Local reference praxeologies |${\protect\mathfrak{p}}_{\protect\boldsymbol{iM}}=\left[{T}_{iM},{\tau}_{iM},{\theta}_{iM},{\varTheta}_M\ \right],i\in \left\{1,\dots, 5\right\}$| from the mathematics for economics textbook
| Task (|$\boldsymbol{T}$|) . | Technique (|$\boldsymbol{\tau}$|) . | Technology (|$\boldsymbol{\theta}$|) . |
|---|---|---|
| |${\boldsymbol{T}}_{\mathbf{1}\boldsymbol{M}}$| | |${\boldsymbol{\tau}}_{\mathbf{1}\boldsymbol{M}}$| | |${\boldsymbol{\theta}}_{\mathbf{1}\boldsymbol{M}}$| |
| Maximize and/or minimize |$f\left(x,y\right)$| s.t |$g\left(x,y\right)=c$| graphically. | 1. Plot curves of |$f\left(x,y\right)$| and |$g\left(x,y\right)$| 2. Identify critical points (graphically) - Tangency points - Boundary points 3. Compare values of the points found in step 2. | 1. Analogy of a landscape (|$f\left(x,y\right)$|) where movement is constraint to a road (|$g\left(x,y\right)=c$|) which explains why the slopes should be equal. 2. A continuous, bounded function subject to a constraint has extrema points. 3. The Extreme Value Theorem guarantees that such a function attains max/min values within its feasible region. |
| |${\boldsymbol{T}}_{\mathbf{2}\boldsymbol{M}}$| | |${\boldsymbol{\tau}}_{\mathbf{2}\boldsymbol{M}}$| | |${\boldsymbol{\theta}}_{\mathbf{2}\boldsymbol{M}}$| |
| Maximize and/or minimize |$f\left(x,y\right)$| s.t |$g\left(x,y\right)=c$| using calculus. | 1. Define |$L\left(x,y,\lambda \right)=f\left(x,y\right)-\lambda \left(g\left(x,y\right)-c\right)$| 2. Find partial derivatives, |${L}_x^{\prime }$| and |${L}_y^{\prime }$| 3. Solve $\left\{\begin{array}{@{}c}{L}^{\prime }x=0\\{}{L}^{\prime }y=0\\{}g\left(x,y\right)=c\end{array}\right.$ 4. Compare values: - Interior points from step 3 - Boundary points | 1. Utilizes graphical analysis from |${\mathfrak{p}}_{1M}$| to verify the calculus-based optimization process. 2. Mathematical existence proof is provided to support the conditions under which the optimization process is valid: Suppose |$\left({x}^{\ast },{y}^{\ast}\right)$| is a point satisfying |$g\left(x,y\right)=c$| and a stationary point for |$f\left(x,y\right)\to \exists \lambda$|such that the Lagrange condition holds for |$\left({x}^{\ast },{y}^{\ast },\lambda \right)$|. 3. Continuity of partial derivatives is assumed, which ensures the existence of tangency points where the slopes of |$f\left(x,y\right)$| and |$g\left(x,y\right)$| are equal. |
| |${\boldsymbol{T}}_{\mathbf{3}\boldsymbol{M}}$| | |${\boldsymbol{\tau}}_{\mathbf{3}\boldsymbol{M}}$| | |${\boldsymbol{\theta}}_{\mathbf{3}\boldsymbol{M}}$| |
| Give an approximation of the increase/decrease in maximum/minimum value of the objective function from a change in |$c$| of the constraint. | 1. Perform steps 1 to 3 from |${\tau}_{2M}$| to determine the Lagrange multiplier |$\lambda .$| 2. Interpret the value of |$\lambda$| as the rate of change of the extreme value of the objective function with respect to a unit change in c. | 1. The solution obtained by Lagrange’s method depends on the value |$\mathrm{c}:$||$\mathrm{m}\left(\mathrm{c}\right)=\mathrm{f}\left(\mathrm{x}\left(\mathrm{c}\right),\mathrm{y}\left(\mathrm{c}\right)\right).$|2. 2. The derivative |$\frac{\mathrm{dm}}{\mathrm{dc}}=\mathrm{\lambda}$| represents the rate of change of the optimal value |$\mathrm{m}\left(\mathrm{c}\right)$| with respect to the constrained parameter (c). |
