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Chapter 6

Introduction to Calculus

Archimedes of Syracuse (c.287 BC c.212 BC ) Bhaskara II (1114 1185)

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6.0 Calculus

2

Calculus is the mathematics of change.Two major branches: Differential calculusand Integral calculus, which arerelated by the Fundamental Theorem of Calculus.

Differential calculus determines varying rates of change. It is appliedto problems involving acceleration of moving objects (from a flywheelto the space shuttle), rates of growth and decay, optimal values, etc.

Integration is the "inverse" (or opposite) of differentiation. Itmeasures accumulations over periods of change. Integration can findvolumes and lengths of curves, measure forces and work, etc. Olderbranch: Archimedes (c. 287212 BC) worked on it.

Applications in science, economics, finance, engineering, etc.

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6.0 Calculus: Early History

3

The foundations of calculus are generally attributed toNewtonandLeibniz, though Bhaskara II is believed to have also laid the basis of it.The Western roots go back to Wallis, Fermat, Descartes and Barrow.

Q: How close can two numbers be without being the same number?Or, equivalent question, by considering the difference of two numbers:How small can a number be without being zero?

Fermats and Newtons answer: The infinitessimal, a positive quantity,smaller than any non-zero real number.

With this concept differential calculus developed, by studying ratiosin which both numerator and denominator go to zero simultaneously.

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6.1 Comparative Statics

Comparative statics: It is the study of different equilibrium statesassociated with different sets of values of parameters andexogenous variables.

Static equilibrium analysis: we start with y* =f(x)

Comparative static equilibrium analysis: y1* - y0* =f(x1) -f(x0)

(subscripts 0 and 1: initial & subsequent points in time)

Issues:

Quantitative & qualitative of change or

Magnitude & direction

The rate of change --i.e., the derivative (Y/ G)

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We uses differential calculus to study what happens to anequilibrium in an economic model when something changes.Example: Macroeconomic Model

Given

Y = C + I0+ G

0

C = a+ b (Y-T)

T =d + tY

Solving for Y => Y* = (a bd + I0 + G0)/(1-(b(1-t)))

Question: What happens to Y* when an exogenous variable, say G0,changes in the model?

??;01

*0

*1*

0*1 =

=

GG

YYYY

6.1 Comparative Statics : Application

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6.2 Rate of Change & the Derivative

Difference quotient:Let y =f(x)- Evaluate y=f(x) at two points: x0and x1: y0=f(x0) and y1=f(x1)- Define: x= x1- x0 => x1= x0+ x

y= y1- y0 => y=f(x0 + x) -f(x0)- Then, we define the difference quotientas:

Q: What happens tof(.)when x changes by a very small amount?

That is, we want to describe the small-scale behavior of a function

( ) ( )

x

xfxxf

x

y o

+=

0

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x

yxf

=)('

Difference quotient:

To describe the small scale behavior of a function, we use thederivative.

Derivative(based on Newtons and Leibnizs approach):Lets take xas an infinitesimal (a positive number, but smaller thanany positive real number) in the difference quotient:

( ) ( )x

xfxxf

x

y o

+=

0

Another interpretation: Infinitesimals are locations which are notzero, but which have zero distance from zero.


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The derivative captures the small-scale behavior of a function i.e.,

what happens tof(.)when x changes by a very small amount.

But, the infinitesimal approach was not elegant. Definition andmanipulation of infinitesimals was not very precise.

DAlembert started to think about vanishing quantities. He sawthe tangent to a curve as a limit of secant lines. This was arevolutionary, though graphical, approach:

As the end point of the secant converges on the point of tangency, itbecomes identical to the tangent in the limit.


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As the lim of x0, then f(x) measures the tangent (rise/run) off(x) at the initial point A. Again, the secant becomes the tangent.

This is the traditional motivation in calculus textbooks.

x

y

y

x0 x1 x2

A

C B

Graphical interpretation of the derivative:Slope = tangent of the function at x=x0:

Jean d'Alembert (17171783)


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( ) ( )dx

dy

x

xfxxf

x

yxf

xx=

=

=

00

00limlim)('

The graphical interpretation is subject to objections like those seenin Zenos paradoxes (say, Achilles and the Tortoise).

Based on the work of Cauchy and Weierstrass, we use limits toreplace infinitesimals. Limits have a precise definition and nice

properties. Then, we have an easier definition to work with:

Derivative(based on Weierstrasss approach):Lets take limits (x0) in the difference quotient we get the

derivative:


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VelocitySupposef(t) is the position of an object at time t, where a t b.

The average velocity, or average rate of change of fwith respect to t,of the object from time ato time bis:

Average velocity = change in position/change in time= [f(b) -f(a)] /(b-a)

If b= a +t => average velocity = [f(a + t) -f(a)]/t

Instantaneous velocityThe instantaneous velocity, or the instantaneous rate of change of fwith respect to t, at t = ais

lim t0[f(a +t) -f(a)]/t (provided the limitexists)

6.2 Rate of Change & the Derivative: Physics

Interpretation

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Product of LaborSupposeQ=f(L) is a production function, where L is the only input,labor. Let L1= L0+L,then, we define:

Average product of labor = Change in production/change in labor

= [f(L1) -f(L0)] /(L1- L0)

Let L0= 0 andf(L0)=0. Define L1=L= 1 hour=> labor productivity =f(L1) /hour

Marginal product of laborThe marginal product of labor at L= L0is

lim L0[f(L0+L) -f(L0)]/L (if the limit exists)

6.2 Rate of Change & the Derivative: Economic

Interpretation

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bx

yxf

bx

bxaxxba

x

y

x

yb

bbxay

x=

=

=

+++=

===

=+=

0lim)(')4

)()()3

in xchange

yinchange

run

rise)2

slope)1

6.2 Rate of Change & the Derivative: Linear

function

What are the secant slope and the derivative for a linear function?

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( )

( ) ( )

( ) ( )

00

0

2

0

2

0

2

0

2

0

2

0

2

00

2

00

2

6lim)('

36434363

4343

43

43

43:Example

xx

yxf

xx

x

xxxxx

x

xxx

x

y

xxxxf

xxf

xy

x

=

=

+=

+++=

+=

+=+

=

=

6.2 Rate of Change & the Derivative: Power

function

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Example (continuation):y= 3x2 4 (red)

Evaluate difference quotient at x=3, when x= 4y/x= 6x+ 3x => y/x= 30

y= 30x 67, secant through pts (3,f(3), 7, f(7)) (blue)


function

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Now, evaluate derivative at x=3y= 3x2 4 (red)f (x=3)=18y = 18x 31, tangent at point (3,f(3)) (blue)


function

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( ) ( )dx

dy

x

xfxxf

x

yxf

xx=

+=

=

00

00limlim)('

Standard Calculus:Based on Weierstrasss approach, using limits instead ofinfinitesimals. The infinitely small behavior of the function is foundby taking the limiting behavior for smaller and smaller numbers.Limits are the easiest way to provide rigorous foundations for

calculus, and for this reason they are the standard approach.

Nikolai Luzin (1883 1950, Russia) "They won't fool me: it's simply theratio of infinitesimals, nothing else."

6.2 Standard & Non Standard Calculus

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( ) ( )

+=

x

xfxxfstxf 00)('

Non-Standard Calculus:Due to the work of Abraham Robinson (1918-1974,Germany/Israel), infinitesimals received a precise definition.Robinson used the hyperreals, *R, (an extension of the real numbers).that contains numbers greater than anything of the form

1 + 1 +... + 1.Such a number is infinite, and its inverse is infinitesimal. Then,

for an infinitesimal x. st(.) denotes the standard part function, whichassociates to every finite hyperreal the unique real infinitely close to it roughly speaking, the part that ignores the error term (i.e., the term

with x).

