jQuery UI Dialog - Default functionality

CUBIC. Local graph near a regular bounded input

that is an input for which height-sign comes from \(h^0\) and slope-sign comes from \(h^1\) and concavity-sign comes from \(h^2\).
As opposed to critical bounded inputs which are inputs for which height-sign comes from \(h^1\) or higher, or slope-sign comes from \(h^2\) or higher, or concavity-sign comes from \(h^3\) or higher.

At this time, only the first three buttons, namely the local graph button, the declare button and the execute button, work.
At the bottom of this page is the---very clunky---animation of the top of the page. I hope that my next attempt will give better results.
Clicking on a band opens it and closes the previous one. The first one is already open.
Finally, give your browser a bit of time to compile the math. You will know when it's done because the math will then look really good.

GIVEN: The function \(CUBIC\) specified by the global input-output rule

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

WANTED: The jQuery UI Dialog functionality
The local graph of a given function \(f\) near a given bounded input \(x_0\) is the graph of the approximate input-output rule of \(f\) for inputs near \(x_0\) namely: \[ x_{0}+h\xrightarrow{\hspace{1mm}f\hspace{1mm}}f(x_{0}+h)= [A_0]h^{0}+ [A_1]h^{1} + [A_2]h^{2} +[...] \]
of \(CUBIC\) for inputs near \(+2\) .
PLANNING AHEAD: What is going to be your overall line of attack to get what is wanted?
  1. Get the global graph of \(CUBIC\) and highlight the local graph near \(+2\) ?
  2. Get the local graph near \(\infty\) of the power function \( x\xrightarrow{\hspace{1mm}a\hspace{1mm}}a(x)= -4x^{3} \) ?
  3. Get the local input-output rule near \(+2\) from the global input-output rule of \(CUBIC\)?
  4. Get the global graph of the power function \( x\xrightarrow{\hspace{1mm}a\hspace{1mm}}a(x)= -4x^{3} \) ?

DO:
  1. In the given global input-output rule, jQuery UI Dialog functionality
    To declare an input is to say which specific input is to be used. It is coded as follows \[\left. \right|_{x\gets \text{specific input}} \]
    the input to be \( \bbox[1pt,yellow]{ \text{near }+2 } \) :

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

  2. jQuery UI Dialog functionality
    To execute a declaration is actually to replace \(x\) by the given input.
    the declaration.

    \( \hspace{10mm} \bbox[1pt,yellow]{ +2+h } \xrightarrow{\hspace{1mm} CUBIC\hspace{1mm}} CUBIC( \bbox[1pt,yellow]{ +2+h } ) = \bbox[1pt,00FF99]{ -4 } {\large (} \bbox[1pt,yellow]{ +2+h } {\large )}^{3}\hspace{-1mm} \bbox[1pt,00FF99]{ -10 } {\large (} \bbox[1pt,yellow]{ +2+h } {\large )}^{2}\hspace{-1mm} \bbox[1pt,00FF99]{ +7 } {\large (} \bbox[1pt,yellow]{ +2+h } {\large )} \bbox[1pt,00FF99]{ -15 } \)

  3. \(CUBIC(+2+h)\) using the for \( {\large (} x_{0}+h {\large )}^{3} \text{ and for } {\large (} x_{0}+h {\large )}^{2} \hspace{-1mm} : \)

    \( \hspace{10mm} +2+h\xrightarrow{\hspace{1mm} CUBIC\hspace{1mm}} CUBIC(+2+h) = \bbox[1pt,00FF99]{ -4 } {\LARGE ( } \bbox[1pt,yellow]{ (+2)^{3}+3\cdot(+2)^{2}h+3\cdot(+2)h^{2}+h^{3} } {\LARGE )} \bbox[1pt,00FF99]{ -10 } {\LARGE (} \bbox[1pt,yellow]{ (+2)^{2}+2\cdot(+2)h+h^{2} } {\LARGE )} \bbox[1pt,00FF99]{ +7 } {\LARGE (} \bbox[1pt,yellow]{ +2+h } {\LARGE )} \bbox[1pt,00FF99]{ -15 } \)

