document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=877009>Question 877009</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help with the following! I've been trying to crack it for days.\r<BR>\n" );
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document.write( "Real is investigating the rate of change of the function y=cosx on the interval XE[0,2pi] .  He determines the instantaneous rate of change at x=0,pi, and 2pi by inspection and believes that the IROC at the roots of y=cosx could be relevant in determining an equation r(x) to predict the instantaneous rate of change of the function y=cosx on the interval xE[0,2pi] . Find the equation that represents the IROC for y=cosx  and use it to determine the exact instantaneous rate of change at  x=pi/4. \r<BR>\n" );
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document.write( "Thank you in advanced! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #529150 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=529150\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%28x%29=-sin%28x%29 ALIGN=MIDDLE  ALT=\"r%28x%29=-sin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%28pi%2F4%29=-sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"r%28pi%2F4%29=-sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , but how can you show a reasoning for that solution, and what reasoning should you use?<BR>\n" );
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document.write( "Not knowing what level math class the problem came from, and what kind of reasoning appeals to the instructor who made up the problem, I do not see what Real is thinking.<BR>\n" );
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document.write( "Maybe the students in your math class have been given \"memorizable\" formulas and tips to calculate derivatives. In that case you would know that the IROC is the derivative and that the derivative of cosine is -sine.<BR>\n" );
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document.write( "Maybe your class is not into derivatives yet, and this problem is a \"warmup\" to the concept of derivatives. In that case, you may be expected to calculate the IROC as a limit.<BR>\n" );
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document.write( "It could also be that you are deep into calculus and Taylor series. In that case, maybe you could use that and Real's belief \"that the IROC at the roots of y=cosx could be relevant in determining an equation r(x) to predict the instantaneous rate of change of the function y=cosx on the interval xE[0,2pi] \".<BR>\n" );
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document.write( "Alternatively, students who never heard of limits or calculus can visualize the IROC as a velocity. THat sound more real than Real.<BR>\n" );
document.write( "Talking about velocities, I will make the independent variable <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the time in seconds, and my <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > will just indicate the position of a moving point, P. (Sorry, I like speeds and rates of change as function of time, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and if we are going to visualize something, a small dose of realism helps).<BR>\n" );
document.write( "Then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%5Bx%5D=cos%28t%29 ALIGN=MIDDLE  ALT=\"P%5Bx%5D=cos%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the x-coordinate of the shadow on the x-axis of a point circling the origin in the x-y plane at constant speed, and its velocity is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%28t%29=-sin%28t%29 ALIGN=MIDDLE  ALT=\"r%28t%29=-sin%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
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document.write( "AS A LIMIT:<BR>\n" );
document.write( "For any value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the average ROC for the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%5B+x+%2C%22 ALIGN=MIDDLE  ALT=\"%22%5B+x+%2C%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Bd ALIGN=MIDDLE  ALT=\"x%2Bd\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%5D%22 ALIGN=MIDDLE  ALT=\"%22%5D%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28x%2Cd%29=%28cos%28x%2Bd%29-cos%28d%29%29%2F%28%28x%2Bd%29-x%29=%28cos%28x%29%2Acos%28d%29-sin%28x%29sin%28d%29-cos%28x%29%29%2Fd ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%28x%29=lim%28d-%3E0%2CR%28x%2Cd%29%29 ALIGN=MIDDLE  ALT=\"r%28x%29=lim%28d-%3E0%2CR%28x%2Cd%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From there you could prove that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%28x%29=-sin%28x%29 ALIGN=MIDDLE  ALT=\"r%28x%29=-sin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > several ways, but you may need to use theorems about limits that were taught in class, like<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28d-%3E0%2Csin%28d%29%2Fd%29=1 ALIGN=MIDDLE  ALT=\"lim%28d-%3E0%2Csin%28d%29%2Fd%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "AS A SHADOW:<BR>\n" );
document.write( "A point P starts at position (1,0) in the x-y plane, when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "It moves counterclockwise, in a circle centered at the origin.<BR>\n" );
document.write( "The radius of that circle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in whatever units of length we are using).<BR>\n" );
document.write( "The angular velocity pf P is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in radians per second).<BR>\n" );
document.write( "The position of point P is a vector that is changing direction with time. <BR>\n" );
document.write( "Its x-component is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%5Bx%5D=cos%28t%29 ALIGN=MIDDLE  ALT=\"P%5Bx%5D=cos%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The speed, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S ALIGN=MIDDLE  ALT=\"S\"   DISABLED_style= \"max-width:90%; height:auto;\" > of point P is a number and is equal the radius times the angular velocity, so it is also equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in units of length per second).<BR>\n" );
document.write( "The velocity of P is a vector that is changing direction with time,<BR>\n" );
document.write( "always perpendicular to the radius from the origin to P,<BR>\n" );
document.write( "always with a magnitude <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=1 ALIGN=MIDDLE  ALT=\"S=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Its component in the x-direction given by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5Bx%5D=S%2A%28-sin%28x%29%29=-sin%28x%29 ALIGN=MIDDLE  ALT=\"v%5Bx%5D=S%2A%28-sin%28x%29%29=-sin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Compared to point P, the projection of point P on the x-axis,<BR>\n" );
document.write( "has the same x-coordinate,<BR>\n" );
document.write( "and the same x-component of velocity.<BR>\n" );
document.write( "The IROC of the x-coordinate of P's shadow,<BR>\n" );
document.write( "which varies with time <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%5Bx%5D=cos%28t%29 ALIGN=MIDDLE  ALT=\"P%5Bx%5D=cos%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . <BR>\n" );
document.write( "(between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=2pi ALIGN=MIDDLE  ALT=\"t=2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > seconds and for as long as we let P move)<BR>\n" );
document.write( "is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%28t%29=-sin%28t%29 ALIGN=MIDDLE  ALT=\"r%28t%29=-sin%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "I know I name the variables differently, but a function is the same, no matter what you call the variables.<BR>\n" );
document.write( "You can change the names of the variables. I don't care what names I'm called and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%5Bx%5D+and+%7B%7B%7Bt ALIGN=MIDDLE  ALT=\"P%5Bx%5D+and+%7B%7B%7Bt\"   DISABLED_style= \"max-width:90%; height:auto;\" > do not mind if you call them <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > respectively.<BR>\n" );
document.write( "I did not want to use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > , because I am more familiar with an x-y plane that with a u-v set of coordinates.  </SPAN>\n" );
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