document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203202>Question 1203202</A>: <I><SPAN STYLE=\"word-break: break-all;\"> This question below is part 2 of a question I had recently posted on algebra.com ( please see the previous question here on algebra.com about average rate of change (https://www.algebra.com/my/your-answer.mpl?question=1203157)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<a  rel=nofollow  rel=nofollow HREF=\"https://ibb.co/WsDMpsZ\"><IMG STYLE=\"max-width:80%;\" src=\"https://i.ibb.co/x3mB83W/IMG-5749.jpg\" alt=\"IMG-5749\" border=\"0\"></a><br /><a target='_blank'  rel=nofollow  rel=nofollow HREF='https://imgbb.com/'>image upload</a><br />\r<BR>\n" );
document.write( "\n" );
document.write( "A request: I haven't learned to solve this using derivatives so please solve using another method ( I use a different formula that doesn't require derivatives) \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Thank you in advance!<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #838574 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=838574\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'll use x in place of theta<BR>\n" );
document.write( "And use y in place of T\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Here are two resources to check out<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/03%3A_Derivatives/3.05%3A_Derivatives_of_Trigonometric_Functions\">https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/03%3A_Derivatives/3.05%3A_Derivatives_of_Trigonometric_Functions</a><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://math.libretexts.org/Courses/Mount_Royal_University/MATH_1200%3A_Calculus_for_Scientists_I/1%3A_Limit__and_Continuity_of_Functions/1.7%3A_Limit_of__Trigonometric_functions\">https://math.libretexts.org/Courses/Mount_Royal_University/MATH_1200%3A_Calculus_for_Scientists_I/1%3A_Limit__and_Continuity_of_Functions/1.7%3A_Limit_of__Trigonometric_functions</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The first link talks about the derivative of trig functions. <BR>\n" );
document.write( "What we're interested in mainly is that<BR>\n" );
document.write( "derivative of sine = cosine<BR>\n" );
document.write( "or in terms of symbols<BR>\n" );
document.write( "d/dx[ sin(x) ] = cos(x)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Rephrased another way<BR>\n" );
document.write( "y = sin(x) leads to dy/dx = cos(x)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The second link is a supplement to explain two important trig limit identities.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If y = (2V/g)*sin(x) then dy/dx = (2V/g)*cos(x)<BR>\n" );
document.write( "The (2V/g) portion is a constant.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Plug in x = pi/6 to get:<BR>\n" );
document.write( "dy/dx = (2V/g)*cos(x)<BR>\n" );
document.write( "dy/dx = (2V/g)*cos(pi/6)<BR>\n" );
document.write( "dy/dx = (2V/g)*sqrt(3)/2<BR>\n" );
document.write( "dy/dx = (V/g)*sqrt(3)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Use the unit circle to determine that cos(pi/6) = sqrt(3)/2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer: The instantaneous rate of change is (V/g)*sqrt(3) when theta = pi/6\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Edit: \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I just realized you wrote that you haven't learned derivatives just yet.<BR>\n" );
document.write( "Another approach would be to find the average rate of change over the interval [pi/6, pi/4] as done so in this link<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.algebra.com/algebra/homework/word/misc/Miscellaneous_Word_Problems.faq.question.1203157.html\">https://www.algebra.com/algebra/homework/word/misc/Miscellaneous_Word_Problems.faq.question.1203157.html</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "pi/6 = 0.523599 approximately<BR>\n" );
document.write( "pi/4 = 0.785398 approximately\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the interval [pi/6, pi/4] is approximately [0.523599, 0.785398]\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then shrink that interval down to something like [0.523599, 0.7] then to [0.523599, 0.6]<BR>\n" );
document.write( "The idea is to keep shrinking the interval such that pi/6 = 0.523599 is kept constant as the left endpoint. Only the right endpoint will move closer and closer to pi/6\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "As this right side approaches pi/6, the average rate of change will approach the instantaneous rate of change.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );