Lecture series one

1
Lecture series one

• Revision of Calculus

2
The real number system used in economics and
financial analysis
Integers J
Fractions F
Rational Numbers Q
Irrational Numbers
Real Numbers R
3
Basic set theory concepts

• A set is a collection of objects
• Sets can be written up by enumeration or
  description
• Enumeration
• Description

4
Basic set theory concepts cntnd.

• Membership/non-membership of a set can be denoted
  by ? (?) , e.g., if A1,2,5 ? 1?A, while 3 ? A.
• Equal Sets contain exactly the same elements
  (order does not matter) A1,2,5, B5,2,1
• Subsets are sets contained in another set if
  A1,2,5 and D1,2,5,6,8, A is a subset of D
  A?D

5
Basic set theory concepts cntnd.

• Union of sets a new set contains all elements
  that belong to either of the original sets, e.g.,
  if A1, 2,5 and D1,2,4,6, 8, then
  A?D1,2,4,5,6,8
• Intersection of sets A new set containing only
  the common elements of the original sets, e.g, if
  A2, 5, 9, D2,4,5,6,8, then A?D2,5
• Complement set the complement set of A is à ,
  containing all the elements of the universal set,
  which is not in A, e.g., if A1, 2,3 and U1,
  2, 3, 5, 6, 8, then à 5,6,8

6
Basic set theory concepts cntnd.

• Commutative law
• Associative law
• Distributive law
• Example A4,5, B3,6,7, C2,3

7
The Cartesian product

• Cartesian Product
• For two sets A and B, let A ? B be the set of
  pairs (a,b) where a?A and b ?B.
• That is A ? B(a,b) a?A and b ?B
• A ? B is called the Cartesian product of A and
  B

8
The Cartesian product cntnd.

• For example, if A1,2,3 and B?,?, then
• A?B is (1, ?), 2, ?),(3, ?),(1, ?),(2,
  ?),(3, ?)

9
The Cartesian product cntnd.

• The set ???, often denoted by ?2 is
  (x,y)x,y??. This is the Cartesian plane

(x,y)
10
Basic set theory concepts cntnd.

• René Descartes
• Born 31 March 1596 in La Haye, Touraine, France
• Died 11 Feb 1650 in Stockholm, Sweden
• Philosopher whose work
• La géometrie includes application of algebra
  to geometry from which we now have Cartesian
  geometry

11
Relations and functions

• Since any ordered pair associates a y value with
  an x value, any collection of ordered pairs-any
  subset of the Cartesian product-will constitute a
  relation between y and x.
• If the relation is such that for each x value
  there exists only one corresponding y value, y is
  said to be a function of x.

12
Relations and functions cntnd.

• Each function has
• S
  T
• A source S (which is a set), also called the
  domain of the function
• A target T (which is a set), also called the
  co-domain, or range of the function
  

f
. . .
. . .
S
T
13
Relations and functions cntnd.

• Example 1 A set (x,y)y2x is a set of ordered
  pairs including for e.g, (1,2), (0,0) is a
  relationship whose graphical counterpart is the
  set of points lying on the straight line y2x.
• Example2 A set(x,y)y?x which consists of
  ordered pairs (1,0),(1,1) and (1,-4) constitutes
  another relationship.

14
Relations and functions cntnd.
y2x
yx
y?x
15
Relations and functions cntnd.

• Example The total cost C of a firm per day is a
  function of its daily output Q C1507Q. The
  firm has a capacity limit of 100 units per day.
  What are the domain and the range of the cost
  function?

16
Relations and functions cntnd.

• Solution
• DomainQ0?Q ?100
• RangeC150 ?C ?850

17
Types of functions

• Constant function they have a constant value for
  every x, e.g., yf(x)1
• Example In national-income models, when
  investment is
• exogenously given, we may have an
  investment function of the
• form I100,00

y
x
18
Types of functions contnd.

• Polynomial functions y is the nth degree
  polynomial function of x ya0a1xa2x2a3x3.an
  xn. For the simplest case of a linear function
  with a01 and a12, we have y12x

1
-0.5
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Types of functions contnd.

• Example 1 of non-linear function quadratic
  function
• ya0a1xa2x2. For a010, a1-7, a21
  y10-7xx2

• If a0a1xa2x20,then x1,2
• If a1-4a0a2gt0, there are 2 real solutions (x1,x2)
• If a1-4a0a20, there is 1 real solution (x1 x2)
• If a1-4a0a2lt0, there is no real solution

20
Types of functions contnd.

• Example 2 of non-linear function cubic function
• For n3, ya0a1xa2x2a3x3, e.g, a05, a17,
  a2-1, a3-2 y57x-x2-2x3

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Types of functions contnd.

• Exponential functions yf(t)bt, bgt1

y
1
t
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Types of functions contnd.

• The curve of y covers all the positive values of
  y in its range, therefore any positive value of y
  must be expressible as some unique power of
  number b.
• Importantly, even if the base is changed to some
  other real number greater than 1, the same value
  of the function holds. Hence it is possible to
  express any positive number y as a power of any
  base bgt1.

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Types of functions contnd.
y
ybt
yb2t
t
24
Types of functions contnd.

• The preferred base e2.71828
• Motivation 1

25
Types of functions contnd.

• Motivation 2 financial application

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Types of functions contnd.

• Suppose that, starting with a principle of 1 we
  find a hypothetical banker to offer us the
  unusual interest rate of 100 (1 interest per
  year). If the interest is compounded once a year,
  the value of our asset at the end of the year is
  2
• V(1)initial principal(1interest
  rate)1(11/1)2

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Types of functions contnd.

• If the interest is compounded semi-annually
• V(2)(150)(150)(11/2)2
• Analogically V(3)(11/3)3 , V(4)(11/4)4
• In the limiting case, when interest is compounded
  continuously throughout the year, the value of
  our asset at the end of the year will be e.

28
Types of functions contnd.

• Of course, the assumptions of neither a one
  dollar deposit nor 100 interest rate are
  realistic.
• Our general interest compounding formula is
  therefore
• And we find the asset value in a generalized
  continuous compounding process to be

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Types of functions contnd.

• In the reverse case, when we want to find the
  present value of an asset, we have

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Examples

• Example one A principal of 10 is invested at
  12 interest for one year. Determine the future
  value if the interest is compounded (a) annually
  (b) semi-annually (c) quarterly (d) monthly (e)
  weekly
• Example two Determine the rate of interest
  required for a principal of 1000 to produce a
  future value of 4000 after 10 years compounded
  continuously
• Example three Find the present value of 1000 in
  four years time if the discount rate is 10
  compounded (a) semi-annually, (b) continuously.

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Types of functions contnd.

• Logarithms
• ybt ?tlogby
• Graphically

yet
tlogey
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Types of functions contnd.

• Common log and natural log

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Rules of exponents

• x01, for x?0 (e.g. 1001, 5001)
• x1x
• x2x?x, x3x?x?x and so on.
• x-n1/xn, for x ?0 (e.g., x-31/x3)
• xn? xmxnm (e.g, x2 ? x3x5)
• xn/xmxn-m (e.g., x10/x3x7)
• (xn)mxn ? m
• xn ? yn(xy)n
• xn/yn(x/y)n

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Rules of logarithms

• lne1
• ln10
• ln(xy)lnxlny
• ln(x/y)lnx-lny
• lnxnnlnx

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What comes next

• Functions with one unknown variable
• Rate of change and the derivative
• Second and higher order derivatives
• Functions with several unknown variables
• Partial differentiation
• Unconstrained optimization of functions with more
  than one unknown variable

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The nature of comparative statics cntnd

• Geometrically, the rate of change corresponds to
  the slope of the function
• It is constant in the case of a linear function
  and differs in the case of a non-linear function

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Rate of change and the derivative cntnd.

• Example 1 Linear function-constant slope
• If yf(x)5x2, find y for x0,1,2.
• For x0 ? yf(0)5(0)22
• For x1 ? yf(1)5(1)27
• For x2 ? yf(2)5(2)212

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Rate of change and the derivative cntnd.

• We notice that the amount by which y changes, as
  x changes by a given amount remains constant it
  is always equal to 5. In the above example
• ?x1-01 ??y7-25
• ?x2-11 ??y12-75
• In other words, ?x is always 1 and ?y is always
  5, or the slope of the function Slope ?y / ?x
  5/15 is constant.

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Rate of change and the derivative cntnd.

• Example 2 Non-linear function variable slope
• If Qf(P)15-P2, find the quantity demanded Q
  for prices P1,2,3.
• For P1? Qf(1)15-1214
• For P2? Qf(2)15-2211
• For P3? Qf(3)15-326

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Rate of change and the derivative cntnd.

• Example 2 Non-linear function variable slope
• If Qf(P)15-P2, find the quantity demanded Q
  for prices P1,2,3.
• For P1? Qf(1)15-1214
• For P2? Qf(2)15-2211
• For P3? Qf(3)15-326

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Rate of change and the derivative cntnd.

• We can vary the change of x and obtain different
  changes in y. The concept of derivative is
  associated with very small changes in x.
  Specifically, the derivative of yf(x) is the
  limit value of its slope, as ?x gets very closet
  to 0

The derivative of the function yf(x) is equal to
the rate of change of y (i.e. ?y/ ?x) when the
change in x is very small (close to 0). It is
denoted by or f(x), reading f Prime x.
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Rate of change and the derivative cntnd.

• Power function rule
• e.g. n1 yx1x, dy/dx1x1-1x01
• n2 yx2, dy/dx2x2-12x12x
• n-3yx-3, dy/dx-3x-3-1-3x-4
• Generalized power function rule
• e.g., a2, n4 y2x4, dy/dx2(4x4-1)8x3

If yf(x)xn, dy/dxf(x)nxn-1
If yf(x)xn, dy/dxanxn-1
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Rate of change and the derivative cntnd.

• Constant function rule
• e.g.yf(x)2, dy/dxf(x)0 yf(x)-1000,
  dy/dx0
• Sum-difference rule
• e.g. y7x24x3

If yf(x)c (constant), dy/dx0
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Rate of change and the derivative cntnd.

• Product rule
• y(2x25x)(3x-2)

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Rate of change and the derivative cntnd.

• Quotient rule
• y(4x23x)/(2x1)

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Rate of change and the derivative cntnd.

• Natural logarithm rule
• Exponential rule

If ylnx (xgt0), dy/dx1/x
If yax, dy/dxaxlna For ae2.718 yex?dy/dxex
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Rate of change and the derivative cntnd.

• Chain rule
• Example 7 Chain rule and power functions
• (i) y(3x2-5x2)10 yu 10, where u
  3x2-5x2

If yf(u), uh(x), i.e.
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Summing up

• Power function rule If yf(x)xn,
  dy/dxf(x)nxn-1
• Constant function rule If yf(x)c (constant),
  dy/dx0
• Sum-difference rule
• Product rule
• Quotient rule

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Summing up.

• Exponential rule If yax, dy/dxaxlna
• Logarithmic rule If ylnx (xgt0), dy/dx1/x
• Chain rule If yf(u), uh(x), i.e.

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Rate of change and the derivative cntnd.

• Find the derivatives of the following functions

51
Rate of change and the derivative cntnd.

• Recall that the first order derivative of a
  function yf(x) is equal to the rate of change of
  y (i.e. dy/dx) when the rate of change of x is
  very small

52
Rate of change and the derivative cntnd.

• Differentiating the first order derivative gives
  the second order derivative (differentiating the
  second order derivative gives the third order
  derivative and so on)

53
Rate of change and the derivative cntnd.

• Example Find the first through fifth derivatives
  of the function
• yf(x)4x4-x317x23x-1

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Rate of change and the derivative cntnd.

• Just as the first order derivative denotes the
  rate of change of a function, the second order
  derivative denotes the rate of change of the
  first derivative

As x increases f(x)gt0 the value of the
function increases f(x)lt0 the value of the
function decreases f(x)0 the value of the
function remains constant f(x)gt0 the slope of
the function increases f(x)lt0 the slope of the
function decreases f(x)0 the slope of the
function remains the same
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The basics of optimization

• Example 1 yf(x)x22x1?f(x)2x2 and
  f(x)2gt0 increasing slope as x increases

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The basics of optimization cntnd.

• Example 2 yf(x)-x2-2x-2, f(x)-2x-2,
  f(x)-2lt0

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The basics of optimization cntnd

• One of the main uses of calculus in economics,
  finance and econometrics involves the
  optimization (finding the minimum or the maximum)
  of a given function, called the objective
  function
• In example 1, the lowest point in the functions
  graph is called a minimum
• In example 2, the highest point in the functions
  graph is called a maximum

58
The basics of optimization cntnd

• Consider the general objective function yf(x).
  To find the mimimum/maximum, we need to examine
  the following two conditions
• - First order necessary condition f(x)0
  (calculate the first derivative, set it to zero
  and solve the resulting equation to find the
  values of x that satisfy it)
• - Second order sufficient condition
• f(x)lt0 max at xx0
• f(x)gt0 min at xx0
• f(x)0 point of inflection

59
Examples

• Example 1 Find the stationary points of the
  function yf(x)5x2-20x
• Example 2 Find the optimal points of the
  function y-5x2250x-1125

60
What follows

• So far we have focused upon the simplest case of
  functions yf(x), i.e. functions with a single
  independent variable x e.g., y2x23x10
• However, in economics, finance and econometrics
  most functional relationships involve more than
  two variables. Hence, we should learn how to
  apply the techniques of differential calculus to
  such multivariate functions. We begin by
  examining the topic of partial differentiation.

