This preview shows pages 1–7. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 = 21 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 = 21 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 illdened. 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?...
View
Full
Document
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.
 Fall '09
 JeffreyA.Miron

Click to edit the document details