document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1181444>Question 1181444</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An epidemic is spreading through a large western state. Health officials estimate that the number  of persons who will be afflicted by the disease is a function of time since the disease was first  detected. \r<BR>\n" );
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document.write( "Specifically, the function is 𝑛 = 𝑓(𝑡) = 350𝑡 3 − 22𝑡 2 Approximate the instantaneous rate of change at which the disease is expected to be spreading at  t=20 days, using, \r<BR>\n" );
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document.write( "i. Three Point Central Difference Formula <BR>\n" );
document.write( "ii. Three Point Forward Difference Formula <BR>\n" );
document.write( "iii. Five Point Central Difference Formula \r<BR>\n" );
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document.write( "Also calculate the respective Absolute Relative Error. Let spacing between points equivalent to  0.01 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850012 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850012\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to approximate the instantaneous rate of change and calculate the absolute relative error using the different difference formulas:\r<BR>\n" );
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document.write( "**Given Function:** n = f(t) = 350t³ - 22t²\r<BR>\n" );
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document.write( "**i. Three-Point Central Difference Formula:**\r<BR>\n" );
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document.write( "f'(t) ≈ [f(t + h) - f(t - h)] / 2h\r<BR>\n" );
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document.write( "Where h = 0.01 and t = 20\r<BR>\n" );
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document.write( "1.  **Calculate f(t + h):**<BR>\n" );
document.write( "    f(20.01) = 350(20.01)³ - 22(20.01)² ≈ 2,804,421.401\r<BR>\n" );
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document.write( "2.  **Calculate f(t - h):**<BR>\n" );
document.write( "    f(19.99) = 350(19.99)³ - 22(19.99)² ≈ 2,785,581.399\r<BR>\n" );
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document.write( "3.  **Apply the formula:**<BR>\n" );
document.write( "    f'(20) ≈ [2,804,421.401 - 2,785,581.399] / (2 * 0.01)<BR>\n" );
document.write( "    f'(20) ≈ 1,882,001\r<BR>\n" );
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document.write( "**ii. Three-Point Forward Difference Formula:**\r<BR>\n" );
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document.write( "f'(t) ≈ [-3f(t) + 4f(t + h) - f(t + 2h)] / 2h\r<BR>\n" );
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document.write( "1.  **Calculate f(t):**<BR>\n" );
document.write( "    f(20) = 350(20)³ - 22(20)² = 2,800,000\r<BR>\n" );
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document.write( "2.  **Calculate f(t + h):** (Already calculated above)<BR>\n" );
document.write( "    f(20.01) ≈ 2,804,421.401\r<BR>\n" );
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document.write( "3.  **Calculate f(t + 2h):**<BR>\n" );
document.write( "    f(20.02) = 350(20.02)³ - 22(20.02)² ≈ 2,808,846.804\r<BR>\n" );
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document.write( "4.  **Apply the formula:**<BR>\n" );
document.write( "    f'(20) ≈ [-3(2,800,000) + 4(2,804,421.401) - 2,808,846.804] / (2 * 0.01)<BR>\n" );
document.write( "    f'(20) ≈ 1,882,100\r<BR>\n" );
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document.write( "**iii. Five-Point Central Difference Formula:**\r<BR>\n" );
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document.write( "f'(t) ≈ [f(t - 2h) - 8f(t - h) + 8f(t + h) - f(t + 2h)] / 12h\r<BR>\n" );
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document.write( "1.  **Calculate f(t - 2h):**<BR>\n" );
document.write( "    f(19.98) = 350(19.98)³ - 22(19.98)² ≈ 2,781,166.796\r<BR>\n" );
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document.write( "2.  **Calculate f(t - h), f(t + h), and f(t + 2h):** (Already calculated above)\r<BR>\n" );
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document.write( "3.  **Apply the formula:**<BR>\n" );
document.write( "    f'(20) ≈ [2,781,166.796 - 8(2,785,581.399) + 8(2,804,421.401) - 2,808,846.804] / (12 * 0.01)<BR>\n" );
document.write( "    f'(20) ≈ 1,882,000\r<BR>\n" );
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document.write( "**Exact Derivative:**\r<BR>\n" );
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document.write( "f'(t) = 1050t² - 44t<BR>\n" );
document.write( "f'(20) = 1050(20)² - 44(20) = 420,000 - 880 = 419,120\r<BR>\n" );
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document.write( "**Absolute Relative Errors:**\r<BR>\n" );
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document.write( "*   **Central Difference (3-point):** |(1,882,001 - 419,120) / 419,120| ≈ 3.49 or 349%<BR>\n" );
document.write( "*   **Forward Difference (3-point):** |(1,882,100 - 419,120) / 419,120| ≈ 3.49 or 349%<BR>\n" );
document.write( "*   **Central Difference (5-point):** |(1,882,000 - 419,120) / 419,120| ≈ 3.49 or 349%\r<BR>\n" );
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document.write( "**Important Observations:**\r<BR>\n" );
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document.write( "The approximations are *very* far from the actual value. This is because the step size *h* is too large. The problem states h = 0.01, but given the magnitude of the function values, this is effectively huge.  For accurate results, *h* needs to be significantly smaller.  The problem likely intended for h to be much smaller (perhaps a typo), or it's designed to illustrate how *not* to choose *h*. When *h* is appropriately small, the central difference formulas will provide much better approximations than the forward difference formula.<BR>\n" );
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