6.2 Standard & Non Standard Calculus

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In the one dimensional case, we define the derivative as:

6.2 Differentials

( ) ( )x

xfxxfxf

x

+=

00

0lim)('

Equivalently, we can write

f(x0+x)=f(x0) +xf(x)

+ rx(x),where rx(x) represent the remainder, which is of smaller orderthanx. That is,

( ).0lim

0=

x

xrxx

The quantityf(x0+x) -f(x0) is composed of two terms:- xf(x), the part proportional to the change in x(x)- rx(x), an error, which gets smaller with x.

The expression df(x) = xf(x) is called the (first) differential off.

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Figure 6.9 Differential Approximation and ActualChange of a Function

The differential df(x) = xf(x) is the linear part of the incrementf(x0+x) -f(x0). This is expressed by geometrically replacing thecurve at point x0by its tangent.

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6.2 Differentials and Approximations

The differential is used to linearly approximate changes in f(x). Theerrori.e., the quality of the approximation- depends on thecurvature of isf(x)and, of course, the magnitude of x.

For very smallx, the approximation should be good, regardless off(x). But, when is xvery small?

An interesting case is when x=1. In this case, df(x) =f(x). Then,the first derivative approximates the change in the function peradditional unit of x.

In the production function example,f(L)measures the additionaloutput that can be produced with an additional unit of labor.

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Figure 6.11 Differential Approximation forBeta with Different Functions

Note: Since x=1 in all cases, dy=f(x). As the function has morecurvature, the linear approximation becomes less precise.

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Recall the solution to Y (income) in the Macroeconomic model:

We have a linear function in I (investment) and G (governmentspending) . Assume I is fixed, then we have y=f(G).

Comparative Static Question:

What happens to Y* when G increases by G? We approximate theanswer by:

0

)1(1

1)('where

);('*

>

=

tb

Gf

GfGY

)()1(1

1),( 00 GIbda

tbGIfY ++

==

6.2 Differentials and Approximations: Example

=> if G=$1, then dY=f(G).

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6.2 Multivariate Calculus: Partial Differentiation

1...ni,limgeneral,In

)xw.r.t.derivativepartial(

),,,(),,,(limlim

),,,(

0

11

1

1

21211

01

0

21

11

=

+=

=

i

iix

nn

xx

n

fx

y

x

y

fx

y

x

xxxfxxxxf

x

y

xxxfy

i

It is straightforward to extend the concepts of derivative anddifferential to more than one variable. In this case,ydepends onseveral variables: x1, x2, , xn.

The derivative of yw.r.t. one of the variables while the othervariables are held constant- is calledpartial derivative.

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6.2 Partial derivatives: Example

( )

( )

==

=

=

=

1

1

LAKQ:functionProduction

LAKLQMPL

LAKK

QMPK

Interpretation: As usual:MPL: Marginal product of labor

MPK: Marginal product of capital

We use them to linearly approximate the change in production Q inthe face of unitary changes, one at a time, of inputs. When L and Kchange simultaneously, we need to use the total derivative:

Q MPL L+ MPK K

Example:: Cobb-Douglas production function

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6.3 Concept of Limit: Preliminaries

Definition:N-ballLet c be a point in Rnand rbe a positive number. The set of all points x Rn

whose distance is less than ris called an n-ballof radius rand center c. It is

usually denoted by B(c) or B(c,r). Thus:

B(c,r)={x: x Rn,x-c

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Definition: Boundary Point(d) if every n-ball B(x) contains at least one point of S and at least one point of

the complement of S, then xis called a boundary pointof S. The set of all

boundary points of S is called the boundary of S.

Note: Every non-isolated boundary point of a set SRis an accumulationpoint of S. An accumulation point is never an isolated point

Example: Lets look at the interval (0, 4).

- The boundary of (0, 4) is the set consisting of the two elements {0, 4}. - Theinterior of the set (0, 4) is the set (0, 4) -i.e., itself.

