3_MathReview_PartialsOptimization

3_MathReview_PartialsOptimization -...

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Self-Clicker Question (P): Is your cell phone on? 1. Yes 2. No 10% 90% “Optimization hinders evolution.” -Alan J. Perlis ECON 410 Math Review –Partial Derivatives & Optimization
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ECON 410 Math Review –Partial Derivatives & Optimization
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3 Class 1 – Math Review
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4 Class 1 – Math Review What is a derivative? Hours of Practice per week Grade 1 2 3 4 75 80 85 90 How much will my function increase (or decrease) if I increase my variable by a little bit.
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5 Class 1 – Math Review What is a partial derivative? Hours of Practice per week Grade 1 2 3 4 75 80 85 90 How much will my function increase (or decrease) if I increase my variable by a little bit in a specific direction .
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6 Class 1 – Math Review Partial Derivative Notation: The partial derivatives of a function, f(x1,x2,x3, …), are 2 1 3 , , ,... df dx 2 3 1 1 2 1 2 1 2 ( , ,...), ( , ,...), ( , ,...),. .. x x x x x x x x x f f f 1 1 2 2 1 2 3 1 2 ( , ,...), ( , ,...), ( , .. x x x x x f f f x
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7 Class 1 – Math Review Partial Derivative Rule: When we are taking the derivative in a specific direction , all other directions are fixed. Thus, to take a partial derivative with respect to a specific direction (variable), treat all other directions (variables) as constant.
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8 Class 1 – Math Review Example: Find the partial derivatives of f(x,y)=xy fx(x,y): f(x,y)=x y fx(x,y)= y fy(x,y): f(x,y)= x y fy(x,y)= x constant
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Self-Clicker Question (P): Find all partial derivatives of f(x1,x2,x3)=x1x2x3 1 2 3 4 5 0% 1% 2% 95% 1. f1=f2=f3=1 2. f1=f2=f3=x1x2x3 3. f1=x2x3, f2=x1x3, f3=x1x2 4. f1=x1, f2=x2, f3=x3 5. f1=f2=f3=x2x3+x1x3+x1x2
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Self-Clicker Question (P): Find all partial derivatives of f(x,y)=x2-y2 1 2 3 4 5 1% 84% 0% 13% 2% 1. df/dx=2x, df/dy=2y 2. df/dx=2x, df/dy=-2y 3. df/dx=2x-2y, df/dy=2x-2y 4. df/dx=-2y, df/dy=2x 5. df/dx=2y, df/dy=2x
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Group-Clicker Question (2 pts): Find all partial derivatives of f(x,y)=x2-ln(z)y2 1 2 3 4 5 1% 0% 92% 6% 1. df/dx=2x, df/dy=-2y, df/dz=1/z 2. df/dx=2x, df/dy=-2y, df/dz=-y2 3. df/dx=2x-ln(z)y2, df/dy=-2yln(z), df/dz=-y2/z 4. df/dx=2x, df/dy=-2yln(z), df/dz=-1/z 5. df/dx=2x, df/dy=-2yln(z), df/dz=-y2/z
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12 Class 1 – Math Review
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13 Class 1 – Math Review Something behaves according to a functional specification What is the best choice ?
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