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Course description

References

iv iv Chapter 1. Applications of Diﬀerentiation

1.1. Mean value theorems of diﬀerential calculus

1.2. Using diﬀerentials and derivatives

1.3. Extreme Values

iii

1

1

5

7

Course description

Application of diﬀerentiation. Taylor theorem. Mean Value theorem of diﬀerential calculus. Methods of integration. Applications of integration.

References

1. Calculus: A complete course by Robert A. Adams and Christopher

Essex.

2. Fundamental methods of mathematical economics by Alpha C. Chiang.

3. Schaum’s outline series: Introduction to mathematical economics by Edward T. Dowling

iv

CHAPTER 1

Applications of Diﬀerentiation

1.1. Mean value theorems of diﬀerential calculus

Theorem 1.1.1 (Mean Value Theorem).

Suppose that the function f is continuous on the closed ﬁnite interval [a, b] and that it is diﬀerentiable on the interval (a, b). Then ∃ a point c ∈ (a, b) such that f (b) − f (a)

= f (c) . b−a It means that the slope of the chord joining the points (a, f (a)) and (b, f (b)) is equal to the slope of the tangent line to te curve y = f (x) at the point

(c, f (c)) so that the two lines are parallel.

Fig 2.28

Example 1.1.1.

√

Verify the conclusion of the mean value theorem for f (x) = x on the interval [a, b], where a ≤ x ≤ b.

Solution. We are to show that ∃ c ∈ (a, b) such that f (b) − f (a)

= f (c) b−a 1

XEQ 201 so long as f is continuous on [a, b] and is diﬀerentiable on (a, b). Now

√

√

1

f (x) = 2√x , f (a) = a, f (b) = b.

√

√

√

√

1

b− a b− a

1

∴ √ =

= √

=√

√ .

√

√

√

b−a

2 c b+ a b− a b+ a

The above equality implies that a < b, we have

√

a=

a+

2

√

a

√

√

c=

√

2

<

b+

2

√ b+ a

2

√

a

√

so that c =

√

2

<

b+

2

√

b

√

2

b+ a

.

2

Since

2

=b

which implies that c ∈ (a, b).

Example 1.1.2.

Show that sin x < x for all x > 0.

Solution. If x > 2π, then sin x ≤ 1 < 2π < x. If 0 < x ≤ 2π, then by

MVT, ∃ c ∈ (0, 2π) such that sin x sin x − sin 0 d =

= [MVT on [0, x]] = sin x x x−0 dx = cos c < 1 x=c which implies that sin x < x in this case too.

increasing decreasing functions

Definition 1.1.1 (Increasing and decreasing functions). Suppose that the function f is deﬁned on an interval I and that x1 and x2 are two points in I.

(a) If f (x2 ) > f (x1 ) whenever x2 > x1 , we say that f is increasing on

I.

(b) If f (x2 ) < f (x1 ) whenever x2 > x1 , we say that f is decreasing on

I.

(c) If f (x2 ) ≥ f (x1 ) whenever x2 > x1 , we say that f is non-decreasing on I.

(d) If f (x2 ) ≤ f (x1 ) whenever x2 > x1 , we say that f is non-increasing on I.

Diagram Fig 2.31

2

XEQ 201

Theorem 1.1.2.

Let J be an open interval and let I be an interval consisting of all points in J and possibly one or both of the end points of J. Suppose that f is continuous on I and diﬀerentiable on J.

(a)

(b)

(c)

(d)

If

If

If

If

f f f f (x) > 0

(x) < 0

(x) ≥ 0

(x) ≤ 0

for for for for all all all all x ∈ J, x ∈ J, x ∈ J, x ∈ J,

then then then then f f f f is is is is increasing on I. decreasing on I. non-decreasing on I. non-increasing on I.

derivative of increasing and decreasing functions

Example 1.1.3.

On what intervals is the function f (x) = x3 − 12x + 1 increasing? On what intervals is it decreasing?

Solution. f (x) = 3x2 − 12 = 3 (x − 2) (x + 2). It follows that f (x) >

0 when x < −2 or x > 2 and f (x) < 0 when −2 < x < 2. Therefore f is increasing on the intervals (−∞, −2) and (2, ∞) and is decreasing on the interval (−2, 2).

Diagram Fig 2.32

Example 1.1.4.

Show that f (x) = x3 is increasing on any interval.

Solution. Let x1 , x2 be any two real numbers satisfying x1 < x2 . Since f (x) = 3x2 > 0 for all x = 0, we have that f (x1 ) < f (x2 ) if either x1 < x2 ≤ 0 or 0 ≤ x1 < x2 . If x1 < 0 < x2 , then f (x1 ) < 0 < f (x2 ). Thus f is increasing on every interval.

Theorem 1.1.3.

