document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=911969>Question 911969</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A particle moves on a vertical line so that it's coordinate in feet at time (t) is s(t) = 4t^3 - 12t^2 +7   ,    t >= 0\r<BR>\n" );
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document.write( "a)<BR>\n" );
document.write( "Find the Velocity function\r<BR>\n" );
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document.write( "b)<BR>\n" );
document.write( "Find the Acceleration function\r<BR>\n" );
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document.write( "c)<BR>\n" );
document.write( "When is the particle moving upward?\r<BR>\n" );
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document.write( "d)<BR>\n" );
document.write( "When is the particle moving downward?\r<BR>\n" );
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document.write( "e)<BR>\n" );
document.write( "What is the total distance that the particle travels in the time interval 0 <= t <= 3<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #553472 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=553472\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A particle moves on a vertical line so that it's coordinate in feet at time (t) is s(t) = 4t^3 - 12t^2 +7   ,    t >= 0\r<BR>\n" );
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document.write( "a)<BR>\n" );
document.write( "Find the Velocity function\r<BR>\n" );
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document.write( "b)<BR>\n" );
document.write( "Find the Acceleration function\r<BR>\n" );
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document.write( "c)<BR>\n" );
document.write( "When is the particle moving upward?\r<BR>\n" );
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document.write( "d)<BR>\n" );
document.write( "When is the particle moving downward?\r<BR>\n" );
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document.write( "e)<BR>\n" );
document.write( "What is the total distance that the particle travels in the time interval 0 <= t <= 3<BR>\n" );
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document.write( "a)<BR>\n" );
document.write( "Find the Velocity function\r<BR>\n" );
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document.write( "Differentiate the position function s(t) to get v(t) = s'(t) the velocity function\r<BR>\n" );
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document.write( "s(t) = 4t^3 - 12t^2 +7\r<BR>\n" );
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document.write( "s'(t) = 3*4t^2 - 2*12t^1 +0\r<BR>\n" );
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document.write( "s'(t) = 12t^2 - 24t\r<BR>\n" );
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document.write( "v(t) = 12t^2 - 24t\r<BR>\n" );
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document.write( "The velocity function is v(t) = 12t^2 - 24t\r<BR>\n" );
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document.write( "b)<BR>\n" );
document.write( "Find the Acceleration function\r<BR>\n" );
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document.write( "Differentiate the velocity function v(t) to get the acceleration function a(t) = v'(t). This is equivalent to finding the second derivative of s(t)\r<BR>\n" );
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document.write( "v(t) = 12t^2 - 24t\r<BR>\n" );
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document.write( "v'(t) = 2*12t^1 - 1*24t^0\r<BR>\n" );
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document.write( "v'(t) = 24t - 24*1\r<BR>\n" );
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document.write( "v'(t) = 24t - 24\r<BR>\n" );
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document.write( "a(t) = 24t - 24\r<BR>\n" );
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document.write( "The acceleration function is a(t) = 24t - 24\r<BR>\n" );
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document.write( "c)<BR>\n" );
document.write( "When is the particle moving upward?\r<BR>\n" );
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document.write( "The particle moves upward when v(t) > 0, essentially when the velocity is positive.\r<BR>\n" );
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document.write( "Graph v(t) = 12t^2 - 24t to get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-10%2C10%2C-25%2C25%2C%0D%0A++graph%28500%2C500%2C-10%2C10%2C-25%2C25%2C0%2C12x%5E2-24x%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We see that the velocity function is positive when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+%3C+0 ALIGN=MIDDLE  ALT=\"t+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+%3E+2 ALIGN=MIDDLE  ALT=\"t+%3E+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. The inequality <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+%3C+0 ALIGN=MIDDLE  ALT=\"t+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > does not make sense in this case because negative time is not possible. So only focus on <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+%3E+2 ALIGN=MIDDLE  ALT=\"t+%3E+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The particle is moving upward when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+%3E+2 ALIGN=MIDDLE  ALT=\"t+%3E+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (beyond 2 seconds)\r<BR>\n" );
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document.write( "d)<BR>\n" );
document.write( "When is the particle moving downward?\r<BR>\n" );
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document.write( "Use the same graph in part c) to find that the velocity is negative when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3Ct%3C2 ALIGN=MIDDLE  ALT=\"0%3Ct%3C2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This is when the particle is moving downward.\r<BR>\n" );
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document.write( "e)<BR>\n" );
document.write( "What is the total distance that the particle travels in the time interval 0 <= t <= 3\r<BR>\n" );
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document.write( "D = total distance the particle travels in the time interval 0 <= t <= 3\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large D = \int_{0}^{3}|v(t)|dt\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large D = \int_{0}^{2}|v(t)|dt+\int_{2}^{3}|v(t)|dt\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large D = -\int_{0}^{2}(12t^2 - 24t)dt+\int_{2}^{3}(12t^2 - 24t)dt\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large D = -(4t^3 - 12t^2)|_{0}^{2}+(4t^3 - 12t^2)|_{2}^{3}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large D = -((4(2)^3 - 12(2)^2)-(4(0)^3 - 12(0)^2))+((4(3)^3 - 12(3)^2)-(4(2)^3 - 12(2)^2))\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large D = 32\">\r<BR>\n" );
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document.write( "The total distance traveled is 32 feet. </SPAN>\n" );
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