- No points of either set are isolated, and each point of the set is anaccumulation point. The same is true, incidentally for each of the sets (0, 4), [0,

4), (0, 4], and [0, 4].

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6.3 Concept of Limit: Topology

Definition: Open and Closed setsA set S in Rnis said to be:

(a) openif all its points are interior points

(b) closedif it contains all its accumulation points

(c) boundedif there is a real number r>0 and a point cin Rn

such that S liesentirely within the n-ball B(c,r)

(d) compactif it is closed and bounded

Examples: Let A be an interval in R. For a

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6.3 Concept of Limit: Topology

Properties:- Every union of open sets is again open.

- Every intersection of closed sets is again closed.

- Every finite intersection of open sets is again open

- Every finite union of closed sets is again closed.Note: Open sets in Rare generally easy. Closed sets can get complicated.

Cantor Middle Third Set

Start with the unit interval S0= [0, 1].Remove from S0the middle third. Set S1= S0\(1/3, 2/3)

Remove from S1the 2 middle thirds. Set S2=S1\{ (1/9, 2/9) U (7/9, 8/9)}.

Continue, where Sn+1=Sn\{ middle thirds of subintervals of Sn}.

Then, the Cantor set Cis defined as C= Sn

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The Cantor set C is an indication of the complicated structure of closed setsin the real line. C has the following properties:- C is compact (i.e. closed and bounded)

- C is perfect i.e., it is close and every point of C is an accumulation point of

C.

- C is uncountable (since every non-empty perfect set is uncountable).

- C has length zero, but contains uncountably many points.

- C does not contain any open set.

This set is used to construct counter-intuitive objects in real analysis or to

show lack of generalization of some results. For example, Riemann integration

does not generalize to all intervals.

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6.3 Concept of Limit: Functions

Definition: Functions

Let S and T be two sets. If with each element xin S there is associated exactly

one elementyin T, denotedf(x), thenfis said to be afunctionfrom S to T. We

write

f: S T,

and say thatfis defined on S with values in T. The set S is called the domainoff; the set of all values offis called the rangeoff, and it is a subset of T. T is

called the targetor codomain.

The imageoffis defined asimag(f)= {tT : there is an sS withf(s) = t}.

If C is a subset of the range T, then thepreimage, or inverse image, of C under the

functionfis the set defined as

f

-1

(C) = {x

S :f(x)

C }

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Example: Domain and image off :X Y

fis a function from domainXto codomain Y. The smaller oval inside Yis the

image off.

Peter G. Lejeune Dirichlet (1805 1859, Germany)

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A functionf: S Rdefined on a set S with values in Ris called real-valued.A functionf: S Rm(m>1) whose values are points in Ris called a vectorfunction.

A vector function is bounded if there is a real number M such that

f (x) M for all xin S.

A functionffrom StoTcan be classified into three groups:

- One-to-oneif wheneverf(s) = f(w),then s=w. Also called injections.

- Ontoif for all tT there is an sS such thatf(s) = t. Also called surjections.

- Bijectionif it is one-to-one and onto -i.e., bijections are functions that are

injective and surjective.

Examples: A linear function is a bijection. A periodic function is not one-to-

one. Say,g(x) = cos(x)is neither one-to-one nor onto in R.

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6.3 Concept of Limit: Inverse Functions

When f: S T is one-to-one on a set Cin S, there is a function fromf(C) back to C,which assigns to each tf(C) the unique point in Cwhich mapped to it. This map is called the inverse of fon Cand it writtenas:

f-1:f(C) C.

Examples:

- Letf: R T, sayf = 3x+2 => f-1: (y-2)/3

- The logarithm is the inverse of the exponential function.- The demand function q=D(p), under the usual assumptions, has as theinverse functionp=D-1(q),which is called the inverse demand function.

