document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1014593>Question 1014593</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this problem, I have answered most of the questions, but I would really appreciate it if you would look over my answers and tell me if I am wrong and help me correct the mistakes. I mostly need help with e) and g). \r<BR>\n" );
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document.write( "Use the function 2e^(-5x^2) to answer each question. \r<BR>\n" );
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document.write( "a) State the domain:<BR>\n" );
document.write( "I wrote the domain as x is an element of all real numbers\r<BR>\n" );
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document.write( "b) Determine the intercepts, if any:<BR>\n" );
document.write( "I found that there was no x-intercept, and a y-intercept at (0,2)\r<BR>\n" );
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document.write( "c) Discuss the symmetry of the graph:<BR>\n" );
document.write( "I said the graph is symmetric with respect to the y-axis because the function is even. \r<BR>\n" );
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document.write( "d) Find any asymptotes: <BR>\n" );
document.write( "I found no vertical asymptote, and a horizontal asymptote at y=0. \r<BR>\n" );
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document.write( "e) Determine the intervals of increase and decrease:<BR>\n" );
document.write( "I need help with this. \r<BR>\n" );
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document.write( "f) What is the maxima and/or minima:<BR>\n" );
document.write( "Maxima is 2 at x=0, no minima value. \r<BR>\n" );
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document.write( "g) Where is the curve concave upward and downward:<BR>\n" );
document.write( "I need help with this. \r<BR>\n" );
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document.write( "h) Locate the points of inflection:<BR>\n" );
document.write( "I don't know if this is right or wrong but i calculated them as <BR>\n" );
document.write( "( -(1/sqrt(10)), 2/sqrt(e) ) and ( 1/sqrt(10), 2/sqrt(e) )\r<BR>\n" );
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document.write( "i) Graph the function:<BR>\n" );
document.write( "I've already graphed it. \r<BR>\n" );
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document.write( "Thank-you, your help is always appreciated! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #630977 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=630977\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "Everything you did answer is spot-on correct.\r<BR>\n" );
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document.write( "<b>Part e</b>\r<BR>\n" );
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document.write( "Intervals of increase/decrease.  Intuitively, an interval of increase is any interval where the function gets bigger as you go from left to right.  An interval of decrease is an interval where the function gets smaller as you move from left to right.  Precisely speaking, A function is increasing on an interval if, and only if, for all pairs of numbers <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x_1\ <\ x_2\"> in the interval, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f(x_1)\ <\ f(x_2)\">. The way to find the intervals is to find the intervals where the first derivative is positive and the intervals where the first derivative is negative.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{\text{d}}{\text{d}x}2e^{-5x^2}\ =\ -20xe^{-5x^2}\">\r<BR>\n" );
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document.write( "So we need the solution interval for:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ -20xe^{-5x^2}\ >\ 0\">\r<BR>\n" );
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document.write( "But since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large e^{-5x^2}\ >\ 0\"> for the entire domain, we only have to solve:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ -20x\ >\ 0\">\r<BR>\n" );
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document.write( "Which, as a quick examination of the graph would suggest, is the interval <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(-\infty, 0\right)\">.\r<BR>\n" );
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document.write( "Similarly, your function decreases on <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(0, \infty)\">\r<BR>\n" );
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document.write( "In point of fact, since the terms \"increasing\" and \"decreasing\" are only defined on an interval, you can't actually exclude a single point in the definition of that interval.  See discussion:  <a  rel=nofollow HREF=\"http://teachingcalculus.com/2012/11/02/open-or-closed/\"><u>Open or Closed?</u> by Lin McMullin</a>\r<BR>\n" );
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document.write( "Based on that, your intervals should be:  Increasing on <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(-\infty, 0\right\\">] and Decreasing on [<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left\.0, \infty\right)\">\r<BR>\n" );
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document.write( "<b>Part g</b>\r<BR>\n" );
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document.write( "Concavity is determined by the sign of the second derivative.\r<BR>\n" );
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document.write( "So you need to solve:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 200x^2\ -\ 20\ \geq\ 0\">\r<BR>\n" );
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document.write( "for the two concave up intervals, and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 200x^2\ -\ 20\ \leq\ 0\">\r<BR>\n" );
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document.write( "for the concave down interval.\r<BR>\n" );
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document.write( "As you might suspect, the critical points are the two inflection points that you found in Part h.\r<BR>\n" );
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document.write( "Here is the graph to compare with your effort:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/display-illustration.mpl?tutor=solver91311&illustration_name=IncreasingDecreasingConcavityIntervals.jpg\">\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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