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Math Review - Math Review(to fill in the blanks please...

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1 Math Review (to fill in the blanks please watch the corresponding Podcast; feel free to watch as many times as you want) I. Calculus A. Introduction Let y=f(x) Spse f(x) looks like: Derivative ( ) x x f x x f x f dx dy x Δ - Δ - Δ ) ( ) ( 0 0 0 lim Think change in y over change in x where the change in x is really small
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2 B. Rules for differentiation 1. Constant (multiplied by a function) ( ) ( ) x ag x f y = = ( ) ( ) x g a x f dx dy = = 2. Sum of two functions ( ) ( ) ( ) x h x g x f y + = = ( ) ( ) ( ) x h x g x f dx dy + = = 3. Product Rule ( ) ( ) ( ) x h x g x f y = = ( ) ( ) ( ) ( ) ( ) x h x g x h x g x f dx dy + = = 4. Quotient Rule ( ) ( ) ( ) x h x g x f y = = ( ) ( ) ( ) ( ) ( ) ( ) [ ] 2 x h x h x g x h x g x f dx dy - = = 5. Chain Rule ( ) ( ) ( ) [ ] x h g y x h z z g y = = = , ( ) ( ) [ ] ( ) x h x h g dx dz dz dy x f dx dy = = =
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3 C. Rules for special functional forms 1. Constant ( ) a x f y = = ( ) 0 = = x f dx dy 2. Power function ( ) b ax x f y = = ( ) 1 - = = b abx x f dx dy 3. Logs (note even when economists say logs they mean natural logs) ( ) ( ) x g x f y ln = = ( ) ( ) ( ) x g x g x f dx dy = = 4. Exponents (a) ( ) ( ) x g a x f y = = ( ) ( ) ( ) a x g a x f dx dy x g ln = = (b) ( ) ( ) x g e x f y = = ( ) ( ) ( ) ( ) ( ) x g e e x g e x f dx dy x g x g = = = ln
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4 D. Examples 1. ( ) 5 2 3 7 2 3 + + + = = x x x x f y ( ) ( ) ( ) 2 3 2 7 3 1 2 + + = = x x x f dx dy 2. ( ) ( ) b ax x f y - = = ln ( ) b ax a x f dx dy - = = 3. ( ) e x x f y 4 5 2 - = = ( ) ) 10 ( 4 5 2 x x f dx dy e x - = = 4. ( ) ( ) 7 ln 2 1 + = = x x x f y ( ) ( ) 7 1 7 ln 2 1 2 1 2 1 + + + = = - x x x x x f dx dy 5. ( ) 3 1 2 + + = = x x x f y ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 3 1 6 3 1 6 2 3 1 1 3 2 + - + = + - - + = + + - + = = x x x x x x x x x x x x f dx dy 6. ( ) ( ) 3 2 , 2 - = = = = x x g z z z f y a dx dy find Note that the derivative often depends on the value of x (“where you are on the curve”) - When is this not true (when is the derivative always the same)?
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5 E. Second derivatives ( ) 2 2 dx y d dx dy dx d x f Differentiate twice (change in a change) ( ) ( ) ( ) 2 3 4 36 12 3 x x
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