document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208077>Question 1208077</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The following question is from my college algebra textbook. It is preview of calculus 1.\r<BR>\n" );
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document.write( "Let m_sec = slope of the secant line.\r<BR>\n" );
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document.write( "I am to use m_sec = [(f(x + h) - f(x)]/h. It is called the difference quotient of f.\r<BR>\n" );
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document.write( "Given f(x) = 2x^2 + x:\r<BR>\n" );
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document.write( "A. Express the slope of the secant line of the given function in terms of x and h. Be sure to simplify your answer.<BR>\n" );
document.write( "I found m_sec to be 4x + 2h + 1.\r<BR>\n" );
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document.write( "B. Find m_sec for h = 0.5 at x = 1. What value does m_sec approach as h tends to 0?\r<BR>\n" );
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document.write( "I don't understand part B.\r<BR>\n" );
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document.write( "C. Find the equation of the secant line at x = 1 with h = 0.01.\r<BR>\n" );
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document.write( "I don't know how to find the equation of the secant line for part C. </SPAN>\n" );
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document.write( "Part A is correct. Nice work.\r<BR>\n" );
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document.write( "Part B<BR>\n" );
document.write( "m_sec = 4x+2h+1<BR>\n" );
document.write( "m_sec = 4(1)+2(0.5)+1<BR>\n" );
document.write( "m_sec = 6<BR>\n" );
document.write( "As h gets closer to 0, the 2h will also get closer to 0.<BR>\n" );
document.write( "This will mean the 2h is so small that it's practically zero (even if we technically don't land on this value exactly), and it goes away.\r<BR>\n" );
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document.write( "Therefore m_sec approaches <font color=red>4x+1</font> as h tends to 0.\r<BR>\n" );
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document.write( "A bit of a spoiler alert: You'll learn later in Calculus that this is the derivative of 2x^2+x.<BR>\n" );
document.write( "The derivative is very useful in many applications. One of which is finding the slope of a tangent line.\r<BR>\n" );
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document.write( "Part C<BR>\n" );
document.write( "m_sec = 4x+2h+1<BR>\n" );
document.write( "m_sec = 4(1)+2(0.01)+1<BR>\n" );
document.write( "m_sec = 5.02 is the slope of the secant line at x = 1 where h = 0.01\r<BR>\n" );
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document.write( "f(x) = 2x^2 + x<BR>\n" );
document.write( "f(1) = 2*1^2 + 1<BR>\n" );
document.write( "f(1) = 3<BR>\n" );
document.write( "The point (x,y) = (1,3) is on the f(x) curve.<BR>\n" );
document.write( "Meaning that x = 1 and y = 3 pair up together.\r<BR>\n" );
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document.write( "y = mx+b<BR>\n" );
document.write( "3 = 5.02*1+b<BR>\n" );
document.write( "3 = 5.02+b<BR>\n" );
document.write( "b = 3-5.02<BR>\n" );
document.write( "b = -2.02<BR>\n" );
document.write( "The equation of the secant line is <font color=red>y = 5.02x-2.02</font><BR>\n" );
document.write( "To help verify you can plug x = 1 into this equation and you should get y = 3.\r<BR>\n" );
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document.write( "Another approach to finding the secant line is to use the point-slope template.<BR>\n" );
document.write( "y-y1 = m(x-x1)<BR>\n" );
document.write( "y-3 = 5.02(x-1)<BR>\n" );
document.write( "y-3 = 5.02x-5.02<BR>\n" );
document.write( "y = 5.02x-5.02+3<BR>\n" );
document.write( "<font color=red>y = 5.02x-2.02</font><BR>\n" );
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