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mathGraphical3-07

# mathGraphical3-07 - ARE211 Fall 2007 LECTURE#9 THU PRINT...

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ARE211, Fall 2007 LECTURE #9: THU, SEP 27, 2007 PRINT DATE: OCTOBER 12, 2007 (GRAPHICAL3) Contents 2. Graphical Overview of Optimization Theory (cont) 1 2.6. Level Sets, upper and lower contour sets and Gradient vectors (cont) 1 2.7. Quasiconcavity, quasiconvexity 3 2.8. Strict Quasiconcavity 5 2.9. Constrained Optimization: Several Variables 10 2. Graphical Overview of Optimization Theory (cont) 2.6. Level Sets, upper and lower contour sets and Gradient vectors (cont) Vectors : Recall that a vector in R n is an ordered collection of n scalars. A vector in R 2 is often depicted as an arrow. Properly the base of the arrow should be at the origin, but often you see vectors that have been “picked up” and placed elsewhere. Example below. Gradient vectors : When economists draw level sets through a point, they frequently attach arrows to the level sets. These arrows are pictorial representation of the gradient vector , i.e., the slope of f at x , f prime ( x ). Its components are the partial derivatives of the function f , evaluated at x , i.e., ( f 1 ( x ) , · · · , f n ( x )) 1

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ARE211, Fall 2007 3 its length is a measure of the steepness of the function at that point (i.e., the steeper the function, the longer is the arrow.) as we’ve seen it is perpendicular to the level set at the point x it points inside the upper contour set. Note Well: It could point into the upper contour set, but then go out the other side! as we’ve seen, it points in the direction of steepest ascent of the function.

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mathGraphical3-07 - ARE211 Fall 2007 LECTURE#9 THU PRINT...

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