If f is continuous on an interval I and f (x) = 0 at every interior point of

I, then f (x) = C, a constant on I.

3

derivetive of constant function

XEQ 201

derivative at interior extreme point Theorem 1.1.4.

If f is deﬁned on an open interval (a, b) and achieves a maximum (or a minimum) at the point c ∈ (a, b), and if f (c) exists, then f (c) = 0.

Values of x where f (x) = 0 are called critical points of the function f .

Theorem 1.1.5 (Rolle’s Theorem).

Suppose that the function g is continuous on the closed ﬁnite interval [a, b] and if it is diﬀerentiable on the open interval (a, b). If g (a) = g (b), ∃ a point c ∈ (a, b) such that g (c) = 0.

Theorem 1.1.6 (The Generalized Mean Value Theorem).

If functions f and g are both continuous on [a, b] and diﬀerentiable on (a, b), and if g (x) = 0 for every x ∈ (a, b), then ∃ a number c ∈ (a, b) such that f (c) f (b) − f (a)

=

. g (b) − g (a) g (c)

Exercise 1.1.

1. Illustrate the MVT by ﬁnding any points in the open interval (a, b) where the tangent line is parallel to the chord joining (a, f (a)) and

(b, f (b)).

(a) f (x) = x2 on [a, b]; Ans: c = b+a .

2

2

(b) f (x) = x3 − 3x + 1 on [−2, 2]; Ans: c = ± √3 .

2. Show that tan x > x for 0 < x < π .

2

3. Find intervals of increase and decrease of the following functions

2

(a) f (x) = x3 − 4x + 1. Ans: Increasing on −∞, − √3 and

2

√ ,∞

3

2

2

; decreasing on − √3 , √3 .

2

(b) f (x) = x2 − 4 . Ans: Increasing on (−2, 0) and (2, ∞); decreasing on (−∞, −2) and (0, 2).

(c) f (x) = x3 (5 − x)2 . Ans: Increasing on (−∞, 3) and (5, ∞); decreasing on (3, 5).

(d) f (x) = x + sin x. Ans: Increasing on (−∞, ∞).

4

XEQ 201

1.2. Using diﬀerentials and derivatives

Suppose dx is regarded as a new independent variable called the diﬀerential of x we can deﬁne a new dependent variable dy, called the diﬀerentail of y as a function of x and dx by dy dx = f (x) dx. dx For example if y = x2 , then dy = 2xdx means the same thing as dy/dx = 2x.

If f (x) = 1/x, then df (x) = − 1/x2 dx.

If y is a function of x, y = f (x), then denoting a small change in x by dx instead of ∆x, the corresponding small change in y, ∆y is approximated by the diﬀerential dy, i.e. dy =

∆y ≈ dy = f (x) dx.

Diagram Fig 2.25

Example 1.2.1.

Without using a scientiﬁc calculator, determine by a pproximately how much the value of sin x increases as x increases from π/3 to (π/3) + 0.006. To 3 decimal places, what is the value of sin ((π/3) + 0.006)?

Solution. If y = sin x, then dx = 0.006. Therefore

dy dx = cos x. Now x =

π

3

≈ 1.0472 and

dy π 1 dx = cos xdx = cos

· 0.006 = (0.006) = 0.003. dx 3

2

This means that the change in the value of sin x is approximately 0.003. Now dy =

sin

π π + 0.006 ≈ sin

+ 0.003 = · · · = 0.869(3 d. p.).

3

3

5

XEQ 201

Suppose changes in x are measured with respect to the size of x, then

relative change in x =

dx dx and percentage change in x = 100 . x x

Diﬀerentials and point elasticity1

For a demand function Q = f (P ), the elasticity is deﬁned as relative change in Q

∆Q/Q

=

.

∆P/P relative change in P point elasticity

If the change in P is inﬁnitesimal, then the expressions ∆Q and ∆P reduce to the diﬀerentials dP and dQ. In that case the elasticity measure assumes the sense of point elasticity of the demand function which is denoted by εd .

Thus

dQ/Q dQ/dP εd ≡

=

. dP/P Q/P

The numerator in the right hand is the derivative (or marginal2) function of the demand function while the denominator is the average function of the demand function. Thus the point elasticity is a ratio of the two functions.

In general, for any given total function y = f (x), the point elasticity of y

w.r.t. x is dy/dx marginal function εyx =

=

. y/x average function

The absolute value of the point elasticity measure is used in deciding whether the function is elastic at a particular point. In the case of a demand function,

elastic

|εd | > 1

The demand is of unit elasticity if |εd | = 1

inelastic |εd | < 1. at a given point.

1Elasticity is the measure of how an economic variable responds to change in another

variable. An elastic variable is one which responds more than proportionally to changes in other variables. An inelastic variable is one which changes less than proportionally in response to changes in other variables.