| |${\boldsymbol{T}}_{\mathbf{4}\boldsymbol{M}}$| | |${\boldsymbol{\tau}}_{\mathbf{4}\boldsymbol{M}}$| | |${\boldsymbol{\theta}}_{\mathbf{4}\boldsymbol{M}}$| |
| Investigate the existence of max/min for |$f\left(x,y\right)$| s.t |$g\left(x,y\right)=c$|. | 1. Check whether the constraint set defined by |$g\left(x,y\right)=c$| is bounded. 2. If it is, |$f\left(x,y\right)$| attains both a maximum and a minimum value on this set. | 1. The technologies |${\theta}_{1M}$| and |${\theta}_{2M}$| 2. If the constraint set is bounded and the function |$f\left(x,y\right)$| is continuous, then by the Extreme Value Theorem |$f\left(x,y\right)$| attains both a maximum and a minimum value on this set. |
| |${\boldsymbol{T}}_{\mathbf{5}\boldsymbol{M}}$| | |${\boldsymbol{\tau}}_{\mathbf{5}\boldsymbol{M}}$| | |${\boldsymbol{\theta}}_{\mathbf{5}\boldsymbol{M}}$| |
| Find the slope of the level curve at some point |$\left({x}^{\ast },{y}^{\ast}\right)$|. | 1. Calculate the partial derivatives |${f}_x^{\prime}\left({x}^{\ast },{y}^{\ast}\right)$|and |${f}_y^{\prime}\left({x}^{\ast },{y}^{\ast}\right)$|. 2.The slope |$s$| at a specific point |$\left({x}^{\ast },{y}^{\ast}\right)$| can be calculated as |$s=-\frac{f_x^{\prime}\left({x}^{\ast },{y}^{\ast}\right)}{f_y^{\prime}\left({x}^{\ast },{y}^{\ast}\right)}$|. | 1. The partial derivatives show how the function |$f\left(x,y\right)$| changes with respect to each variable. 2. The slope |$s$| is the rate of change along the level curve passing through |$\left({x}^{\ast },{y}^{\ast}\right)$|. 3. Informal graphical example in earlier chapter |
| Task (|$\boldsymbol{T}$|) . | Technique (|$\boldsymbol{\tau}$|) . | Technology (|$\boldsymbol{\theta}$|) . |
|---|---|---|
| |${\boldsymbol{T}}_{\mathbf{1}\boldsymbol{M}}$| | |${\boldsymbol{\tau}}_{\mathbf{1}\boldsymbol{M}}$| | |${\boldsymbol{\theta}}_{\mathbf{1}\boldsymbol{M}}$| |
| Maximize and/or minimize |$f\left(x,y\right)$| s.t |$g\left(x,y\right)=c$| graphically. | 1. Plot curves of |$f\left(x,y\right)$| and |$g\left(x,y\right)$| 2. Identify critical points (graphically) - Tangency points - Boundary points 3. Compare values of the points found in step 2. | 1. Analogy of a landscape (|$f\left(x,y\right)$|) where movement is constraint to a road (|$g\left(x,y\right)=c$|) which explains why the slopes should be equal. 2. A continuous, bounded function subject to a constraint has extrema points. 3. The Extreme Value Theorem guarantees that such a function attains max/min values within its feasible region. |
| |${\boldsymbol{T}}_{\mathbf{2}\boldsymbol{M}}$| | |${\boldsymbol{\tau}}_{\mathbf{2}\boldsymbol{M}}$| | |${\boldsymbol{\theta}}_{\mathbf{2}\boldsymbol{M}}$| |
