document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=996014>Question 996014</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let g(x) = 8x + sin 8x for 0 &#8804; x < 2&#960;. \r<BR>\n" );
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document.write( "The function g will have ____critical points in [0,2pi) \r<BR>\n" );
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document.write( "List in increasing order the first two critical points in [0, 2&#960;). Enter DNE for any empty answer blank. <BR>\n" );
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document.write( "This problem is driving me bonkers! I guess I am  confused with how the domain works with this . I got to cos(8x) =-1 part and then did a U-sub to find out what 8x is, I know cos(u)=-1 then u must be pi. I got pi/8 but I am confused beyond belief on how to find the increasing order of this or what it means. Like how it gets to be 3pi/8, 5pi/8 etc.\r<BR>\n" );
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document.write( "Thank you for any and all help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #614629 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=614629\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> g(x) = 8x + sin(8x)<BR>\n" );
document.write( "g ' (x) = 8 + 8*cos(8x)<BR>\n" );
document.write( "0 = 8 + 8*cos(8x)<BR>\n" );
document.write( "8*cos(8x) = -8<BR>\n" );
document.write( "cos(8x) = -8/8<BR>\n" );
document.write( "cos(8x) = -1<BR>\n" );
document.write( "8x = arccos(-1)<BR>\n" );
document.write( "8x = pi + 2pi*n ... where n is an integer<BR>\n" );
document.write( "x = (pi + 2pi*n)/8\r<BR>\n" );
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document.write( "If we made a table with n as the input, x as the output, then...\r<BR>\n" );
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document.write( "<tr><th>n</th><th>x (exact)</th><th>x (approximate)</th></tr><tr><TD >-1</TD><TD >-pi/8</TD><TD >-0.39270</TD></tr><tr><TD >0</TD><TD >pi/8</TD><TD >0.39270</TD></tr><tr><TD >1</TD><TD >3pi/8</TD><TD >1.17810</TD></tr><tr><TD >2</TD><TD >5pi/8</TD><TD >1.96350</TD></tr><tr><TD >3</TD><TD >7pi/8</TD><TD >2.74889</TD></tr><tr><TD >4</TD><TD >9pi/8</TD><TD >3.53429</TD></tr><tr><TD >5</TD><TD >11pi/8</TD><TD >4.31969</TD></tr><tr><TD >6</TD><TD >13pi/8</TD><TD >5.10509</TD></tr><tr><TD >7</TD><TD >15pi/8</TD><TD >5.89049</TD></tr><tr><TD >8</TD><TD >17pi/8</TD><TD >6.67588</TD></tr>\n" );
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document.write( "When n = -1, the value of x is -0.39270 which is outside the interval [0,2pi). The same applies to when n = 8 (x = 6.67588). So we only focus on when n = 0 all the way through n = 7.\r<BR>\n" );
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document.write( "We can see that there are 8 critical points (7-0+1 = 8) since there are 8 x values in the interval [0,2pi).\r<BR>\n" );
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document.write( "The first two critical points are x = pi/8 and x = 3pi/8<BR>\n" );
document.write( "pi/8 = 0.39270 approx<BR>\n" );
document.write( "3pi/8 = 1.17810 approx <BR>\n" );
document.write( "This corresponds to n = 0 and n = 1 respectively.\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "The function g will have <u> &nbsp; &nbsp; <font color=\"red\">8</font> &nbsp; &nbsp; </u> critical points in [0,2pi) <BR>\n" );
document.write( "List in increasing order the first two critical points in [0, 2pi). Enter DNE for any empty answer blank. <BR>\n" );
document.write( "x = <font color=\"red\">pi/8</font><BR>\n" );
document.write( "x = <font color=\"red\">3pi/8</font>\r<BR>\n" );
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document.write( "If the book is seeking approximate x values (instead of exact), then the two answers for x are<BR>\n" );
document.write( "x = <font color=\"red\">0.39270</font><BR>\n" );
document.write( "x = <font color=\"red\">1.17810</font> </SPAN>\n" );
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