2Marginal denotes the rate of change of a quantity with respect to a variable on which it depends. 6

XEQ 201

Example 1.2.2.

Find εd for the demand function Q = 100 − 2P . Determine the point elasticity at P = 25.

Solution.

dQ dP = −2 and εd =

Q

P

=

100−2P

.

P

Therefore

−2

P

=

.

(100 − P ) /P

P − 50

Thus

= · · · = −1.

εd

P =25

Therefore the demand is of unit elasticity when P = 25.

Exercise 1.2.

1. Use diﬀerentials to determine approximate change in the values of the given function as its argument changes from the given value to the given amount. What is the approximate value of the function after the change?

(a) y = 1/x as x increases from 2 to 2.01.

(b) h (t) = cos (πt/4) as t increases from 2 to 2 + (1/10π).

2. Find the approximate percentage changes in the given function that will result from an increase of 2% in the value of x.

(a) y = x2

(b) y = 1/x2

3. Given the consumption function C = a + bY (with a > 0; 0 < b <

1);

(a) Find its marginal function and its average function.

(b) Find the income elasticity of consumption εCY , and determine its sign, assuming Y > 0.

(c) Show that the consumption is inelastic at all positive income levels. 1.3. Extreme Values

Maximum and Minimum Values

Definition 1.3.1. Function f has an absolute maximum value f (x0 ) at x0 in its domain if f (x) ≤ f (x0 ) holds for every x in the domain of f .

7

absolute extreme values

XEQ 201

Similarly, f has an absolute minimum value f (x0 ) at x0 in its domain if f (x) ≥ f (x0 ) holds for every x in the domain of f .

Remark 1. extreme value is unique, can occur at several points existence of extreme value not guaranteed

existence of extreme values for closed ﬁnite intervals local extreme values 1. A function will have only one absolute maximum (or minimum) value if it exists. However the value can occur at many points. For example, f (x) = sin x has absolute maximum of 1 but it occurs at every point π + 2nπ, n ∈ Z.

2

2. A function need not have any extreme value. The function f (x) =

1

x becomes arbitrarily large as x approaches 0 from the right, and so has no ﬁnite absolute maximum value.

Theorem 1.3.1.

If the domain of the function f is a closed, ﬁnite interval or a union of ﬁnitely many such intervals, and if f is continuous on that domain, then f must have an absolute maximum value and absolute minimum value.

Definition 1.3.2. Function f has a local maximum value (loc. max.) f (x0 ) at the point x0 in its domain provided ∃ a number h > 0 such that f (x) ≤ f (x0 ) whenever x is in the domain of f and |x − x0 | < h.

Similarly, f has a local minimum value (loc. min.) f (x1 ) at the point x1 in its domain provided ∃ a number h > 0 such that f (x) ≥ f (x1 ) whenever x is in the domain of f and |x − x1 | < h.

Diagram Fig 4.17

From the above ﬁgure we see that local extreme values can occur at any of the following points.

8

XEQ 201 critical, singular or end points

(i) critical points of f ; points x ∈ D (f ) where f (x) = 0.

(ii) singular points of f ; points x ∈ D (f ) where f (x) is not deﬁned.

(iii) endpoints of the domain of f ; points that do belong to D (f ) but are not interior points of D (f ).

In the ﬁgure above, x1 , x3 , x4 are critical points, x2 and x5 are singular points and a and b are endpoints.

Theorem 1.3.2.

If the function f is deﬁned on an interval I and has a local maximum (or local minimum) value at the point x = x0 in I, then x0 must be either a critical point of f , a singular point of f , or and endpoint of I.

Example 1.3.1.

Find the maximum and minimum values of the function g (x) = x3 − 3x2 −

9x + 2 on the interval −2 ≤ x ≤ 2.

Solution. Since g is a polynomial it can never have a singular point.

For critical points we calculate g (x) = 3x2 − 6x − 9 = 3 (x + 1) (x − 3) = 0.

Thus x = −1 or x = 3. But x = 3 is not in the domain of g and so we ignore it. We then investigate the endpoints x = −2 and x = 2 and critical point x = −1. g (−2) = 0, g (−1) = 7, g (2) = −20

The maximum value of g on −2 ≤ x ≤ 2 is at the critical point x = −1, and the minimum value is at the endpoint x = 2.

Diagram Fig 4.19

9

exteme values occur at critical, singular or end points

XEQ 201

Example 1.3.2.

Find the maximum and minimum values of h (x) = 3x2/3 −2x on the interval

[−1, 1].

Solution. The derivative of h is h (x) = 2x−1/3 − 2.

