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Unformatted text preview: constrained problems. You must look at your data or a graph to make that decision. Example: Find the Extrema of where Let and your constraint ( 3 equations and 3 unknowns ) Thus Same Example by Substitution: Find the Extrema of where Another New Example: Maximize where Let and your constraint Thus leads to a maximum value: ( 4 equations and 4 unknowns ) Theorem: The Lagrange Multiplier constraint. is the rate of change of the Extreme Value with respect to t where is the Proof: Let be the Extreme Value. where Thus We know Thus Since Thus we have shown: An Example of this Theorem in Use: Cobb-Douglas Production Function: The maximal output will increase by 31,750 if the available money is increased by 1000 and allocated optimally. Two Different Methods to Find Extrema on a Bounded Region: Method 1 (Substitution) bounded by is a Minimum Method 2 (Lagrange Multipliers) Visual of The Lagrange Multiplier Method: Minimize subject to the constraint Let us assume we found the minimum at Then this point is located on the level curve for We know So the two level curves at AND on the level curve for for and share the same tangent line SUMMARY 12.1 Finding the equation of a plane / Knowing the 7 different surfaces 12.2 z = f(x,y) / Contour Plots / Density Plots /Specific Level Curves (Using Mathematica for these) 12.3 Show Limit Does Not Exist by two different paths / Show Limit Exists using Theorem 2.1 Determining where a function is continuous especially piecewise defined functions 12.4 Computing Partial Derivatives Symbolically, with a Table, and by Contour Plots / Using Mathematica to graph a surface, an intersecting plane, the curve of intersection, the tangent line at a specific point on the space curve 12.5 Using the Chain Rule and Implicit Differentation 12.6 Computing a Directional Derivative / Deriving the Max rate of change occurs in the direction of the gradient vector / Finding Max and Min rates of change / Using the gradient to find a tangent plane and a normal line to a surface 12.7 Finding tangent planes and normal lines for a surface / Linear Approximations / Calculating the Total Differential Approximating the Sag in a Beam / Finding where a function is not Differentiable by looking at the Partials Take Home Possibility: Show a function is Differentiable 12.8 Finding Local Extrema / Finding Critical Points and using the Discriminant to Classify them / Finding the Regression Line using the Least Squares Method / Finding Absolute Extrema 12.9 Constrained Optimization by the Method of Lagrange Multipliers / Optimization with an Inequality Constraint Optimization with TWO constraints...
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This note was uploaded on 03/05/2014 for the course MATH 101c taught by Professor Loukianoff,v during the Spring '08 term at Ohlone.

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