document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1019853>Question 1019853</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A function f is defined for all real numbers and has the following properties.<BR>\n" );
document.write( "• f(1) = 2<BR>\n" );
document.write( "• f(3) = 10<BR>\n" );
document.write( "• f'(2)=4<BR>\n" );
document.write( "• For all real numbers a and h, f(a+h)&#8722;f(a)=kah+nh^2&#8722;2nh (where k and n are<BR>\n" );
document.write( "constants).\r<BR>\n" );
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document.write( "a) Find the value of k for this function.<BR>\n" );
document.write( "b) Find the value of n for this function.<BR>\n" );
document.write( "c) Find a formula for f'(x) (this should just be in terms of x, it should not depend on n or k).<BR>\n" );
document.write( "d) Using your answer to part c, find a formula for f(x) (this should just be in terms of x, it should not depend on n or k). Hint: Find a function g(x) such that g'(x) = f'(x).Then there is a theorem that there exists a (unique) real number c such that f(x) = g(x) + c.\r<BR>\n" );
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document.write( "How do I isolate k and n so that I'm able to find out what they equal? I'm 100% stuck on this problem, and I'm not even sure where to start at any help would be appreciated, thank you for your time. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #635896 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=635896\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We start with the difference f(3) - f(2) = k*2*1 +n -2n = 2k - n = 6, where we let a = 2 and h = 1.<BR>\n" );
document.write( "==> 2k - n = 6   (Equation A)<BR>\n" );
document.write( "Similarly, f(2) - f(1) = k*1*1 +n -2n = k - n = 2, where we let a = 1 and h = 1.\r<BR>\n" );
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document.write( "==> k-n = 2    (Equation B)<BR>\n" );
document.write( "Solving for k and n from Equations A and B yields k = 4 and n = 2.\r<BR>\n" );
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document.write( "With these values in mind, we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%2Bh%29+-+f%28x%29+=+4xh+%2B+2h%5E2+-4h ALIGN=MIDDLE  ALT=\"f%28x%2Bh%29+-+f%28x%29+=+4xh+%2B+2h%5E2+-4h\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where we replaced a by x.<BR>\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28f%28x%2Bh%29+-+f%28x%29%29%2Fh+=+4x+%2B+2h+-+4 ALIGN=MIDDLE  ALT=\"%28f%28x%2Bh%29+-+f%28x%29%29%2Fh+=+4x+%2B+2h+-+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, after dividing both sides by h.<BR>\n" );
document.write( "==> f'(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28h-%3E0%2C+%28f%28x%2Bh%29+-+f%28x%29%29%2Fh%29+=+lim%28h-%3E0%2C+%284x%2B2h-4%29%29+=+4x-4 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now f'(x) = 4x - 4 ==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+2x%5E2+-4x+%2Bc ALIGN=MIDDLE  ALT=\"f%28x%29+=+2x%5E2+-4x+%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > for some constant c.\r<BR>\n" );
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document.write( "We can use any one of the initial values given above to find the value of c.  In particular, f(1) = 2, and so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+=+2%2A1%5E2+-+4%2A1+%2Bc ALIGN=MIDDLE  ALT=\"2+=+2%2A1%5E2+-+4%2A1+%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which gives c = 4.\r<BR>\n" );
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document.write( "Therefore, f(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+-+4x%2B4 ALIGN=MIDDLE  ALT=\"2x%5E2+-+4x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Problem solved. </SPAN>\n" );
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