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6.3 Concept of Limit: Composition Functions

Let f: S T andg: V W be two functions. Suppose that T is asubset of V.Then, the composition of f with g is defined as the function:(g f)(x)= g(f(x)) for all x in S.

That is,function compositionis the application of one function to theresults of another. The functionsfandg can be composedby computingthe output ofgwhen it has an argument off(x) instead of x. Intuitively,if zis a functiong(y)andyis a functionf(x), then zis a function h(x).

Example: Definef(x) = x5andg(x)= exp(x). Then, (g f)(x) = exp(x5)

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6.3 Concept of Limit: Sequences

Definition: Sequence

A sequenceof real numbers is a functionf: N R.That is, a sequence can be written asf(1), f(2), f(3), ..... Usually, we will denote

such a sequence by the symbol {aj} where aj= f(j).

Example: The sequence 1, 1/2, 1/3, 1/4, 1/5, ...is written as {1/j}.

Definition: Convergence

A sequence {aj} of real (or complex) numbers is said to convergeto a real (or

complex) number cif for every >0, there is an integerN>0 such that ifj> N,then

| aj- c | < .The number cis called the limit of the sequence {aj} and we write ajc.

If a sequence {aj} does not converge, then we say that it diverges.

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Example: The sequence {1/j} converges to zero.

We need to show that no matter which >0 we choose, the sequence willeventually become smaller than this number. Take any >0. Then, there existsa positive integerNsuch that 1/N Nwe have:

| 1/j - 0 | = | 1/j | < 1/N < , wheneverj > N.This is precisely the definition of the sequence {1/j}converging to zero.

Note: Easy proof. A proper choice ofNis the key.

If {aj} is a convergent sequence, {aj} is bounded, and the limit is unique.

Example: The sequence of Fibonacci numbers is unbounded. Then, the

sequence cannot converge, since a convergent sequence must be bounded.

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Algebra of Convergent Sequences:Let {aj} be a convergent sequence. Then, the sequence is bounded, and the

limit is unique.

Suppose {aj} and {bj} are converging to aand b, respectively. Then

- Their sum converges to a + b, and the sequences can be added term by term.

- Their product converges to a * b, and the sequences can be multiplied termby term.

- Their quotient converges to a/b, provide that b0, and the sequences can bedivided term by term (if the denominators are not zero).

- If an bnfor all n, then a b. (It does not work for strict inequalities).

We know how to work with convergent sequences, we would like to have aneasy criteria to determine whether a sequence converges.

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Definition: Monotonicity

A sequence {aj} is calledmonotone increasing if aj + 1 ajfor allj.

A sequence {aj} is calledmonotone decreasing if aj aj + 1for allj.

Proposition:Monotone Sequences

- If {aj} is a monotone increasing sequence that is bounded above, then the

sequence must converge.

- If {aj} is a monotone decreasing sequence that is bounded bellow, then the

sequence must converge.

Example:

- {j/(j+1)} is monotone increasing & bounded above by 1. It must converge.

- {1/j} is monotone decreasing & bounded bellow by 0. It must converge.

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Often, it is hard to determine the actual limit of a sequence. We want to

have a definition which only includes the known elements of a particularsequence and does not rely on the unknown limit.

Definition: Cauchy Sequence

Let {aj} be a sequence of real (or complex) numbers. We say that {aj} isCauchy if for each >0 there is an integerN>0 such that ifj, k > Nthen

|aj-ak|< . Now, we know what it means for the elements of a sequence to get closer

together, and to stay close together.

Theorem: Completeness Theorem in R.

Let {aj} be a Cauchy sequence in R. Then, {aj} is bounded.

Let {aj} be a sequence in R. {aj} is Cauchy iff it converges to some limit a.

6.3 Concept of Limit: Cauchy Sequence

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By considering Cauchy sequences instead of convergent sequences we do

not need to refer to the unknown limit of a sequence (in effect, both conceptsare the same).

Q: Not all sequences converge. How do we deal with these situation?