Note that x−1/3 is not deﬁned at x = 0 in D (h), so x = 0 is a singular point of h. Also h (x) = 0 at x−1/3 = 1, that is at x = 1, which also happens to be an endpoint of the domain of h. We therefore examine the values of h at endpoints x = −1 and x = 1 and at the singular point x = 0. h (−1) = 5, h (0) = 0, h (1) = 1

The function h has a maximum value 5 at the endpoint x = −1 and a minimum value 0 at the singular point x = 0.

Diagram Fig 4.20

10

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...CONSUMER EQUILIBRIUM The consumer is assumed to be rational, that is, given his income and market prices of the different commodities, he aims at attaining the highest possible satisfaction or utility. However in order to achieve this objective, he must be able to compare the satisfaction of the different baskets of goods he can buy with his income. Utility is the measure of personal satisfaction. Marginal utility is the additional satisfaction that a consumer derives from an additional unit of a commodity or service. Total utility is the total satisfaction derived from consumption of a given combination of commodities. Consumer equilibrium is attained when the consumer purchases a combination of commodities that achieves utility maximization, given his income and price of commodities. The theory of demand tries to analyse consumer behaviour during the process of utility maximization. Once utility maximization is achieved, a consumer cannot increase utility by consuming more of one good and less of another because the marginal utility-price ratio equated. That is to say, utility is maximised when the consumer equates utility to the last amount spent on each good consumed. A summary equation to explain this is as follows: Marginal Utility of Good 1 = Marginal Utility of Good 2 Price of Good 1 Price of Good 2 Sometimes, the rule of consumer equilibrium is not satisfied, which means that utility is not maximized. This utility can be increased by......

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...good and a change in its price. Arc Elasticity – the elasticity of one variable with respect to another between two given points. It is the ratio of the percentage change of one of the variables between the two points to the percentage change of the other variable. Point Elasticity – the relatively responsiveness of a change in one variable to an extremely small change in another variable. The concept of point elasticity comes into play when discussing the elasticity at a specific point on a curve. Income Elasticity – a measure of the relationship between a change in the quantity demanded for a particular good and a change in income. It refers to the sensitivity of the quantity demanded for a certain product in response to a change in consumer incomes. Cross Elasticity – an economic concept that measures the responsiveness in the quantity demand of one good when a change in price takes place in another good. Price Elasticity of Supply – measures the responsiveness of quantity supplied to a change in price. Difference between Point and Arc Elasticity – arc elasticity of one variable with respect to another between two given points while point elasticity measures the change in quantity demanded to a very small change...

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...9/14/2010 Agenda • Course Overview • Valuing Payoff Streams • Intertemporal consumption: Saving and Borrowing • What To Take Away The Structure of the Course Today’s lecture Introduction to Markets Consumers and Producers Consumer Theory and Demand Uncertainty Technology and Production Microeconomics Perfect competition Monopoly and Pricing strategies Market Interaction Game Theory Competitive Strategy Auctions Information in Markets and Agency 1 9/14/2010 Agenda • Course Overview • Valuing Payoff Streams • Intertemporal consumption: Saving and Borrowing • What To Take Away How to value different streams of future payoffs? Would you rather get $5 dollars every year for the next 5 years or $10 dollars today? How does one compare streams of future payoffs? Let’s start by comparing two periods. What is the equivalent of $1 tomorrow? If the interest rate is R I could invest $1 and receive $(1+R) tomorrow. Conversely, I can borrow $1 today under the promise of paying back $(1+R) tomorrow. We will determine the value of a future payment by discounting them at the interest rate that could be earned. The interest rate R often is given on a yearly basis. How to discount a payment of $1 in 5 years time? By compounding we get that the Present Discounted Value (PDV) of $1 in 5 years is 1 (1 R)5 2 9/14/2010 Example So would you rather get $5 dollars every year for the next 5 years or $10 dollars today? It will...

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...Topic: “Elasticity of Demand and Managerial Decision Making” The demand of a commodity depends on the size of the total market or industry demand for the commodity which in turn is the sum of demands for the commodity of the individual consumers in the market. The demand for a commodity arises from the consumers’ willingness and ability to purchase the commodity. Consumer Demand Theory postulates that the quantity demanded of a commodity is a function of the price of the commodity, the consumers’ income, the price of the complementary and substitute commodities and the taste of the consumer. It is also expressed as: Qdx=f(Px, I, Py, T) Where Qdx = Quantity demanded of the commodity X by an individual per time period Px = Price per unity of commodity Y I = Consumer income Py = Price of substitute or complementary commodity T = Consumer taste When the firm increases the price of a commodity, sales generally decline. A manager expects an inverse relationship between the quantity demanded of a commodity and its price. On the other hand, when a consumer’s income rises, he/ she usually purchases more of most commodities, known as normal goods. There are some goods and services, however, which the consumer purchases less as income increases, which are termed as inferior goods. The inverse relationship between the price and the quantity demanded of the commodity per time period is the individual’s demand schedule for the commodity, and the plot of data......

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