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Question
pleas i want a note on;(1) mean value theorem (2) curve sketching (3) cost function and (4) demand function

Answer
Dear innocent:

(1) The mean value theorem states that if a function is continuous on [a,b] then there is some point in the middle of the function that is equal to the average value over the entire region [a,b].

(2) To sketch a curve, usually it's done in two dimensions in rectangular coordinates named x and y.  The x variable goes on the horizontal axis and the y variable goes on the vertical axis.

(3) Usually the cost function is a basically increasing function with time as the x variable and cost as the y variable.  Now and then it may decrease in value, but usually it goes up.

(4) The demand function slopes one way and the supply function slopes the other.  The ideal production is at the point that maximizes return, which is usually near where the two curves cross.  It is not right on that point except in perfect circumstances.

Calculus

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