| Maximize and/or minimize |$f\left(x,y\right)$| s.t |$g\left(x,y\right)=c$| using calculus. | 1. Define |$L\left(x,y,\lambda \right)=f\left(x,y\right)-\lambda \left(g\left(x,y\right)-c\right)$| 2. Find partial derivatives, |${L}_x^{\prime }$| and |${L}_y^{\prime }$| 3. Solve $\left\{\begin{array}{@{}c}{L}^{\prime }x=0\\{}{L}^{\prime }y=0\\{}g\left(x,y\right)=c\end{array}\right.$ 4. Compare values: - Interior points from step 3 - Boundary points | 1. Utilizes graphical analysis from |${\mathfrak{p}}_{1M}$| to verify the calculus-based optimization process. 2. Mathematical existence proof is provided to support the conditions under which the optimization process is valid: Suppose |$\left({x}^{\ast },{y}^{\ast}\right)$| is a point satisfying |$g\left(x,y\right)=c$| and a stationary point for |$f\left(x,y\right)\to \exists \lambda$|such that the Lagrange condition holds for |$\left({x}^{\ast },{y}^{\ast },\lambda \right)$|. 3. Continuity of partial derivatives is assumed, which ensures the existence of tangency points where the slopes of |$f\left(x,y\right)$| and |$g\left(x,y\right)$| are equal. |
| |${\boldsymbol{T}}_{\mathbf{3}\boldsymbol{M}}$| | |${\boldsymbol{\tau}}_{\mathbf{3}\boldsymbol{M}}$| | |${\boldsymbol{\theta}}_{\mathbf{3}\boldsymbol{M}}$| |
| Give an approximation of the increase/decrease in maximum/minimum value of the objective function from a change in |$c$| of the constraint. | 1. Perform steps 1 to 3 from |${\tau}_{2M}$| to determine the Lagrange multiplier |$\lambda .$| 2. Interpret the value of |$\lambda$| as the rate of change of the extreme value of the objective function with respect to a unit change in c. | 1. The solution obtained by Lagrange’s method depends on the value |$\mathrm{c}:$||$\mathrm{m}\left(\mathrm{c}\right)=\mathrm{f}\left(\mathrm{x}\left(\mathrm{c}\right),\mathrm{y}\left(\mathrm{c}\right)\right).$|2. 2. The derivative |$\frac{\mathrm{dm}}{\mathrm{dc}}=\mathrm{\lambda}$| represents the rate of change of the optimal value |$\mathrm{m}\left(\mathrm{c}\right)$| with respect to the constrained parameter (c). |
| |${\boldsymbol{T}}_{\mathbf{4}\boldsymbol{M}}$| | |${\boldsymbol{\tau}}_{\mathbf{4}\boldsymbol{M}}$| | |${\boldsymbol{\theta}}_{\mathbf{4}\boldsymbol{M}}$| |
| Investigate the existence of max/min for |$f\left(x,y\right)$| s.t |$g\left(x,y\right)=c$|. | 1. Check whether the constraint set defined by |$g\left(x,y\right)=c$| is bounded. 2. If it is, |$f\left(x,y\right)$| attains both a maximum and a minimum value on this set. | 1. The technologies |${\theta}_{1M}$| and |${\theta}_{2M}$| 2. If the constraint set is bounded and the function |$f\left(x,y\right)$| is continuous, then by the Extreme Value Theorem |$f\left(x,y\right)$| attains both a maximum and a minimum value on this set. |
| |${\boldsymbol{T}}_{\mathbf{5}\boldsymbol{M}}$| | |${\boldsymbol{\tau}}_{\mathbf{5}\boldsymbol{M}}$| | |${\boldsymbol{\theta}}_{\mathbf{5}\boldsymbol{M}}$| |
| Find the slope of the level curve at some point |$\left({x}^{\ast },{y}^{\ast}\right)$|. | 1. Calculate the partial derivatives |${f}_x^{\prime}\left({x}^{\ast },{y}^{\ast}\right)$|and |${f}_y^{\prime}\left({x}^{\ast },{y}^{\ast}\right)$|. 2.The slope |$s$| at a specific point |$\left({x}^{\ast },{y}^{\ast}\right)$| can be calculated as |$s=-\frac{f_x^{\prime}\left({x}^{\ast },{y}^{\ast}\right)}{f_y^{\prime}\left({x}^{\ast },{y}^{\ast}\right)}$|. | 1. The partial derivatives show how the function |$f\left(x,y\right)$| changes with respect to each variable. 2. The slope |$s$| is the rate of change along the level curve passing through |$\left({x}^{\ast },{y}^{\ast}\right)$|. 3. Informal graphical example in earlier chapter |