A: We change the sequence into a convergent one (extract subsequences) andwe modify our concept of limit (lim supand lim inf).

Definition: Subsequence.

Let {aj} be a sequence. When we extract from this sequence only certainelements and drop the remaining ones we obtain a new sequences consisting

of an infinite subset of the original sequence. That sequence is called a

subsequenceand denoted by {aj,k} (k=1, 2, ...,).

Note: We can think of a subsequence as a composition function.

6.3 Concept of Limit: Subsequences

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Example: Take the sequence {(-1)j} , which does not converge. The sequence

is: {-1, 1, -1, 1,...}

Extract every even number in the sequence, we get: {-1, -1, -1, -1,...}

=> subsequence converges to -1.

Extract every odd number in the sequence, we get: {1, 1, 1, 1,...}

=> subsequence converges to 1.

Note: We can extract infinitely many subsequences from any given sequence

Proposition: Subsequences from Convergent Sequence

Let {aj} be a convergent sequence, then every subsequence of {aj} converges

to the same limit.

Let {aj} be a sequence such that every possible subsequence extracted from

{aj} converge to the same limit, then {aj} also converges to that limit.


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Theorem: Bolzano-Weierstrass

Let {aj} be a sequence of real numbers that is bounded. Then, there exists a

subsequence {aj,k} that converges.

This is one on the most important results of basic real analysis, and

generalizes the above proposition. It explains why subsequences can beuseful, even if the original sequence does not converge.

Example: The sequence {sin(j)} does not converge, but since it is bounded,

we can extract a convergent subsequence.

Note: The Bolzano-Weierstrass theorem does guaranty the existence of thatsubsequence, but it does not say how to obtain it. It can be difficult. We will

extend the concept of limits to deal with divergent sequences.


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Definition: Lim Sup and Lim Inf

Let {aj} be a sequence of real numbers. Define

Aj= inf{aj, aj + 1, aj + 2, ...}

and let c = lim (Aj). Then cis called the limit inferiorof the sequence {aj}.

Let {aj} be a sequence of real numbers. Define

Bj= sup{aj, aj + 1, aj + 2, ...}

and let d = lim (Bj). Then dis called the limit superiorof the sequence .

Summary::

- lim inf(aj) = lim(Aj), whereAj= inf{aj, aj + 1, aj + 2, ...}

- lim sup(aj) = lim(Bj), where Bj= sup{aj, aj + 1, aj + 2, ...}

These limits are often counter-intuitive, they have one very useful property:lim supand lim infalways exist (possibly infinite) for any sequence of real

numbers. Moreover, there is a subsequence converging to cand another to d.

6.3 Concept of Limit: Lim Sup and Lim Inf

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Example 1: Consider {(-1)j}. We find the numbersAj= inf{aj, aj + 1, aj + 2, ...}

A1= inf{-1, 1, -1, 1, ...} = -1

A2= inf{1, -1, 1, -1, ... } = -1

etc. It is clear that lim inf{(-1)j}= -1. (also the infimum)

Similarly, lim sup{(-1)j}= 1. (also the supremum)

Example 2: Consider {1/j}. The sequence is {1, 1/2, 1/3, 1/4, ...}. Then, the

infimum is zero, while the supremum is 1. Lets get theAjand BjA1= inf{1, 1/2, 1/3, 1/4, ...}= 0 & B1=sup{1, 1/2, 1/3, 1/4, ...}= 1

A2= inf{1/2, 1/3, 1/4, 1/5, ...}= 0 & B2=sup{1/2, 1/3, 1/4, 1/5,.}=1/2

A3= inf(1/3, 1/4, 1/5, 1/6, ...} = 0 & B3=sup( 1/3, 1/4, 1/5, 1/6,.}= 1/3

etc. It is clear that lim inf{1/j} = 0. (also the infimum)

etc. It is clear that lim sup{1/j} = 0. (different from the supremum)

