document.write( "Question 1207264: Graph the given arc. Let 𝐶: 𝑧 = 𝑧(𝑡), 0 ≤ 𝑡 ≤ 4, defined by
\n" ); document.write( "𝑧 = {𝑡 − 𝑖𝑡 , 0 ≤ 𝑡 ≤ 2
\n" ); document.write( " 𝑡 − 𝑖 , 2 < 𝑡 ≤ 4 \n" ); document.write( "

Algebra.Com's Answer #845035 by Edwin McCravy(20056) $\"\"$ $\"About$

You can put this solution on YOUR website!

\r\n" );
document.write( "\r\n" );
document.write( "This is a piecewise graph with parameter t.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We will use a two-dimensional Cartesian plane, and identify the \r\n" );
document.write( "point with coordinates (x,y) with the complex number z = x+iy. \r\n" );
document.write( "Here, 'i' is the imaginary unit and is identified with the point \r\n" );
document.write( "with coordinates (0,1).\r\n" );
document.write( "\r\n" );
document.write( "We normally think of an \"arc\" as a curved line.  However in complex\r\n" );
document.write( "analysis we use a more general definition. A set of points in the \r\n" );
document.write( "complex plane is called \"an arc\" if x = x(t) and y = y(t) for a<t<b \r\n" );
document.write( "where x(t) and y(t) are continuous functions of real parameter t. \r\n" );
document.write( "We denote an arc C as z(t)=x(t)+iy(t) for a<t<b.\r\n" );
document.write( "\r\n" );
document.write( "The left part of the graph where t goes from t=0 to t=2,\r\n" );
document.write( "is a line segment from 0-i*0, or the point (0,0) to 2-i*2, or the \r\n" );
document.write( "point (2,-2), where the segment includes both its endpoints.\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "The right part of the graph where t goes from t=2 to t=4,\r\n" );
document.write( "is a line segment from 2-i, or the point (2,-1) to 4-i, or \r\n" );
document.write( "the point (4,-1), where the segment does not include its \r\n" );
document.write( "left endpoint, but does include its right endpoint.\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( " \n" ); document.write( "

\n" );