Local reference praxeologies |${\protect\mathfrak{p}}_{\protect\boldsymbol{iM}}=\left[{T}_{iM},{\tau}_{iM},{\theta}_{iM},{\varTheta}_M\ \right],i\in \left\{1,\dots, 5\right\}$| from the mathematics for economics textbook
| Task (|$\boldsymbol{T}$|) . | Technique (|$\boldsymbol{\tau}$|) . | Technology (|$\boldsymbol{\theta}$|) . |
|---|---|---|
| |${\boldsymbol{T}}_{\mathbf{1}\boldsymbol{M}}$| | |${\boldsymbol{\tau}}_{\mathbf{1}\boldsymbol{M}}$| | |${\boldsymbol{\theta}}_{\mathbf{1}\boldsymbol{M}}$| |
| Maximize and/or minimize |$f\left(x,y\right)$| s.t |$g\left(x,y\right)=c$| graphically. | 1. Plot curves of |$f\left(x,y\right)$| and |$g\left(x,y\right)$| 2. Identify critical points (graphically) - Tangency points - Boundary points 3. Compare values of the points found in step 2. | 1. Analogy of a landscape (|$f\left(x,y\right)$|) where movement is constraint to a road (|$g\left(x,y\right)=c$|) which explains why the slopes should be equal. 2. A continuous, bounded function subject to a constraint has extrema points. 3. The Extreme Value Theorem guarantees that such a function attains max/min values within its feasible region. |
| |${\boldsymbol{T}}_{\mathbf{2}\boldsymbol{M}}$| | |${\boldsymbol{\tau}}_{\mathbf{2}\boldsymbol{M}}$| | |${\boldsymbol{\theta}}_{\mathbf{2}\boldsymbol{M}}$| |
| Maximize and/or minimize |$f\left(x,y\right)$| s.t |$g\left(x,y\right)=c$| using calculus. | 1. Define |$L\left(x,y,\lambda \right)=f\left(x,y\right)-\lambda \left(g\left(x,y\right)-c\right)$| 2. Find partial derivatives, |${L}_x^{\prime }$| and |${L}_y^{\prime }$| 3. Solve $\left\{\begin{array}{@{}c}{L}^{\prime }x=0\\{}{L}^{\prime }y=0\\{}g\left(x,y\right)=c\end{array}\right.$ 4. Compare values: - Interior points from step 3 - Boundary points | 1. Utilizes graphical analysis from |${\mathfrak{p}}_{1M}$| to verify the calculus-based optimization process. 2. Mathematical existence proof is provided to support the conditions under which the optimization process is valid: Suppose |$\left({x}^{\ast },{y}^{\ast}\right)$| is a point satisfying |$g\left(x,y\right)=c$| and a stationary point for |$f\left(x,y\right)\to \exists \lambda$|such that the Lagrange condition holds for |$\left({x}^{\ast },{y}^{\ast },\lambda \right)$|. 3. Continuity of partial derivatives is assumed, which ensures the existence of tangency points where the slopes of |$f\left(x,y\right)$| and |$g\left(x,y\right)$| are equal. |
| |${\boldsymbol{T}}_{\mathbf{3}\boldsymbol{M}}$| | |${\boldsymbol{\tau}}_{\mathbf{3}\boldsymbol{M}}$| | |${\boldsymbol{\theta}}_{\mathbf{3}\boldsymbol{M}}$| |
| Give an approximation of the increase/decrease in maximum/minimum value of the objective function from a change in |$c$| of the constraint. | 1. Perform steps 1 to 3 from |${\tau}_{2M}$| to determine the Lagrange multiplier |$\lambda .$| 2. Interpret the value of |$\lambda$| as the rate of change of the extreme value of the objective function with respect to a unit change in c. | 1. The solution obtained by Lagrange’s method depends on the value |$\mathrm{c}:$||$\mathrm{m}\left(\mathrm{c}\right)=\mathrm{f}\left(\mathrm{x}\left(\mathrm{c}\right),\mathrm{y}\left(\mathrm{c}\right)\right).$|2. 2. The derivative |$\frac{\mathrm{dm}}{\mathrm{dc}}=\mathrm{\lambda}$| represents the rate of change of the optimal value |$\mathrm{m}\left(\mathrm{c}\right)$| with respect to the constrained parameter (c). |
