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

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

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

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

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

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

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

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

7. $$CUBIC(+2+h)$$

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

8. 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:

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:

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:

9. the local graphs from $$0$$ to $$+2$$, and the local graphs near $$+2$$

jQuery UI Accordion - Default functionality

### In the given global input-output rule, jQuery UI Dialog functionality .ui-widget-header, .ui-state-default, ui-button{ background:#b9cd6d; border: 1px solid #b9cd6d; color: #FFFFFF; font-weight: bold; } $(function() {$( "#dialog-2" ).dialog({ autoOpen: false, width: 500, height: 250, modal: false, position: "left top", buttons: { OK: function() {$(this).dialog("close");} }, });$( "#opener-2" ).click(function() { $( "#dialog-2" ).dialog( "open" ); }); }); 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 .ui-widget-header, .ui-state-default, ui-button{ background:#b9cd6d; border: 1px solid #b9cd6d; color: #000000; font-weight: bold; }$(function() { $( "#dialog-3" ).dialog({ autoOpen: false, width: 500, height: 250, modal: false, position: "left top", buttons: { OK: function() {$(this).dialog("close");} }, }); $( "#opener-3" ).click(function() {$( "#dialog-3" ).dialog( "open" ); }); }); 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:

### 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:

### 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: