document.write( "Question 885173: Find the gradient of the tangent to y(x) = x2 at A(1,1), B(-1,1) and C(2,4). \n" ); document.write( "

Algebra.Com's Answer #535005 by KMST(5328) $\"\"$ $\"About$

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WITH CALCULUS:
\n" ); document.write( "In calculus, you are told that the slope/gradient of the tangent at a certain point
\n" ); document.write( "is the value of the derivative of the function at that point.
\n" ); document.write( "The derivative of $\"x%5En\"$ is $\"n%2Ax%5E%28n-1%29\"$ , so
\n" ); document.write( "the derivative of $\"y%28x%29=x%5E2\"$ is
\n" ); document.write( " $\"dy%2Fdx=2x\"$ .
\n" ); document.write( "At point A(1,1), $\"x=1\"$ so $\"dy%2Fdx=2%2A1=highlight%282%29\"$ .
\n" ); document.write( "At point A(-1,1), $\"x=-1\"$ so $\"dy%2Fdx=2%2A%28-1%29=highlight%28-2%29\"$ .
\n" ); document.write( "At point A(2,4), $\"x=2\"$ so $\"dy%2Fdx=2%2A2=highlight%284%29\"$ .
\n" ); document.write( "
\n" ); document.write( "WITHOUT CALCULUS:
\n" ); document.write( "If you are not studying calculus,
\n" ); document.write( "you could write the equation of a line passing through the point given with slope $\"m\"$ ,
\n" ); document.write( "and then you would find the value of $\"m\"$ that gives that line
\n" ); document.write( "exactly one intersection point with $\"y%28x%29=x%5E2\"$ .
\n" ); document.write( "You would do that for each point.
\n" ); document.write( "
\n" ); document.write( "For point A(1,1), a point with $\"x%5BA%5D=red%281%29\"$ and $\"y%5BA%5D=blue%281%29\"$ ,
\n" ); document.write( "the line would be $\"y-blue%281%29=m%28x-red%281%29%29\"$
\n" ); document.write( "Then, knowing that the tangent intersects the graph of $\"y%28x%29=x%5E2\"$ at only one point,
\n" ); document.write( "you would know that there should be just one solution to $\"x%5E2-blue%281%29=m%28x-red%281%29%29\"$ .
\n" ); document.write( " $\"x%5E2-1=m%28x-1%29\"$ --> $\"x%5E2-1=mx-m\"$ --> $\"x%5E2-1-mx%2Bm=0\"$ --> $\"x%5E2-mx%2B%28m-1%29=0\"$
\n" ); document.write( "has just one solution when
\n" ); document.write( " $\"%28-m%29%5E2-4%2A1%2A%28m-1%29=0\"$ --> $\"m%5E2-4m-4=0\"$ --> $\"%28m-2%29%5E2=0\"$ --> $\"m-2=0\"$ --> $\"m=highlight%282%29\"$ .
\n" ); document.write( "
\n" ); document.write( "For point B(-1,1), a point with $\"x%5BA%5D=red%28-1%29\"$ and $\"y%5BA%5D=blue%281%29\"$ ,
\n" ); document.write( "the line would be $\"y-blue%281%29=m%28x-red%28-1%29%29\"$ .
\n" ); document.write( "The equation to find the intersection point of that line and $\"y%28x%29=x%5E2\"$ is
\n" ); document.write( " $\"x%5E2-blue%281%29=m%28x-red%28-1%29%29\"$ --> $\"x%5E2-1=m%28x%2B1%29\"$ --> $\"x%5E2-1=mx%2Bm%29\"$ --> $\"x%5E2-1-mx-m=0%29\"$ --> $\"x%5E2-mx%2B%28-m-1%29=0%29\"$ .
\n" ); document.write( "For that equation to have just one solution, it must be
\n" ); document.write( " $\"%28-m%29%5E2-4%2A1%2A%28-m-1%29=0\"$ --> $\"m%5E2%2B4m%2B4=0\"$ --> $\"%28m%2B2%29%5E2=0\"$ $\"m%2B2=0\"$ --> $\"m=highlight%28-2%29\"$ .
\n" ); document.write( "
\n" ); document.write( "For point C(2,4), a point with $\"x%5BA%5D=red%282%29\"$ and $\"y%5BA%5D=blue%284%29\"$ ,
\n" ); document.write( "the line would be $\"y-blue%284%29=m%28x-red%282%29%29\"$ .
\n" ); document.write( "The equation to find the intersection point of that line and $\"y%28x%29=x%5E2\"$ is
\n" ); document.write( " $\"x%5E2-blue%284%29=m%28x-red%282%29%29\"$ --> $\"x%5E2-4=m%28x-2%29\"$ --> $\"x%5E2-4=mx-2m%29\"$ --> $\"x%5E2-4-mx%2B2m=0%29\"$ --> $\"x%5E2-mx%2B%282m-4%29=0%29\"$ .
\n" ); document.write( "For that equation to have just one solution, it must be
\n" ); document.write( " $\"%28-m%29%5E2-4%2A1%2A%282m-4%29=0\"$ --> $\"m%5E2-8m%2B16=0\"$ --> $\"%28m-4%29%5E2=0\"$ $\"m-4=0\"$ --> $\"m=highlight%284%29\"$ . \n" ); document.write( "

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