6.3 Concept of Limit: Lim Sup and Lim Inf

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6.3 Concept of Limit: Definition

Definition: LimitLetf: S Rm. Let cbe an accumulation point of S. Suppose thereexists a point bin Rmwith the property that for every >0 there is a>0 such that

f (x) - b < for all xin S, xc, for which x-c < .Then, we say the limitoff (x) is b, as xtends to c, and we write

Note: This is the (,)-definitionof limit, introduced by Bolzano/Cauchyand perfected by Weierstrass.

bxfcx

=

)('lim

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6.3 Concept of Limit: Right- & Left-hand Limit

The limit (f(x), x

c, direction) function attempts to compute thelimiting value off(x) as xapproaches cfrom left, c-(the left-hand limit) orright, c-+(the right-hand limit).

When the left-hand and the right-hand limits are equal, say to L, wesay the limit exists and equals L.

If q =f(v), what value does q approach as v N? Answer: LAs vN from either side, q L.

Then, both the left-side limit and the

right side-limit are equal.

Therefore, lim q= L.

q

L

vN

v

N N v

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6.4 Evaluation of a Limit

To take a limit, substitute successively smaller values that tend toNfrom both the left and right sides sinceNmay not be in thedomain of the function.

If vis in both the numerator and denominator remove it fromeither depending on the function

Taking limits sometimes is not straightforward.

Example: Given q= (2v+ 5)/(v+ 1), find the limit of qas v+.Dividing the numerator by denominator:

2lim

1

32

1

52

=

++=

+

+=

+q

vv

vq

v

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6.4 Limit Theorems

If q = av+b, => = aN+b

If q= g(v) = b, => = b

If q= v, => =N

If q= vk, => =Nk

)(lim)(lim)(lim 2121 qqqqNvNvNv

=

qNv

lim

qNv

lim

qNv

lim

qNv

lim

)(lim)(lim)(lim 2121 qqqqNvNvNv

=

)(lim/)(lim)/(lim 2121 qqqq NvNvNv =

Example: Find lim (1+v)/(2 + v) as v0

( )

( ) 2

1

2lim

1lim

0

0

2

1 =

+

+=

v

v

L

L

v

v

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6.4 LHospitals Rules

Iffandgare differentiable in a neighborhood of x=c, andf(c)=g(c)=0,then

provided the limits exist.

Note: The same result holds for one-sided limits.

Iffandgare differentiable and limxf(x) = limxg(x) = , then

provided the last limit exists.

In other situations L'Hospital's rules may also apply, but often aproblem can be rewritten so that one of these two cases will apply.

)('

)('lim

)(

)(lim

xg

xf

xg

xf

cxcx =

)('

)('lim

)(

)(lim

xg

xf

xg

xf

xx =

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6.4 Limit Jokes

qNv

lim

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Definition: Continuous function

When a function q=f(v) possesses a limit as vtends to the pointNin the domain and

When this limit is also equal tof(N) --i.e., the value of thefunction at v=N--, then the function is continuous inN.

Note: Continuity preserves limits

Requirements for continuity

N must be in the domain of the function

f, q = f(v)f has a limit as vN

limit equalsf(N) in value

( )

)()(lim

exits)(lim

defined

Nfvf

vf

Nf

Nv

Nv

=

6.5 Continuityand Differentiability of a Function

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Alternative definition: Continuous functionLetf: SR be a real valued function on a set S in Rn. Let cbe apoint in S. We say thatfis continuousat cif for every >0, thereexists a >0 such that

f(c+u)f(c)

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If cis an accumulation (limit) point of S, the definition of continuityimplies that

limu0f(c +u) = f(c).

Intuition is tricky: Geometry seems to show that iffis continuous

at c, it must be continuous near c. This is wrong!

Example: The Dirichlet function

Letf: RR defined by

f(x) = x if xis rational= 0 if xis irrational

is continuous at x=0, but at no other point.


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Almost all the basic functions in mathematical econ models areassumed to be continuous.

