5E2%2B%282x%2Aexpr%28dx%2Fdt%29%29%5E2\"\r\n" );
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document.write( "\"4=%28dx%2Fdt%29%5E2%2B4x%5E2%2A%28dx%2Fdt%29%5E2\"\r\n" );
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document.write( "\"4=%28dx%2Fdt%29%5E2%281%2B4x%5E2%29\"\r\n" );
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document.write( "\"4%5E%22%22%2F%281%2B4x%5E2%29=%28dx%2Fdt%29%5E2\"\r\n" );
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document.write( "\"dx%2Fdt=sqrt%284%5E%22%22%2F%281%2B4x%5E2%29%29\"\r\n" );
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document.write( "\"dx%2Fdt=2sqrt%281%5E%22%22%2F%281%2B4x%5E2%29%29\"\r\n" );
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document.write( "\"q=expr%281%2F2%29x\"\r\n" );
document.write( "\"dq%2Fdt+=+expr%281%2F2%29%28dx%2Fdt%29\"\r\n" );
document.write( "\"dq%2Fdt+=+expr%281%2F2%29%282sqrt%281%5E%22%22%2F%281%2B4x%5E2%29%29%29\"\r\n" );
document.write( "\"dq%2Fdt+=+sqrt%281%5E%22%22%2F%281%2B4x%5E2%29%29\"\r\n" );
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document.write( "Evaluating that when x=2,\r\n" );
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document.write( "\"%22%22=%22%22\"\"1%2Fsqrt%2817%29\" units per second.\r\n" );
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document.write( "Edwin

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