derivative.tex March 8, 2012
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Economic Applications of Di¤erential Calculus The concept of a derivative slopes, continuity, limits and derivatives
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Rules of di¤erentiation functions of single variables economic applications marginal revenue, marginal cost, etc. partial derivatives
economic applications of partial derivatives marginal products, elasticities, Shepard’s Lemma and the conditional input demand function, macro models and market models
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Total di¤erentials de…ned economic applications isoquants, isocost lines, indi¤erence curves and budget lines
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The concept of a derivative slopes, continuity, limits and derivatives Consider the single variable function f (x)
A derivative of f (x) at x = x0 can be interpreted as the the slope of the straight line that is tangent to f (x) at x = x0 Example: Draw a graph with a tangent to y = 3x:5 at x = 4. The tangent function is 3 + :75x, so the derivative of this function at x = 4 is :75.
y 10 8 6 4 2 0 0
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4
5
When x = 4, y = 6
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9
10
x
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With this in mind, let’s start our search for a derivative by considering the slope of a function between two values of x
De…nition: The slope (or gradient or rate of change) of a function f between two points x0 and x1 (where x0 , x1 2 X) is de…ned as slope =
f (x1 ) x1
f (x0 ) x0
Note that since x was assumed a scalar, x0 and x1 denote speci…c values of that scalar. Now consider the de…nition of the slope in a slightly di¤erent notation. De…ne x1 = x0 + t. In which case slope =
f (x0 + t) t
f (x0 )
t is the distance from x0 to x1 Is the slope de…ned when t = 0? No So, we cannot use the concept of a slope, by itself, to determine the slope of a function at a particular point because the slope is not de…ned when t = 0. Example of a slope: f (x) = 1 + 2x + 3x2 Find the slope of f (x) between x1 = 0 and x2 = 1 f (1) f (0) 6 1 = =5 1 0 1 Back to our discussion of what is a derivative and where does it come from. De…ne x1 = x0 + t. In which case slope(x0 ; t) =
f (x0 + t) t
3
f (x0 )
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To understand he concept of a derivative, in addition to slopes, one needs to understand the concept of continuity antd limits.
Start with an informal de…nition of a limit: lim f (x) = c
x!x0
means that as x tends to (approaches) x0 , the function f (x) tends to (approaches) c Formal de…nition of a limit: lim f (x) = c
x!x0
where c is a …nite constant, means the following given any positive number " > 0, there exists a positive number > 0 (whose value may depend on ") such that if jx0 xj < , then jf (x)
cj < "
Some examples: lim f (x) = 4
x!3
j3
means that if, for example, " = :01, there exists some xj < then jf (x) 4j < :01.
> 0 such that if
Note that if the limit exists the same would be true for " = :0000000000000001.
What is lim f (x) = :1x1=2 . It is :2 x!4
Consider the graph of f (x) = :1x1=2 in the neighborhood of x = 4. As x approaches 4 from above or below, f (x) appoaches 0:2.
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y 0.20 0.15 0.10 0.05 0.00 0
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2
5
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5
x
Now consider the function y=
3x 3:5
:5x
if if
x1
y -5
-4
-3
-2
-1
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3
4
5
x
-5
-10
-15
What is lim
3:5 not de…ned at x = 1. x!1
3x :5x
if if
x1
That is, a function can have a limit at x0 even if the the function is not de…ned at that point. This property of limits is critical in the de…nition of a derivative
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What is the limit of f (x) at x = 1 if f (x) = 1 if 0
x
