jQuery UI Dialog - Default functionality

# CUBIC slope

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: 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 Slope-sign.

DO:
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 :

$$\langle \diagdown, \diagdown \rangle$$

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