    4. \( \hspace{76mm} = \bbox[1pt,00FF99]{ -4 } {\LARGE (} \hspace{2mm} \bbox[1pt,yellow]{ +8 } \hspace{1mm} \bbox[1pt,yellow]{ } \hspace{7mm} \bbox[1pt,yellow]{ +12h } \hspace{9mm} \bbox[1pt,yellow]{ +6h^{2} } \hspace{6mm} \bbox[1pt,yellow]{ +h^{3} } \hspace{0mm} {\LARGE )} \bbox[1pt,00FF99] { -10 } {\LARGE (} \hspace{1mm} \bbox[1pt,yellow]{ +4 } \hspace{8mm} \bbox[1pt,yellow]{ +4h } \hspace{5mm} \bbox[1pt,yellow]{ +h^{2} } \hspace{1mm} {\LARGE )} \bbox[1pt,00FF99]{ +7 } {\LARGE (} \bbox[1pt,yellow]{ +2+h } {\LARGE )} \bbox[1pt,00FF99] { -15 } \)

    \( \hspace{76mm} = \hspace{6mm} {\LARGE (} \hspace{0mm} \bbox[1pt,F4F4F4]{ -32 } \hspace{7mm} \bbox[1pt,F4F4F4]{ -48h } \hspace{7mm} \bbox[1pt,F4F4F4]{ -24h^{2} } \hspace{5mm} \bbox[1pt,F4F4F4]{ -4h^{3} } \hspace{0mm} {\LARGE )} \hspace{3mm} + \hspace{3mm} {\LARGE (} \hspace{0mm} \bbox[1pt,F4F4F4]{ -40 } \hspace{4mm} \bbox[1pt,F4F4F4]{ -40h } \hspace{3mm} \bbox[1pt,F4F4F4]{ -10h^{2} } \hspace{0mm} {\LARGE )} \hspace{3mm} + \hspace{2mm} {\LARGE (} \hspace{0mm} \bbox[1pt,F4F4F4]{ +14 } \hspace{1mm} \bbox[1pt,F4F4F4]{ +7h } \hspace{0mm} {\LARGE )} \bbox[1pt,F4F4F4]{ -15 } \)

    \( \hspace{76mm} = \hspace{6mm} {\LARGE (} \hspace{0mm} \bbox[1pt,F4F4F4]{ -32h^{0} } \hspace{2mm} \bbox[1pt,F4F4F4]{ -48h^{+1} } \hspace{2mm} \bbox[1pt,F4F4F4]{ -24h^{+2} } \hspace{2mm} \bbox[1pt,F4F4F4]{ -4h^{+3} } \hspace{0mm} {\LARGE )} \hspace{3mm} + \hspace{3mm} {\LARGE (} \hspace{0mm} \bbox[1pt,F4F4F4]{ -40h^{0} } \hspace{0mm} \bbox[1pt,F4F4F4]{ -40h^{+1} } \hspace{0mm} \bbox[1pt,F4F4F4]{ -10h^{+2} } \hspace{0mm} {\LARGE )} \hspace{3mm} + \hspace{2mm} {\LARGE (} \hspace{0mm} \bbox[1pt,F4F4F4]{ +14h^{0} } \hspace{0mm} \bbox[1pt,F4F4F4]{ +7h^{+1} } \hspace{0mm} {\LARGE )} \bbox[1pt,F4F4F4]{ -15h^{0} } \)

  4. \(CUBIC(+2+h)\) in terms of \(h^{0}\), \(h^{1}\), \(h^{2}\), \(h^{3}\) :

    \( \hspace{10mm} +2+h \xrightarrow{ \hspace{1mm} CUBIC \hspace{1mm} } CUBIC(+2+h) = {\LARGE [} \hspace{0mm} \bbox[1pt,F4F4F4]{ -32 } \hspace{0mm} \bbox[1pt,F4F4F4]{ -40 } \hspace{0mm} \bbox[1pt,F4F4F4]{ +14 } \hspace{0mm} \bbox[1pt,F4F4F4]{ -15 } \hspace{0mm} \hspace{0mm} {\LARGE ]} h^{0} + {\LARGE [} \bbox[1pt,F4F4F4]{ -48 } \hspace{0mm} \bbox[1pt,F4F4F4]{ -40 } \hspace{0mm} \bbox[1pt,F4F4F4]{ +7 } \hspace{0mm} {\LARGE ]} h^{+1} +{\LARGE [} \hspace{0mm} \bbox[1pt,F4F4F4]{ -24-10 } {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{+2} } +{\LARGE [} \hspace{2mm} \bbox[1pt,F4F4F4]{ -4 } \hspace{2mm} {\LARGE ]} h^{+3} \)

