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class2_review_ - Econ 51 Winter 2011 Class#2 Welcome back...

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1 Econ 51, Winter 2011 Class #2 • Welcome back! •New TA: Sanaa Nadeem •TA sections and OH schedule: still waiting for approval (will update once we get approval). •No after-class office hours today: Makeup OH is next Monday 10th, 10:00-11:50 in Econ. Building 239. TAs stay today for a while after class, so ask questions. • Today: Review of Math and Econ 50 Sunday, January 9, 2011
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2 Math Review Sunday, January 9, 2011
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3 Overview of The Math Review Differentiation Rules of Differentiation Probability and Statistics Expected Value, Variance, Covariance and Correlation Optimization Unconstrained and Constrained Optimizations; First- and Second-Order Conditions Sunday, January 9, 2011
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Differentiation Sunday, January 9, 2011
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5 What is differentiation? Let f(x) be a function from the set of real numbers to itself. The derivative is the “marginal rate of change” of a function, i.e., the derivative of f(x) at x is defined as lim t 0 [f(x+t) - f(x)]/t The derivative of f(x) is denoted by df(x)/dx (pronounced “D F D X”) or f’(x) . Example: If u(x) is a utility function, then u’(x) is the “marginal utility,” i.e., the rate of increase in utility of per- unit increase in consumption x . Example 2: If c(x) is a firm’s cost function, then c’(x) is the “marginal cost,” i.e., per-unit change of production cost. The word “ differentiation ” means finding a derivative of a function. Sunday, January 9, 2011
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6 Some Differentiation Rules Derivatives for well-known functions: f(x) is a constant function, i.e., f(x)=c for all x f’(x)=0. f(x)=x n for a constant n f’(x)=nx n-1 f(x)=e x f’(x)=e x f(x)=ln(x) f’(x)=1/x (remember ln(x) is the logarithm, also written as log(x) ) Sunday, January 9, 2011
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7 Some Differentiation Rules, continued Some useful facts: Let f(.), g(.), h(.) be functions. (note: f(.) is just a way to write a generic function; we use f(.) instead of f(x) when we don’t want to write x explicitly) h(x)=f(x)+g(x) h’(x)=f’(x)+g’(x) h(x)=f(x)g(x) h’(x)=f’(x)g(x)+f(x)g’(x) h(x)=f(x)/g(x) h’(x)=[f’(x)g(x)-f(x)g’(x)]/(g(x)) 2 h(x)=f(g(x)) h’(x)=f’(g(x))g’(x) Sunday, January 9, 2011
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8 Derivatives of Functions of Multiple Variables When a function depends on more than one variable, it makes sense to consider the rate of change (derivative) with respect to each of them separately: called “ partial derivatives. Write f/ x, f/ y etc. instead of df/dx or f’(x) (and pronounced “partial F partial X” etc). f/ x is the partial derivative of function f with respect to x , for example. Rule: the same as before. Just treat as constants all variables with respect to which you are NOT differentiating. Examples: f(x,y)= x 2 +y 4 f(x,y)/ x = 2x, f (x,y)/ y = 4y 3 f(x,y)= x 2 y 4 f(x,y)/ x = 2xy 4 , f(x,y)/ y = 4x 2 y 3 f(x,y)= y 4 f (x,y)/ x = 0, f (x,y)/ y = 4y 3 Sunday, January 9, 2011
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Probability and Statistics Sunday, January 9, 2011
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10 Expected Value
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