61
Partial differentiation

• Consider the following multivariate function with
  n independent variables zf(x1, x2, x3. xn)
• The assumption of independence implies that
  each xi can vary by itself without affecting the
  others. For example, a change in the value of x1,
  ? x1 while x2, x3. xn remain constant (i.e. ?
  x20, ? x30. ? xn0) will produce a
  corresponding change in z (?z).
• If we take the limit of the rate of change
  of z w.r.t x1 (i.e. ?z/ ? x1 )
• as the change in x becomes very small, the
  limit will constitute the partial derivative of z
  with respect to x1. This partial derivative is
  symbolized by fx1 or dz/dx1.

62
Partial differentiation cntnd.

• Techniques of partial differentiation
• To partially differentiate a multivariate
  function we allow only one variable to vary,
  while all others remain constant.
• e.g. in zf(x1, x2) we have to treat x2 as
  constant.

In general, if zf(x1, x2, x3. xn), then the
partial derivative of y w.r.t xi is given by

63
Partial differentiation cntnd.

• Find the first-order partial derivatives of
• Example 1 zf(x1, x2) x12-2x2
• Example 2 zf(x,y)x4exlny

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Partial differentiation cntnd.

• We already saw that function zf(x,y) can give
  rise to two first -order partial derivatives fx,
  and fy By differentiating fx, and fy w.r.t. x
  and y, we can obtain four second -order partial
  derivatives.

65
Partial differentiation cntnd.

• The partial derivatives fxy and fyx are called
  the cross partial derivatives, because they
  measure the rate of change of the first-order
  partial derivative with respect to the other
  variable.
• Youngs theorem implies that as long as the two
  cross-partial derivatives are both continuous,
  they will be identical fxy fyx

66
Partial differentiation cntnd.

• Find the first and second order partial
  derivatives of
• Example 1 zf(x,y)2x4y5xy33xy10
• Example 2 zx4exlny

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What follows

• In the univariate function case we developed
  optimality conditions based on the first and
  second derivative
• This is also possible in the multivariate case!

68
Mutivatiate optimization

• Analogically to the univariate case, the first
  order necessary condition for optimum is
• dzfxdxfydy0
• It amounts to fx0, fy0, for arbitrary
  values of dx and dy, not both zero. Solving the
  resulting system of simultaneous equations (fx0,
  fy0) gives the stationary point (x0, y0).

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Multivariate optimization cntnd.

• Once again, the sufficient condition for
  optimality is based on the second differential.
  In the multivariate case this is
• where

70
Multivariate optimization cntnd.

• The second-order sufficient condition for a
  minimum of zf(x,y) is that the second total
  differential of z is positive d2zgt0, which is
  equivalent to fxxgt0, fyygt0, and fxxfyygt(fxy)2
• The second-order sufficient condition for a
  maximum of zf(x,y) is that the second total
  differential of z is negative d2zlt0, which is
  equivalent to fxxlt0, fyylt0, and fxxfyygt(fxy)2

71
Multivariate optimization cntnd.
72
Geometric representation
C
y
If the slope of the line segment AC is smaller
than the slope of the tangent AB, the function is
concave
f(x2)
B
f(x1)
A
f
x
x1
x2
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Geometric representation cntnd.
z
f(?(x)(1- ?)y
?f(x)(1- ?)f(y)
f(y)
f(x)
y
(x1,x2)
(y1,y2)
x
When ?f(x)(1- ?)f(y)? f(?(x)(1- ?)y we have a
maximum
?(x)(1- ?)y
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Geometric representation cntnd.

• The opposite is true in the case of convex
  univariate and multivariate functions

y
C
x
B
A
y
x
z
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Example

• Find any local minima or maxima of the functions
  zf(x,y)x2xy2y23
• Find and classify the stationary points of the
  following function
• f(x,y)x3y3-3x-3y

76
Optimization with equality constraints

• So far we have dealt with finding free extrema of
  an objective function. An useful example, that we
  shall explore during our third series of lectures
  is the classical linear regression model.
• However, most choices in economics and finance
  involve optimization under constraints. A case in
  point is the portfolio optimization model.

77
Optimization with equality constraints cntnd.
Free maximum
Constrained maximum
constraint
78
Optimization with equality constraints cntnd.

• Step1 Define the Lagrangian function
• Lf(x,y)?c-g(x,y)
• Step 2 Employ the necessary conditions with
  respect to x, y and ? to find the stationary
  value of L
• The resulting system of equations is solved for
  the three unknowns x, y and ?.
• We should point out that these are the
  first-order necessary conditions for optimality,
  the second-order conditions are complicated and
  will not be discussed here.

79
Optimization with equality constraints cntnd.

• ? is known as the Lagrange multiplier. It
  measures the approximate change in the stationary
  value of the dependent variable z, due to a
  one-unit increase in the constraint c.
• Example 1 Use the Lagrange method to find any
  stationary points for zf(x,y)-2x2y2, subject
  to y-2x -1
• Step 1

80
Optimization with equality constraints cntnd.

• Step 2

81
Examples

• Example 1 L4KLL2?(105-K-2L)
• Example 2 z2x2-xy s.t. xy12

82
Integral calculus

• So far, we were interested in finding the optimum
  of an objective function
• In what follows, our focus will be on the
  opposite problem the delineation of the time
  path of some variable (or its primitive
  function), on the basis of a known pattern of
  change.

83
Integral calculus contnd.

• Example Let net investment I be defined as the
  rate of change of the capital stock, so that
  IdK/dt. Where I(t) denotes the flow of money,
  measured in pounds per year, and K(t) is the
  amount of capital accumulation at time t as a
  result of this investment flow. If we integrate
  this function we will find the capital stock.

84
Integral calculus contnd.

• It is easy to recognize that, if we know the
  function KK(t) to begin with, the derivative can
  be found by differentiation.
• In the problem confronting us now, the shoe is on
  the other foot we need to uncover the primitive
  function from the derivative!

85
Integral calculus contnd.

• We can differentiate primitive function F(x) to
  find the derivative function f(x) dF(x)/dxf(x).
• We can also integrate the derivative function
  f(x) to find the primitive function, F(x)
• The primitive function, F(x) is referred to as
  the integral (antiderivative) of the derivative
  function, f(x).
• Note that while the primitive function F(x)
  produces a unique derivative f(x), the derivative
  function is traceable to an infinite number of
  possible integrals, due to the presence of the
  arbitrary constant of integration, c.

86
Integral calculus contnd.

• ? is known as the integral sign. f(x) integrand
  (i.e. the function to be integrated), dx
  differential of x.
• The integral ?f(x)dx is known as the indefinite
  integral, because it has no definite numerical
  value. Since it is equal to F(x)c, it is implied
  that it varies along with x. Thus, like the
  derivative, an indefinite integral is itself a
  function of x.

87
Rules of integration

• The power rule

88
Rules of integration cntnd.

• Logarithmic rule
• Exponential rule

89
Rules of integration cntnd.

• Scalar multiplication rule
• The integral of a sum and difference

90
Rules of integration cntnd.

• Example 1
• Example 2

91
Rules of integration cntnd.

• Example 3

92
The definite integral

• Contrary to the indefinite integrals that have no
  definite numerical value (since they are
  functions of a variable), definite integrals
  possess definite numerical value.
• Definite integrals are evaluated between two
  points in the domain of the function f(x) upper
  limit (b) and lower limit (a).

93
The definite integral cntnd.

• The calculation of definite integrals proceeds in
  two steps
• Step 1 Find the primitive function F(x). Note
  that the arbitrary constant of integration may be
  skipped since it will drop out in the next step.
• Step 2 Substitute xa to find F(a) and xb to
  find F(b) and calculate their difference
  F(b)-F(a)

94
The definite integral cntnd.

• Example
• Step 1
• Step 2

95
Properties of the definite integral
96
The definite integral as an area under a function

• The definite integral has a definite value, equal
  to F(b)-F(a) which
• can be interpreted geometrically to measure
  the area under f(x).
• For example, the definite integral
  measures the area below the graph of yf(x)x
  between 0 and 1.

y
yx
1
x
1
97
Example

• Find the area under the graph of
• f(x)4x3-3x24x2 between x1 and x2