document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=870605>Question 870605</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Are there any *paid* calculus tutors online?  I need to hire one for a few questions.\r<BR>\n" );
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document.write( "The derivative of a function at a point a is given by\r<BR>\n" );
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document.write( "	lim (f(a+h) – f(a))<BR>\n" );
document.write( "f’ (a) =h->0<BR>\n" );
document.write( "                 h\r<BR>\n" );
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document.write( "(that's not showing up right.  It is f'(a) = lim/h is approaching 0; then parentheses with f, all over h)\r<BR>\n" );
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document.write( "Explain where this limit comes from and provide meaning to what “f’(a)” represents. It is encouraged to interpret this limit geometrically/graphically. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #524977 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=524977\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is from the definition of the derivative \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://www.sciweavers.org/tex2img.php?eq=%5Clim_%7Bh%5Cto%200%7D%5Cfrac%7Bf%28x%2Bh%29-f%28x%29%7D%7Bh%7D&bc=White&fc=Black&im=jpg&fs=18&ff=arev&edit=0\" align=\"center\" border=\"0\" alt=\"\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}\" width=\"231\" height=\"65\" />\r<BR>\n" );
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document.write( "where x = a. So instead of calculating f'(x), which is a general algebraic expression, you are calculating f'(a) which is a single number.\r<BR>\n" );
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document.write( "Geometrically, or graphically, f'(a) is the slope of the tangent line to f(x) at the point x = a on the function f(x). Put another way, we have some tangent line y = mx+b where m = f'(a) and the tangent line goes through the point (a, f(a)). \r<BR>\n" );
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document.write( "Here is a good animation that shows what's going on.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF=\"http://math.bu.edu/people/tkohl/teaching/spring2013/secant.html\">http://math.bu.edu/people/tkohl/teaching/spring2013/secant.html</a>\r<BR>\n" );
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document.write( "Visually, h is the horizontal distance from point P to Q. The secant line is going through points P,Q. <BR>\n" );
document.write( "As Q gets closer to P, that secant line is slowly becoming a tangent line. <BR>\n" );
document.write( "It is only a tangent line when P = Q since tangent lines only cross the function (locally) at one point. </SPAN>\n" );
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