CUBIC slope

\( x\xrightarrow{\hspace{1mm}CUBIC\hspace{1mm}} CUBIC(x) = -4x^{3}-10x^{2}+7x-15 \)

1. the inputs to be \( +2+h \) :
   

   \( \left.x\right|_{x\gets\bbox[yellow]{+2+h}}\xrightarrow{\hspace{1mm} CUBIC\hspace{1mm}}\left. CUBIC(x)\right|_{x\gets\bbox[yellow]{+2+h}} =\left. -4x^{3}-10x^{2}+7x-15\right|_{x\gets\bbox[yellow]{+2+h}} \)

2. Carry out the replacement of \(x\) by \( +2+h \) :
   

   \( +2+h \) \( \xrightarrow{\hspace{1mm} CUBIC\hspace{1mm}} CUBIC( \) \( +2+h \) \( ) = \) \( -4 \) \( {\large (} \) \( +2+h \) \( {\large )}^{3}\hspace{-1mm} \) \( \)   \( -10 \) \( {\large (} \) \( +2+h \) \( {\large )}^{2}\hspace{-1mm} \)   \( +7 \) \( {\large (} \) \( +2+h \) \( {\large )} \)   \( -15 \)

3. Expand using the for \( {\large (}\hspace{10mm}{\large )}^{3}\hspace{-1mm} \)   and for \( {\large (}\hspace{10mm}{\large )}^{2}\hspace{-1mm} \)  :
   

   \( +2+h\xrightarrow{\hspace{1mm} CUBIC\hspace{1mm}} CUBIC(+2+h) = \) \( -4 \) \( {\LARGE ( } \) \( (+2)^{3}+3\cdot(+2)^{2}h+3\cdot(+2)h^{2}+h^{3} \) \( {\LARGE )} \)   \( -10 \) \( {\LARGE (} \) \( (+2)^{2}+2\cdot(+2)h+h^{2} \) \( {\LARGE )} \)   \( +7 \) \( {\LARGE (} \) \( +2+h \) \( {\LARGE )} \)   \( -15 \)

4. To get Slope-sign, focus on the \( h^{+1} \) s:
   

   \( +2+h\xrightarrow{\hspace{1mm} CUBIC\hspace{1mm}} CUBIC(+2+h) = \) \( -4 \) \( {\LARGE (} \hspace{13mm} \) \( +3\cdot(+2)^{2}h\hspace{-1mm} \) \( \hspace{36mm} {\LARGE )} \)   \( -10 \) \( {\LARGE (} \hspace{14mm} \) \( +2\cdot(+2)h \hspace{-1mm} \) \( \hspace{11mm} {\LARGE )} \)   \( +7 \) \( {\LARGE (} \hspace{10mm} \) \( +h \hspace{-1mm} \) \( {\LARGE )} \hspace{0mm} \)   \( -15 \)

   
   

   \( \hspace{64mm} = \) \( -4 \) \( {\LARGE (} \hspace{25mm} \) \( +12h \) \( \hspace{23mm} {\LARGE )} \)   \( -10 \) \( {\LARGE (} \hspace{33mm} \) \( +4h \) \( \hspace{10mm} {\LARGE )} \)   \( +7 \) \( {\LARGE (} \hspace{10mm} \) \( +h \) \( {\LARGE )} \)   \( -15 \)

   
   

   \( \hspace{64mm} =\hspace{5mm} {\LARGE (} \hspace{23mm} \) \( -48h \) \( \hspace{34mm} {\LARGE )} + \hspace{2mm} {\LARGE (} \hspace{28mm} \) \( -40h \) \( \hspace{14mm} {\LARGE )} + \hspace{2mm} \) \( {\LARGE (} \hspace{5mm} \) \( +7h \) \( \hspace{5mm} {\LARGE )} \)   \( -15 \)

5. Reorganize \(CUBIC(+2+h)\) in powers of \(h\):
   

   \( +2+h\xrightarrow{\hspace{1mm} CUBIC\hspace{1mm}} CUBIC(+2+h) \) \( = \) \( {\LARGE [} \hspace{2mm} \) \( \hspace{5mm} \hspace{5mm} \hspace{5mm} \) \( \hspace{2mm} {\LARGE ]} \) \( h^{0} \) \( +{\LARGE [} \hspace{0mm} \) \( -48-40+7 \) \( \hspace{0mm} {\LARGE ]} \) \( h^{+1} \) \( {\LARGE [} \hspace{2mm} \) \( \hspace{5mm} \hspace{5mm} \hspace{5mm} \) \( \hspace{2mm} {\LARGE ]} \) \( h^{+2} \) \( +{\LARGE [} \hspace{2mm} \) \( \hspace{5mm} \hspace{5mm} \hspace{5mm} \) \( \hspace{2mm} {\LARGE ]} \) \( h^{+3} \)

   
   

   \( \hspace{64mm} \) \( = {\LARGE [} \hspace{2mm} \) \( \hspace{5mm} \hspace{5mm} \hspace{5mm} \) \( \hspace{2mm} {\LARGE ]} \) \( h^{0} \) \( +{\LARGE [} \hspace{2mm} \) \( \hspace{5mm} -81 \hspace{5mm} \) \( \hspace{2mm} {\LARGE ]} \) \( h^{+1} \) \( +{\LARGE [} \hspace{2mm} \) \( \hspace{5mm} \hspace{5mm} \hspace{5mm} \) \( \hspace{2mm} {\LARGE ]} \) \( h^{+2} \) \( +{\LARGE [} \hspace{2mm} \) \( \hspace{5mm} \hspace{5mm} \hspace{5mm} \) \( \hspace{2mm} {\LARGE ]} \) \( h^{+3} \)

6. Since the coefficient of \(h^1\) in \(CUBIC(+2+h)\) is negative, Slope-sign of \(CUBIC\) near \(+2\) \(=\) \( \langle \diagdown, \diagdown \rangle \)
   
   
   1. the power function \( x\xrightarrow{\hspace{3mm}} -x^{+1} \).
      
      Since the exponent is positive odd and the coefficient is negative, the local graph near \(0\) is:
      
      
      
      
   2. off the local graph of the power function \( x\xrightarrow{\hspace{3mm}} -x^{+1} \) near \(0\):
      
      
      
      
   3. Code the Slope-sign as seen :
      
      \( \langle \diagdown, \diagdown \rangle \)

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