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Unformatted text preview: Some slides for Econ 2450 Continuously updated Nippe Lagerl&f March 19, 2009 1 Math revision Course mostly about using economic models Requires a bit of math to understand Nothing that should be news Numbers Natural numbers: 1 ; 2 ; 3 ::: Integers: ::: & 2 ; & 1 ; ; 1 ; 2 ::: Real numbers: &allnumbers that lie on the real line (e.g. 5 , & 100 , & = 3 : 14 ::: , 1 11 ) Rational numbers: numbers that can be written as a b where a and b are integers Irrational numbers: real numbers that are not rational numbers Inequalities a > b means a is strictly greater than b a & b means a is weakly greater than (i.e., greater than or equal to) b Double inequalities, intervals If a > b and a < c we can write b < a < c or a 2 ( b;c ) Means a lies on the (open) interval ( b;c ) If a & b and a c we can write b a c or a 2 [ b;c ] Means a lies on the (closed) interval [ b;c ] If a > b and a c we can write b < a c or a 2 ( b;c ] Means a lies on the (halfopen) interval ( b;c ] Functions Here: mostly singevariable functions What are functions? & Functions are &rules & Functions describe how one variable depends on an other variable & Same as everyday speak: &the outcome is a function of how much you spend and what you tell them... Graphs are used to illustrate functions in diagrams De&nitions A function f ( x ) of a real variable x with domain D f is a rule which assigns a unique real number to each number in D f D f can be de&ned by a simple statement, e.g. D f = [0 ; 1] The set of values that f ( x ) takes as x varies over D f is called the range of f , and is denoted R f All functions in this course will be given by formulas , e.g. f ( x ) = 2 x 2 f ( x ) = 3 x & 14 f ( x ) = b + ax where a and b are constants, i.e., do not depend on the variable x Exponential and logarithmic functions Power functions take the form f ( x ) = a bx where a and b are constants a is the base ; bx is the exponent If a = e , where e is de&ned from e = lim n !1 & 1 + 1 n n & 2 : 71828 , then we get an exponential function: f ( x ) = e bx The (natural) logarithmic function f ( x ) = ln( x ) is de&ned from e ln( x ) = x Note that: & e bx > for all x & ln( x ) is de&ned only for x > Some important equalities ln( xy ) = ln( x ) + ln( y ) ln( x y ) = ln( x ) & ln( y ) ln( x a ) = a ln( x ) e x + y = e x e y ( e x ) y = e xy Derivatives Many functions can be di/erentiated To di/erentiate a function is the same as taking the deriv ative of the function (but not same as e.g. deriving a function) The derivative of f ( x ) is (usually) denoted f ( x ) and de&ned from f ( x ) = lim h ! f ( x + h ) & f ( x ) h On notation There are many ways to denote derivatives We often denote f ( x ) by some some other variable y , and write y = f ( x ) We can then write f ( x ) as any of the following: y @y @x @f ( x ) @x dy dx df ( x ) dx The notation @f ( x ) =@x is usually referred to as the par tial derivative of f ( x ) with respect to x The notation df ( x ) =dx is usually referred to as the...
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 Spring '10
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