document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1089274>Question 1089274</A>: <I><SPAN STYLE=\"word-break: break-all;\"> p(a,b) is a variable point in the first quadrant on the curve y=e^-x. Q and R are the points where the tangent to the curve at p meets the x- and y - axes respectively. find the maximum area of the triangle OQR, where  O is the origin.<BR>\n" );
document.write( "thank you<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #703632 by </B><A HREF=/tutors/aboutme.mpl?userid=natolino_2017><B>natolino_2017(77)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=natolino_2017><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=703632\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> y = exp(-x)\r<BR>\n" );
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document.write( "b = exp(-a).\r<BR>\n" );
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document.write( "dy/dx = -exp(-x), so dy/dx = -exp(-a) = -b (slope on the tangent line on (a,b))\r<BR>\n" );
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document.write( "Let H: height of the triangle<BR>\n" );
document.write( "    L: Base of the triangle.\r<BR>\n" );
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document.write( "we need H in the terms of a and L in the terms of a.\r<BR>\n" );
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document.write( "-b = (H-b)/(0-a). So H(a) = ab+b = exp(-a)(a+1).\r<BR>\n" );
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document.write( "-b = (b-0)/(a-L). So L(a) = a+1\r<BR>\n" );
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document.write( "A(a) = H(a)*(L(a))/2 (Area of the triangle: Base*height/2)\r<BR>\n" );
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document.write( "A(a) = exp(-a)(a+1)^2/2 with a>0\r<BR>\n" );
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document.write( "A'(a) = 0.5(-exp(-a)(a+1)^2 +2*exp(-a)(1+a)) = 0\r<BR>\n" );
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document.write( " solving for a: a = 1.\r<BR>\n" );
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document.write( "A''(a) = 0.5(-exp(-a)(1-a^2)-2aexp(-a)) = 0.5(exp(-a)(a^2-2a-1)\r<BR>\n" );
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document.write( "A''(1) = -exp(-1) < 0, so the point is a max\r<BR>\n" );
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document.write( "A(1) = 0.5exp(-1)(1+1)^2 = 2exp(-1) [unit^2] (0.736 [unit^2] aprox)\r<BR>\n" );
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document.write( "@natolino_\r<BR>\n" );
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