jQuery UI Dialog - Default functionality

# CUBIC concavity

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 - Default functionality

This is the beginning of the TEXT in the POPUP WINDOW. \begin{align*} x\,large\xrightarrow{\hspace{5mm} AFFINE\hspace{5mm}} AFFINE(x) %&= \left.a \cdot x \oplus b \right|_{x \gets large} &= a \hspace{-0.6mm} \cdot\hspace{-0.6mm} x \hspace{0.6mm}\oplus\hspace{0.6mm} b \\ &= a \hspace{-0.6mm} \cdot\hspace{-0.6mm} x^{+1}+[...] \end{align*} The dialog window can be moved, resized and closed with the 'x' icon.

of $$CUBIC$$ for inputs near $$+2$$ .

PLANNING AHEAD: From the given global input-output rule, get the term in the local input-output rule near $$+2$$ that controls the concavity-sign.

DO:
1. the inputs to be $$\bbox[1pt,yellow]{ +2+h }$$ :

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

2. Carry out the replacement of $$x$$ by $$\bbox[1pt,yellow]{ +2+h } :$$

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

3. Expand 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. To get Concavity-sign, focus on the $$\bbox[1pt,yellow] { h^{2} } \text{s:}$$

$$\hspace{10mm} +2+h\xrightarrow{\hspace{1mm} CUBIC\hspace{1mm}} CUBIC(+2+h) = \bbox[1pt,00FF99]{ -4 } {\LARGE (}\hspace{35mm} \bbox[1pt,yellow]{ +3\cdot(+2)h^{2} } \hspace{11mm} {\LARGE )} \bbox[1pt,00FF99] { -10 } {\LARGE (} \hspace{34mm} \bbox[1pt,yellow]{ +h^{2} } \hspace{-1mm} {\LARGE )} \bbox[1pt,00FF99]{ +7 } {\LARGE (}\hspace{15mm}{\LARGE )} \bbox[1pt,00FF99] { -15 }$$

$$\hspace{76mm} = \bbox[1pt,00FF99]{ -4 } {\LARGE (} \hspace{46mm} \bbox[1pt,yellow]{ +6h^{2} } \hspace{14mm}{\LARGE )} \bbox[1pt,00FF99]{ -10 } {\LARGE (}\hspace{33mm} \bbox[1pt,yellow]{ +h^{2} } {\LARGE )} \bbox[1pt,00FF99]{ +7 } {\LARGE (}\hspace{15mm}{\LARGE )} \bbox[1pt,00FF99]{ -15 }$$

$$\hspace{76mm} = \hspace{7mm} {\LARGE (} \hspace{44mm} \bbox[1pt,F4F4F4]{ -24h^{2} } \hspace{14mm} {\LARGE )} \hspace{2mm} + \hspace{2mm} {\LARGE (} \hspace{29mm} \bbox[1pt,F4F4F4]{ -10h^{2} } {\LARGE )} \bbox[1pt,00FF99]{ +7 } {\LARGE (} \hspace{15mm} {\LARGE )} \bbox[1pt,00FF99]{ -15 }$$

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

$$\hspace{10mm} +2+h \xrightarrow{ \hspace{1mm} CUBIC \hspace{1mm} } CUBIC(+2+h) = {\LARGE [} \hspace{20mm} {\LARGE ]} h^{0} +{\LARGE [} \hspace{20mm} {\LARGE ]} h^{+1} +{\LARGE [} \hspace{0mm} \bbox[1pt,F4F4F4]{ -24-10 } {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{+2} } +{\LARGE [} \hspace{15mm} {\LARGE ]} h^{+3}$$

$$\hspace{77mm} = {\LARGE [} \hspace{20mm} {\LARGE ]} h^{0} +{\LARGE [} \hspace{20mm} {\LARGE ]} h^{+1} +{\LARGE [} \hspace{5mm} \bbox[1pt,F4F4F4]{ -34 } \hspace{5mm} {\LARGE ]} \bbox[1pt,F4F4F4]{ h^{+2} } +{\LARGE [} \hspace{15mm} {\LARGE ]} h^{+3}$$

6. Since the coefficient of $$h^2$$ in $$CUBIC(+2+h)$$ is negative, the Concavity-sign of $$CUBIC$$ near $$+2$$ $$= \langle \cap, \cap \rangle$$

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

2. off the local graph near $$0$$ of the power function $$x\xrightarrow{\hspace{1mm}P\hspace{1mm}}P(x)= -x^{2}$$ :

3. Code the Concavity-sign as seen :

$$\langle \cap, \cap \rangle$$

1. Following is the basic popup. It opens with the page and closes with an x. There is a button to re-open it.