    \( \hspace{77mm} = {\LARGE [} \hspace{16mm} \bbox[1pt,F4F4F4]{ -73 } \hspace{16mm} {\LARGE ]} h^{0} +{\LARGE [} \hspace{10mm} \bbox[1pt,F4F4F4]{ -81 } \hspace{10mm} {\LARGE ]} h^{+1} +{\LARGE [} \hspace{5mm} \bbox[1pt,F4F4F4]{ -34 } \hspace{5mm} {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{+2} } + {\LARGE [} \hspace{2mm} \bbox[1pt,F4F4F4]{ -4 } \hspace{2mm} {\LARGE ]} h^{+3} \)

  5. \(CUBIC(+2+h)\)

    \( \hspace{77mm} = {\LARGE [} \hspace{16mm} \bbox[1pt,F4F4F4]{ -73 } \hspace{16mm} {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{0} } +{\LARGE [} \hspace{10mm} \bbox[1pt,F4F4F4]{ -1 } \hspace{10mm} {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{+1} } +{\LARGE [} \hspace{5mm} \bbox[1pt,F4F4F4]{ -1 } \hspace{5mm} {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{+2} } + {\LARGE [} \hspace{2mm} \bbox[1pt,F4F4F4]{ -1 } \hspace{2mm} {\LARGE ]} h^{+3} \)

  6. \(CUBIC(+2+h)\)

    \( \hspace{77mm} = {\LARGE [} \hspace{16mm} \bbox[1pt,F4F4F4]{ -73 } \hspace{16mm} {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{0} } +{\LARGE [} \hspace{10mm} \bbox[1pt,F4F4F4]{ -1 } \hspace{10mm} {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{+1} } +{\LARGE [} \hspace{5mm} \bbox[1pt,F4F4F4]{ -1 } \hspace{5mm} {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{+2} } +{\LARGE [} \hspace{1mm} \bbox[1pt,F4F4F4]{ . . . } \hspace{1mm} {\LARGE ]} \)

  7. Graph each power function near \(0\)

    1. the power function \( x\xrightarrow{\hspace{1mm}P\hspace{1mm}}P(x)= -1x^{0} \).
      Since the exponent is \(0\) and the coefficient is negative, the local graph of \(P\) near \(0\) is:

      power image

    2. the power function \( x\xrightarrow{\hspace{1mm}Q\hspace{1mm}}Q(x)= -1x^{+1} \).
      Since the exponent is \(+1\) and the coefficient is negative, the local graph of \(Q\) near \(0\) is:

      power image

    3. the power function \( x\xrightarrow{\hspace{1mm}R\hspace{1mm}}R(x)= -1x^{+2} \).
      Since the exponent is positive even and the coefficient is negative, the local graph of \(R\) near \(0\) is:

      power image

  8. the local graphs from \(0\) to \(+2\), and the local graphs near \(+2\)

    power image
    power image
    power image

jQuery UI Accordion - Default functionality

In the given global input-output rule, jQuery UI Dialog functionality
To declare an input is to say which specific input is to be used. It is coded as follows \[\left. \right|_{x\gets \text{specific input}} \]
the input to be \( \bbox[1pt,yellow]{ \text{near }+2 } \) :

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

jQuery UI Dialog functionality
To execute a declaration is actually to replace \(x\) by the given input.
the declaration.

\( \hspace{10mm} \left.x\right|_{x\gets\bbox[1pt,yellow]{+2+h}}\xrightarrow{\hspace{1mm} CUBIC\hspace{1mm}}\left. CUBIC(x)\right|_{x\gets\bbox[1pt,yellow]{+2+h}} =\left. -4x^{3}-10x^{2}+7x-15\right|_{x\gets\bbox[1pt,yellow]{+2+h}} \)
\( \hspace{10mm} \bbox[1pt,yellow]{ +2+h } \xrightarrow{\hspace{1mm} CUBIC\hspace{1mm}} CUBIC( \bbox[1pt,yellow]{ +2+h } ) = \bbox[1pt,00FF99]{ -4 } {\large (} \bbox[1pt,yellow]{ +2+h } {\large )}^{3}\hspace{-1mm} \bbox[1pt,00FF99]{ -10 } {\large (} \bbox[1pt,yellow]{ +2+h } {\large )}^{2}\hspace{-1mm} \bbox[1pt,00FF99]{ +7 } {\large (} \bbox[1pt,yellow]{ +2+h } {\large )} \bbox[1pt,00FF99]{ -15 } \)

\(CUBIC(+2+h)\) using the for \( \hspace{10mm} \left.x\right|_{x\gets\bbox[1pt,yellow]{+2+h}}\xrightarrow{\hspace{1mm} CUBIC\hspace{1mm}}\left. CUBIC(x)\right|_{x\gets\bbox[1pt,yellow]{+2+h}} =\left. -4x^{3}-10x^{2}+7x-15\right|_{x\gets\bbox[1pt,yellow]{+2+h}} \)
\( \hspace{10mm} \bbox[1pt,yellow]{ +2+h } \xrightarrow{\hspace{1mm} CUBIC\hspace{1mm}} CUBIC( \bbox[1pt,yellow]{ +2+h } ) = \bbox[1pt,00FF99]{ -4 } {\large (} \bbox[1pt,yellow]{ +2+h } {\large )}^{3}\hspace{-1mm} \bbox[1pt,00FF99]{ -10 } {\large (} \bbox[1pt,yellow]{ +2+h } {\large )}^{2}\hspace{-1mm} \bbox[1pt,00FF99]{ +7 } {\large (} \bbox[1pt,yellow]{ +2+h } {\large )} \bbox[1pt,00FF99]{ -15 } \) \( {\large (} x_{0}+h {\large )}^{3} \text{ and for } {\large (} x_{0}+h {\large )}^{2} \hspace{-1mm} : \)

\[ \hspace{10mm} +2+h\xrightarrow{\hspace{1mm} CUBIC\hspace{1mm}} CUBIC(+2+h) = \bbox[1pt,00FF99]{ -4 } {\LARGE ( } \bbox[1pt,yellow]{ (+2)^{3}+3\cdot(+2)^{2}h+3\cdot(+2)h^{2}+h^{3} } {\LARGE )} \bbox[1pt,00FF99]{ -10 } {\LARGE (} \bbox[1pt,yellow]{ (+2)^{2}+2\cdot(+2)h+h^{2} } {\LARGE )} \bbox[1pt,00FF99]{ +7 } {\LARGE (} \bbox[1pt,yellow]{ +2+h } {\LARGE )} \bbox[1pt,00FF99]{ -15 } \]

Expand 2

\[ \hspace{10mm} +2+h\xrightarrow{\hspace{1mm} CUBIC\hspace{1mm}} CUBIC(+2+h) = \bbox[1pt,00FF99]{ -4 } {\LARGE ( } \bbox[1pt,yellow]{ (+2)^{3}+3\cdot(+2)^{2}h+3\cdot(+2)h^{2}+h^{3} } {\LARGE )} \bbox[1pt,00FF99]{ -10 } {\LARGE (} \bbox[1pt,yellow]{ (+2)^{2}+2\cdot(+2)h+h^{2} } {\LARGE )} \bbox[1pt,00FF99]{ +7 } {\LARGE (} \bbox[1pt,yellow]{ +2+h } {\LARGE )} \bbox[1pt,00FF99]{ -15 } \] \[ \hspace{76mm} = \bbox[1pt,00FF99]{ -4 } {\LARGE (} \hspace{2mm} \bbox[1pt,yellow]{ +8 } \hspace{1mm} \bbox[1pt,yellow]{ } \hspace{7mm} \bbox[1pt,yellow]{ +12h } \hspace{9mm} \bbox[1pt,yellow]{ +6h^{2} } \hspace{6mm} \bbox[1pt,yellow]{ +h^{3} } \hspace{0mm} {\LARGE )} \bbox[1pt,00FF99] { -10 } {\LARGE (} \hspace{1mm} \bbox[1pt,yellow]{ +4 } \hspace{8mm} \bbox[1pt,yellow]{ +4h } \hspace{5mm} \bbox[1pt,yellow]{ +h^{2} } \hspace{1mm} {\LARGE )} \bbox[1pt,00FF99]{ +7 } {\LARGE (} \bbox[1pt,yellow]{ +2+h } {\LARGE )} \bbox[1pt,00FF99] { -15 } \]