For example, a production function is continuous if a small changein inputs yields a small change in output. (A reasonable assumption.)

If a function fails to be continuous at a point c, then the function iscalled discontinuousat c, (c is called apoint of discontinuity).

Examples: Non-continuous functions:

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We sayf: RnRis differentiableat x* iff(x*) exists. Not every continuous function has a derivative at every point. For

example: f(x)=|x|.

|x|is not differentiable at x=0, the left and the right-handed

limits are different (-1 and +1). Then, there is no unambiguoustangent line defined at x=0.

We need the functionf(.) to be smoothi.e., no kinks.

Iff is differentiable at x*, thenfis continuous at x*. (Converse

is, of course, not true.)

As with continuous functions, differentiable functions can beadded, multiplied, divided, and composed with each other to yield

again differentiable functions.


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A functionf: Rn

Ris continuously differentiableon an open set Uof Rn

if and only if for each x,df/dxiexists for all xin Uand is continuousin x.

This rational function is not defined at v= 2, even though the limit

exists as v2. It is discontinuous and thus does not havecontinuous derivatives --i.e., it is not continuous differentiable.

Thisimagecannotcurrentlybedisplayed.

4

442

23

+=

v

vvvq

This continuous function is not differentiable at x=3 and, thus, doesnot have continuous derivatives (it is not continuously differentiable):

>

=

)3(,1

)3(,5

xwherex

xwherexy


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not continuous:

f(t) = -1 if t= 0f(t) = t2otherwise

continuousbut notdifferentiable:

f(t) = |t|

continuous& differentiable:

f(t) = t2

t

f(t)


x

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For a function to be continuous differentiable

All points in the domain off defined

The limit is taken on the difference quotient at x=x0 as x0 fromboth directions. The continuity condition is necessary, not sufficient.

The differentiability condition (smoothness) is both necessary andsufficient for whetherfis differentiable.

Theorem: Rolles Theorem

Iffis continuous on [a, b] and differentiable on (a, b), andf(a)=f(b)=0,then there exists a number xin (a, b)such thatf'(x) = 0.

Note: An extension of Rolle's theorem that removes the conditions onf(a)andf(b)is the Mean-Value Theorem. These theorems form the

basis for the familiar test for local extrema of a function.


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Here we want to list some functions that illustrate more or lesssubtle points for continuous and differentiable functions.

Dirichlet function: A function that is not continuous at any point in R

Countable discontinuities: A function that is continuous at the irrational

numbers and discontinuous at the rational numbers.

C1function: A function that is differentiable, but the derivative is not

continuous.

6.6 Continuityand Differentiability: Examples

=

=irrationalisif,0

rationalis/if,/1

x

qpxqy

=

=

0if,0

0if),/1sin(2

x

xxxy

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Cnfunction: A function that is n-times differentiable, but not (n+1)-times differentiable

Cinffunction: A function that is not zero, infinitely oftendifferentiable, but the n-th derivative at zero is always zero.

Weierstrass function: A function that is continuous everywhere and

nowhere differentiable in R.

Cantor function: A continuous, non-constant, differentiable function

whose derivative is zero everywhere except on a set of length zero

6.6 Continuityand Differentiability: Examples

3 13 += nxy

=

=

0if,0

0if),/1exp(2

x

xxy

|)4sin(|)4/3(0

xy n

n

n

=

=

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6.7 The Limit and the Quotient Ratio

Let qy/x and vx such that q =f(v) and

qx

y

vx 00

limlim

=

Q: What value does variable qapproach as vapproaches 0?

A: If the function is differentiable, we move from the quotient ratio tothe derivative.

Note: This definition may not work well when x is a vector, say x (y,z).Measuring y is not a problem (the difference between twofunctions), but measuring x (y, z). is not clear.

We need the concept of distance for a vector, a norm.

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6.8 Resolution of a Controversy: Butter Biscuits

or Fruit Chewy Cookies?

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