document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1183710>Question 1183710</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 11) Create a graph of a function given the following information:<BR>\n" );
document.write( "• The instantaneous rate of change at x = 2<BR>\n" );
document.write( "is zero.<BR>\n" );
document.write( "• The instantaneous rate of change at x = 3<BR>\n" );
document.write( "is negative.<BR>\n" );
document.write( "• The average rate of change on the interval 0 <= x <= 4 is zero. </SPAN>\n" );
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document.write( "Average rate of change on [0,4] zero means f(0)=f(4).<br><BR>\n" );
document.write( "Instantaneous rate of change zero at x=2 means there is a local maximum or minimum at x=2.<br><BR>\n" );
document.write( "Those two conditions together are easily satisfied by a function of the form<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=a%28x-2%29%5E2 ALIGN=MIDDLE  ALT=\"f%28x%29=a%28x-2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "To have the instantaneous rate of change negative at x=3, we simply need to make the parabola open downward, which means a is negative.<br><BR>\n" );
document.write( "So one simple function satisfying the given conditions is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+-%28x-2%29%5E2 ALIGN=MIDDLE  ALT=\"f%28x%29+=+-%28x-2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "A graph....<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-2%2C6%2C-10%2C10%2C-%28x-2%29%5E2%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-2%2C6%2C-10%2C10%2C-%28x-2%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(1) f(0)=f(4)=-4<BR>\n" );
document.write( "(2) f'(2)=0<BR>\n" );
document.write( "(3) f'(3)<0<br><BR>\n" );
document.write( "A sinusoidal function can also be found that satisfies the given conditions:<BR>\n" );
document.write( "f(0)=f(4) means the period of the function can be 4<BR>\n" );
document.write( "f'(2)=0 and f'(3)<0 means we want a local maximum at x=2<br><BR>\n" );
document.write( "This function satisfies those conditions:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=cos%28%28pi%2F2%29%28x-2%29%29 ALIGN=MIDDLE  ALT=\"f%28x%29=cos%28%28pi%2F2%29%28x-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "A graph....<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-2%2C6%2C-2%2C2%2Ccos%28%28pi%2F2%29%28x-2%29%29%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-2%2C6%2C-2%2C2%2Ccos%28%28pi%2F2%29%28x-2%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(1) f(0)=f(4)=-1<BR>\n" );
document.write( "(2) f'(2)=0<BR>\n" );
document.write( "(3) f'(3)<0<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );