MATH-UA 211 Lecture Notes - Lecture 9: Product Rule, Quotient Rule, Marginal Cost
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If f(x) is differentiable (derivative exists) at x=a, then f(x) is continuous at x=a: the converse is not true (if f(x) is continuous at x=a ) f ( x)= x at x=0. X ={ x if x>0 f " ( x)={ 1x>0. Xif x 0 o o: the derivative does not exist, similar to continuity at a specific point. Application example: t(x) is the amount of tax a family pays if income is x. F" ( x)=1g "(1)+ 2xg(1: (1)(0)+ 2(-2)= -4, quotient rule f ( x) g(x) d d( x) (g( x) f " ( x) f ( x) g" ( x)) (g( x))2. Findthe slope of atengent line the graph of f (x)= x2 5. F" ( x)= (3x +2) (2x) ( x2 5) (3) (3x 2)2 o o o: plug in 2 ( 8)( 4) ( 1) (3)