| |${\boldsymbol{T}}_{\mathbf{4}\boldsymbol{M}}$| | |${\boldsymbol{\tau}}_{\mathbf{4}\boldsymbol{M}}$| | |${\boldsymbol{\theta}}_{\mathbf{4}\boldsymbol{M}}$| |
| Investigate the existence of max/min for |$f\left(x,y\right)$| s.t |$g\left(x,y\right)=c$|. | 1. Check whether the constraint set defined by |$g\left(x,y\right)=c$| is bounded. 2. If it is, |$f\left(x,y\right)$| attains both a maximum and a minimum value on this set. | 1. The technologies |${\theta}_{1M}$| and |${\theta}_{2M}$| 2. If the constraint set is bounded and the function |$f\left(x,y\right)$| is continuous, then by the Extreme Value Theorem |$f\left(x,y\right)$| attains both a maximum and a minimum value on this set. |
| |${\boldsymbol{T}}_{\mathbf{5}\boldsymbol{M}}$| | |${\boldsymbol{\tau}}_{\mathbf{5}\boldsymbol{M}}$| | |${\boldsymbol{\theta}}_{\mathbf{5}\boldsymbol{M}}$| |
| Find the slope of the level curve at some point |$\left({x}^{\ast },{y}^{\ast}\right)$|. | 1. Calculate the partial derivatives |${f}_x^{\prime}\left({x}^{\ast },{y}^{\ast}\right)$|and |${f}_y^{\prime}\left({x}^{\ast },{y}^{\ast}\right)$|. 2.The slope |$s$| at a specific point |$\left({x}^{\ast },{y}^{\ast}\right)$| can be calculated as |$s=-\frac{f_x^{\prime}\left({x}^{\ast },{y}^{\ast}\right)}{f_y^{\prime}\left({x}^{\ast },{y}^{\ast}\right)}$|. | 1. The partial derivatives show how the function |$f\left(x,y\right)$| changes with respect to each variable. 2. The slope |$s$| is the rate of change along the level curve passing through |$\left({x}^{\ast },{y}^{\ast}\right)$|. 3. Informal graphical example in earlier chapter |
| Task (|$\boldsymbol{T}$|) . | Technique (|$\boldsymbol{\tau}$|) . | Technology (|$\boldsymbol{\theta}$|) . |
|---|---|---|
| |${\boldsymbol{T}}_{\mathbf{1}\boldsymbol{M}}$| | |${\boldsymbol{\tau}}_{\mathbf{1}\boldsymbol{M}}$| | |${\boldsymbol{\theta}}_{\mathbf{1}\boldsymbol{M}}$| |
| Maximize and/or minimize |$f\left(x,y\right)$| s.t |$g\left(x,y\right)=c$| graphically. | 1. Plot curves of |$f\left(x,y\right)$| and |$g\left(x,y\right)$| 2. Identify critical points (graphically) - Tangency points - Boundary points 3. Compare values of the points found in step 2. | 1. Analogy of a landscape (|$f\left(x,y\right)$|) where movement is constraint to a road (|$g\left(x,y\right)=c$|) which explains why the slopes should be equal. 2. A continuous, bounded function subject to a constraint has extrema points. 3. The Extreme Value Theorem guarantees that such a function attains max/min values within its feasible region. |
| |${\boldsymbol{T}}_{\mathbf{2}\boldsymbol{M}}$| | |${\boldsymbol{\tau}}_{\mathbf{2}\boldsymbol{M}}$| | |${\boldsymbol{\theta}}_{\mathbf{2}\boldsymbol{M}}$| |