Expand 3

\[ \hspace{10mm} +2+h\xrightarrow{\hspace{1mm} CUBIC\hspace{1mm}} CUBIC(+2+h) = \bbox[1pt,00FF99]{ -4 } {\LARGE ( } \bbox[1pt,yellow]{ (+2)^{3}+3\cdot(+2)^{2}h+3\cdot(+2)h^{2}+h^{3} } {\LARGE )} \bbox[1pt,00FF99]{ -10 } {\LARGE (} \bbox[1pt,yellow]{ (+2)^{2}+2\cdot(+2)h+h^{2} } {\LARGE )} \bbox[1pt,00FF99]{ +7 } {\LARGE (} \bbox[1pt,yellow]{ +2+h } {\LARGE )} \bbox[1pt,00FF99]{ -15 } \] \[ \hspace{76mm} = \bbox[1pt,00FF99]{ -4 } {\LARGE (} \hspace{2mm} \bbox[1pt,yellow]{ +8 } \hspace{1mm} \bbox[1pt,yellow]{ } \hspace{7mm} \bbox[1pt,yellow]{ +12h } \hspace{9mm} \bbox[1pt,yellow]{ +6h^{2} } \hspace{6mm} \bbox[1pt,yellow]{ +h^{3} } \hspace{0mm} {\LARGE )} \bbox[1pt,00FF99] { -10 } {\LARGE (} \hspace{1mm} \bbox[1pt,yellow]{ +4 } \hspace{8mm} \bbox[1pt,yellow]{ +4h } \hspace{5mm} \bbox[1pt,yellow]{ +h^{2} } \hspace{1mm} {\LARGE )} \bbox[1pt,00FF99]{ +7 } {\LARGE (} \bbox[1pt,yellow]{ +2+h } {\LARGE )} \bbox[1pt,00FF99] { -15 } \] \[ \hspace{76mm} = \hspace{6mm} {\LARGE (} \hspace{0mm} \bbox[1pt,F4F4F4]{ -32 } \hspace{7mm} \bbox[1pt,F4F4F4]{ -48h } \hspace{7mm} \bbox[1pt,F4F4F4]{ -24h^{2} } \hspace{5mm} \bbox[1pt,F4F4F4]{ -4h^{3} } \hspace{0mm} {\LARGE )} \hspace{3mm} + \hspace{3mm} {\LARGE (} \hspace{0mm} \bbox[1pt,F4F4F4]{ -40 } \hspace{4mm} \bbox[1pt,F4F4F4]{ -40h } \hspace{3mm} \bbox[1pt,F4F4F4]{ -10h^{2} } \hspace{0mm} {\LARGE )} \hspace{3mm} + \hspace{2mm} {\LARGE (} \hspace{0mm} \bbox[1pt,F4F4F4]{ +14 } \hspace{1mm} \bbox[1pt,F4F4F4]{ +7h } \hspace{0mm} {\LARGE )} \bbox[1pt,F4F4F4]{ -15 } \]

Expand 4

\[ \hspace{10mm} +2+h\xrightarrow{\hspace{1mm} CUBIC\hspace{1mm}} CUBIC(+2+h) = \bbox[1pt,00FF99]{ -4 } {\LARGE ( } \bbox[1pt,yellow]{ (+2)^{3}+3\cdot(+2)^{2}h+3\cdot(+2)h^{2}+h^{3} } {\LARGE )} \bbox[1pt,00FF99]{ -10 } {\LARGE (} \bbox[1pt,yellow]{ (+2)^{2}+2\cdot(+2)h+h^{2} } {\LARGE )} \bbox[1pt,00FF99]{ +7 } {\LARGE (} \bbox[1pt,yellow]{ +2+h } {\LARGE )} \bbox[1pt,00FF99]{ -15 } \] \[ \hspace{76mm} = \bbox[1pt,00FF99]{ -4 } {\LARGE (} \hspace{2mm} \bbox[1pt,yellow]{ +8 } \hspace{1mm} \bbox[1pt,yellow]{ } \hspace{7mm} \bbox[1pt,yellow]{ +12h } \hspace{9mm} \bbox[1pt,yellow]{ +6h^{2} } \hspace{6mm} \bbox[1pt,yellow]{ +h^{3} } \hspace{0mm} {\LARGE )} \bbox[1pt,00FF99] { -10 } {\LARGE (} \hspace{1mm} \bbox[1pt,yellow]{ +4 } \hspace{8mm} \bbox[1pt,yellow]{ +4h } \hspace{5mm} \bbox[1pt,yellow]{ +h^{2} } \hspace{1mm} {\LARGE )} \bbox[1pt,00FF99]{ +7 } {\LARGE (} \bbox[1pt,yellow]{ +2+h } {\LARGE )} \bbox[1pt,00FF99] { -15 } \] \[ \hspace{76mm} = \hspace{6mm} {\LARGE (} \hspace{0mm} \bbox[1pt,F4F4F4]{ -32 } \hspace{7mm} \bbox[1pt,F4F4F4]{ -48h } \hspace{7mm} \bbox[1pt,F4F4F4]{ -24h^{2} } \hspace{5mm} \bbox[1pt,F4F4F4]{ -4h^{3} } \hspace{0mm} {\LARGE )} \hspace{3mm} + \hspace{3mm} {\LARGE (} \hspace{0mm} \bbox[1pt,F4F4F4]{ -40 } \hspace{4mm} \bbox[1pt,F4F4F4]{ -40h } \hspace{3mm} \bbox[1pt,F4F4F4]{ -10h^{2} } \hspace{0mm} {\LARGE )} \hspace{3mm} + \hspace{2mm} {\LARGE (} \hspace{0mm} \bbox[1pt,F4F4F4]{ +14 } \hspace{1mm} \bbox[1pt,F4F4F4]{ +7h } \hspace{0mm} {\LARGE )} \bbox[1pt,F4F4F4]{ -15 } \] \[ \hspace{76mm} = \hspace{6mm} {\LARGE (} \hspace{0mm} \bbox[1pt,F4F4F4]{ -32h^{0} } \hspace{2mm} \bbox[1pt,F4F4F4]{ -48h^{+1} } \hspace{2mm} \bbox[1pt,F4F4F4]{ -24h^{+2} } \hspace{2mm} \bbox[1pt,F4F4F4]{ -4h^{+3} } \hspace{0mm} {\LARGE )} \hspace{3mm} + \hspace{3mm} {\LARGE (} \hspace{0mm} \bbox[1pt,F4F4F4]{ -40h^{0} } \hspace{0mm} \bbox[1pt,F4F4F4]{ -40h^{+1} } \hspace{0mm} \bbox[1pt,F4F4F4]{ -10h^{+2} } \hspace{0mm} {\LARGE )} \hspace{3mm} + \hspace{2mm} {\LARGE (} \hspace{0mm} \bbox[1pt,F4F4F4]{ +14h^{0} } \hspace{0mm} \bbox[1pt,F4F4F4]{ +7h^{+1} } \hspace{0mm} {\LARGE )} \bbox[1pt,F4F4F4]{ -15h^{0} } \]

\(CUBIC(+2+h)\) in terms of \(h^{0}\), \(h^{1}\), \(h^{2}\), \(h^{3}\) :

\[ \hspace{10mm} +2+h \xrightarrow{ \hspace{1mm} CUBIC \hspace{1mm} } CUBIC(+2+h) = {\LARGE [} \hspace{0mm} \bbox[1pt,F4F4F4]{ -32 } \hspace{0mm} \bbox[1pt,F4F4F4]{ -40 } \hspace{0mm} \bbox[1pt,F4F4F4]{ +14 } \hspace{0mm} \bbox[1pt,F4F4F4]{ -15 } \hspace{0mm} \hspace{0mm} {\LARGE ]} h^{0} + {\LARGE [} \bbox[1pt,F4F4F4]{ -48 } \hspace{0mm} \bbox[1pt,F4F4F4]{ -40 } \hspace{0mm} \bbox[1pt,F4F4F4]{ +7 } \hspace{0mm} {\LARGE ]} h^{+1} +{\LARGE [} \hspace{0mm} \bbox[1pt,F4F4F4]{ -24-10 } {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{+2} } +{\LARGE [} \hspace{2mm} \bbox[1pt,F4F4F4]{ -4 } \hspace{2mm} {\LARGE ]} h^{+3} \]

Reorganize 2

\[ \hspace{10mm} +2+h \xrightarrow{ \hspace{1mm} CUBIC \hspace{1mm} } CUBIC(+2+h) = {\LARGE [} \hspace{0mm} \bbox[1pt,F4F4F4]{ -32 } \hspace{0mm} \bbox[1pt,F4F4F4]{ -40 } \hspace{0mm} \bbox[1pt,F4F4F4]{ +14 } \hspace{0mm} \bbox[1pt,F4F4F4]{ -15 } \hspace{0mm} \hspace{0mm} {\LARGE ]} h^{0} + {\LARGE [} \bbox[1pt,F4F4F4]{ -48 } \hspace{0mm} \bbox[1pt,F4F4F4]{ -40 } \hspace{0mm} \bbox[1pt,F4F4F4]{ +7 } \hspace{0mm} {\LARGE ]} h^{+1} +{\LARGE [} \hspace{0mm} \bbox[1pt,F4F4F4]{ -24-10 } {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{+2} } +{\LARGE [} \hspace{2mm} \bbox[1pt,F4F4F4]{ -4 } \hspace{2mm} {\LARGE ]} h^{+3} \] \[ \hspace{77mm} = {\LARGE [} \hspace{16mm} \bbox[1pt,F4F4F4]{ -73 } \hspace{16mm} {\LARGE ]} h^{0} +{\LARGE [} \hspace{10mm} \bbox[1pt,F4F4F4]{ -81 } \hspace{10mm} {\LARGE ]} h^{+1} +{\LARGE [} \hspace{5mm} \bbox[1pt,F4F4F4]{ -34 } \hspace{5mm} {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{+2} } + {\LARGE [} \hspace{2mm} \bbox[1pt,F4F4F4]{ -4 } \hspace{2mm} {\LARGE ]} h^{+3} \]

\(CUBIC(+2+h)\)

\[ \hspace{10mm} +2+h \xrightarrow{ \hspace{1mm} CUBIC \hspace{1mm} } CUBIC(+2+h) = {\LARGE [} \hspace{0mm} \bbox[1pt,F4F4F4]{ -32 } \hspace{0mm} \bbox[1pt,F4F4F4]{ -40 } \hspace{0mm} \bbox[1pt,F4F4F4]{ +14 } \hspace{0mm} \bbox[1pt,F4F4F4]{ -15 } \hspace{0mm} \hspace{0mm} {\LARGE ]} h^{0} + {\LARGE [} \bbox[1pt,F4F4F4]{ -48 } \hspace{0mm} \bbox[1pt,F4F4F4]{ -40 } \hspace{0mm} \bbox[1pt,F4F4F4]{ +7 } \hspace{0mm} {\LARGE ]} h^{+1} +{\LARGE [} \hspace{0mm} \bbox[1pt,F4F4F4]{ -24-10 } {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{+2} } +{\LARGE [} \hspace{2mm} \bbox[1pt,F4F4F4]{ -4 } \hspace{2mm} {\LARGE ]} h^{+3} \] \[ \hspace{77mm} = {\LARGE [} \hspace{16mm} \bbox[1pt,F4F4F4]{ -73 } \hspace{16mm} {\LARGE ]} h^{0} +{\LARGE [} \hspace{10mm} \bbox[1pt,F4F4F4]{ -81 } \hspace{10mm} {\LARGE ]} h^{+1} +{\LARGE [} \hspace{5mm} \bbox[1pt,F4F4F4]{ -34 } \hspace{5mm} {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{+2} } + {\LARGE [} \hspace{2mm} \bbox[1pt,F4F4F4]{ -4 } \hspace{2mm} {\LARGE ]} h^{+3} \] \[ \hspace{77mm} = {\LARGE [} \hspace{16mm} \bbox[1pt,F4F4F4]{ -73 } \hspace{16mm} {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{0} } +{\LARGE [} \hspace{10mm} \bbox[1pt,F4F4F4]{ -1 } \hspace{10mm} {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{+1} } +{\LARGE [} \hspace{5mm} \bbox[1pt,F4F4F4]{ -1 } \hspace{5mm} {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{+2} } + {\LARGE [} \hspace{2mm} \bbox[1pt,F4F4F4]{ -1 } \hspace{2mm} {\LARGE ]} h^{+3} \]

\(CUBIC(+2+h)\)

\[ \hspace{10mm} +2+h \xrightarrow{ \hspace{1mm} CUBIC \hspace{1mm} } CUBIC(+2+h) = {\LARGE [} \hspace{0mm} \bbox[1pt,F4F4F4]{ -32 } \hspace{0mm} \bbox[1pt,F4F4F4]{ -40 } \hspace{0mm} \bbox[1pt,F4F4F4]{ +14 } \hspace{0mm} \bbox[1pt,F4F4F4]{ -15 } \hspace{0mm} \hspace{0mm} {\LARGE ]} h^{0} + {\LARGE [} \bbox[1pt,F4F4F4]{ -48 } \hspace{0mm} \bbox[1pt,F4F4F4]{ -40 } \hspace{0mm} \bbox[1pt,F4F4F4]{ +7 } \hspace{0mm} {\LARGE ]} h^{+1} +{\LARGE [} \hspace{0mm} \bbox[1pt,F4F4F4]{ -24-10 } {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{+2} } +{\LARGE [} \hspace{2mm} \bbox[1pt,F4F4F4]{ -4 } \hspace{2mm} {\LARGE ]} h^{+3} \] \[ \hspace{77mm} = {\LARGE [} \hspace{16mm} \bbox[1pt,F4F4F4]{ -73 } \hspace{16mm} {\LARGE ]} h^{0} +{\LARGE [} \hspace{10mm} \bbox[1pt,F4F4F4]{ -81 } \hspace{10mm} {\LARGE ]} h^{+1} +{\LARGE [} \hspace{5mm} \bbox[1pt,F4F4F4]{ -34 } \hspace{5mm} {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{+2} } + {\LARGE [} \hspace{2mm} \bbox[1pt,F4F4F4]{ -4 } \hspace{2mm} {\LARGE ]} h^{+3} \] \[ \hspace{77mm} = {\LARGE [} \hspace{16mm} \bbox[1pt,F4F4F4]{ -73 } \hspace{16mm} {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{0} } +{\LARGE [} \hspace{10mm} \bbox[1pt,F4F4F4]{ -1 } \hspace{10mm} {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{+1} } +{\LARGE [} \hspace{5mm} \bbox[1pt,F4F4F4]{ -1 } \hspace{5mm} {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{+2} } + {\LARGE [} \hspace{2mm} \bbox[1pt,F4F4F4]{ -1 } \hspace{2mm} {\LARGE ]} h^{+3} \] \[ \hspace{77mm} = {\LARGE [} \hspace{16mm} \bbox[1pt,F4F4F4]{ -73 } \hspace{16mm} {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{0} } +{\LARGE [} \hspace{10mm} \bbox[1pt,F4F4F4]{ -1 } \hspace{10mm} {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{+1} } +{\LARGE [} \hspace{5mm} \bbox[1pt,F4F4F4]{ -1 } \hspace{5mm} {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{+2} } +{\LARGE [} \hspace{1mm} \bbox[1pt,F4F4F4]{ . . . } \hspace{1mm} {\LARGE ]} \]

the power function \( x\xrightarrow{\hspace{1mm}K\hspace{1mm}}K(x)= -1x^{0} \).

Since the exponent is \(0\) and the coefficient is negative, the local graph of \(K\) near \(0\) is:

power image

the local graph of \( x\xrightarrow{\hspace{1mm}K\hspace{1mm}}K(x)= -1x^{0} \) from \(0\) to \(+2\).

power image

the power function \( x\xrightarrow{\hspace{1mm}L\hspace{1mm}}L(x)= -1x^{+1} \).

Since the exponent is \(+1\) and the coefficient is negative, the local graph of \(L\) near \(0\) is:

power image

the local graph of \( x\xrightarrow{\hspace{1mm}L\hspace{1mm}}L(x)= -1x^{+1} \) from \(0\) to \(+2\).

power image

the local graph of \(L\) to the local graph of \(K\)

power image

the power function \( x\xrightarrow{\hspace{1mm}Q\hspace{1mm}}Q(x)= -1x^{+2} \).

Since the exponent is positive even and the coefficient is negative, the local graph of \(Q\) near \(0\) is:

power image

the local graph of \( x\xrightarrow{\hspace{1mm}Q\hspace{1mm}}Q(x)= -1x^{+2} \) from \(0\) to \(+2\),

power image

the local graph of \(Q\) to the local graph of \(K+L\)

power image