Lecture_00_Math_Review

Lecture_00_Math_Review - Review of Calculus Tools c...

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Unformatted text preview: Review of Calculus Tools c & 2008 Je/rey A. Miron Outline 1. Derivatives 2. Optimization of a Function of a Single Variable 3. Partial Derivatives 4. Optimization of a Function of Several Variables 5. Optimization Subject to Constraints 1 Derivatives The basic tool we need to review is derivatives. The basic, intuititive de&nition of a derivative is that it is the rate of change of a function in response to a change in its argument. Lets take an example and look at it more slowly. Say we have some variable y that is a function of another variable x , e.g., y = f ( x ) For example, we could have y = x 2 or y = 7 x + 3 or y = ln x Graphically, I am just assuming that we have something that looks like the following: 1 Graph: A Standard Di/erentiable Function with a Maximum y = & ( x & 3) 2 + 8 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 x y Now say that we are interested in knowing how y will change if we change x . Let&s say that y is test scores, and x is hours of studying. Assume we are initially at some amount of x , e.g., you have been in the habit of studying 20 hours per week. You want to know how much higher your test scores would be at some other amount of x , x + h . One thing you could do, if you know the formula, is take this alternate x + h , and compute f ( x ) as well as f ( x + h ) . You could then look at the di/erence: f ( x + h ) & f ( x ) This would be the change in y . For some purposes, that might be exactly what you care about. In other instances, however, you might care about not just how much of a change there would be, but how much per amount of change in x , i.e., per h : That is also easy to calculate: 2 f ( x + h ) & f ( x ) h Now look at this graphically: 3 Graph: Calculating the Rate of Change in f Over a Discrete Interval y = 4 x 1 = 2-1 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 x y x x+h f(x) f(x+h) h f(x+h) - f(x) As you can see, we are just calculating the ratio of two legs of a triangle; that ratio is the slope of the line that connects the two points, as seen above. The problem is that this calculation would have a di/erent answer if we calculated it at a di/erent point: 4 Graph: Calculating the Rate of Change at a Di/erent Point y = 4 x 1 = 2-1 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 x y x x+h' f(x) f(x+h) h' f(x+h') - f(x) So, what if we calculated all this, but for a smaller h ? Then, we would get a di/erent rate of changeslope: 5 Graph: Rate of Change as h Shrinks y = 4 x 1 = 2 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 x y h So, let&s think about the limiting case of this. Say we examine lim h ! f ( x + h ) & f ( x ) h At one level, this thing might seem a bit confusing or ill-dened. The numerator obviously goes to zero as h gets small. The denominator also goes to zero. So, why should we expect the limit to converge to anything?...
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This note was uploaded on 09/16/2009 for the course ECONOMICS 1010A taught by Professor Jeffreya.miron during the Fall '09 term at Harvard.

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Lecture_00_Math_Review - Review of Calculus Tools c...

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