| Maximize and/or minimize |$f\left(x,y\right)$| s.t |$g\left(x,y\right)=c$| using calculus. | 1. Define |$L\left(x,y,\lambda \right)=f\left(x,y\right)-\lambda \left(g\left(x,y\right)-c\right)$| 2. Find partial derivatives, |${L}_x^{\prime }$| and |${L}_y^{\prime }$| 3. Solve $\left\{\begin{array}{@{}c}{L}^{\prime }x=0\\{}{L}^{\prime }y=0\\{}g\left(x,y\right)=c\end{array}\right.$ 4. Compare values: - Interior points from step 3 - Boundary points | 1. Utilizes graphical analysis from |${\mathfrak{p}}_{1M}$| to verify the calculus-based optimization process. 2. Mathematical existence proof is provided to support the conditions under which the optimization process is valid: Suppose |$\left({x}^{\ast },{y}^{\ast}\right)$| is a point satisfying |$g\left(x,y\right)=c$| and a stationary point for |$f\left(x,y\right)\to \exists \lambda$|such that the Lagrange condition holds for |$\left({x}^{\ast },{y}^{\ast },\lambda \right)$|. 3. Continuity of partial derivatives is assumed, which ensures the existence of tangency points where the slopes of |$f\left(x,y\right)$| and |$g\left(x,y\right)$| are equal. |
| |${\boldsymbol{T}}_{\mathbf{3}\boldsymbol{M}}$| | |${\boldsymbol{\tau}}_{\mathbf{3}\boldsymbol{M}}$| | |${\boldsymbol{\theta}}_{\mathbf{3}\boldsymbol{M}}$| |
| Give an approximation of the increase/decrease in maximum/minimum value of the objective function from a change in |$c$| of the constraint. | 1. Perform steps 1 to 3 from |${\tau}_{2M}$| to determine the Lagrange multiplier |$\lambda .$| 2. Interpret the value of |$\lambda$| as the rate of change of the extreme value of the objective function with respect to a unit change in c. | 1. The solution obtained by Lagrange’s method depends on the value |$\mathrm{c}:$||$\mathrm{m}\left(\mathrm{c}\right)=\mathrm{f}\left(\mathrm{x}\left(\mathrm{c}\right),\mathrm{y}\left(\mathrm{c}\right)\right).$|2. 2. The derivative |$\frac{\mathrm{dm}}{\mathrm{dc}}=\mathrm{\lambda}$| represents the rate of change of the optimal value |$\mathrm{m}\left(\mathrm{c}\right)$| with respect to the constrained parameter (c). |
| |${\boldsymbol{T}}_{\mathbf{4}\boldsymbol{M}}$| | |${\boldsymbol{\tau}}_{\mathbf{4}\boldsymbol{M}}$| | |${\boldsymbol{\theta}}_{\mathbf{4}\boldsymbol{M}}$| |
| Investigate the existence of max/min for |$f\left(x,y\right)$| s.t |$g\left(x,y\right)=c$|. | 1. Check whether the constraint set defined by |$g\left(x,y\right)=c$| is bounded. 2. If it is, |$f\left(x,y\right)$| attains both a maximum and a minimum value on this set. | 1. The technologies |${\theta}_{1M}$| and |${\theta}_{2M}$| 2. If the constraint set is bounded and the function |$f\left(x,y\right)$| is continuous, then by the Extreme Value Theorem |$f\left(x,y\right)$| attains both a maximum and a minimum value on this set. |
| |${\boldsymbol{T}}_{\mathbf{5}\boldsymbol{M}}$| | |${\boldsymbol{\tau}}_{\mathbf{5}\boldsymbol{M}}$| | |${\boldsymbol{\theta}}_{\mathbf{5}\boldsymbol{M}}$| |
| Find the slope of the level curve at some point |$\left({x}^{\ast },{y}^{\ast}\right)$|. | 1. Calculate the partial derivatives |${f}_x^{\prime}\left({x}^{\ast },{y}^{\ast}\right)$|and |${f}_y^{\prime}\left({x}^{\ast },{y}^{\ast}\right)$|. 2.The slope |$s$| at a specific point |$\left({x}^{\ast },{y}^{\ast}\right)$| can be calculated as |$s=-\frac{f_x^{\prime}\left({x}^{\ast },{y}^{\ast}\right)}{f_y^{\prime}\left({x}^{\ast },{y}^{\ast}\right)}$|. | 1. The partial derivatives show how the function |$f\left(x,y\right)$| changes with respect to each variable. 2. The slope |$s$| is the rate of change along the level curve passing through |$\left({x}^{\ast },{y}^{\ast}\right)$|. 3. Informal graphical example in earlier chapter |
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