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Mathematics for Business

Instructor: Prof. Ken Tsang

Room E409-R11

Email: kentsang@uic.edu.hk

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CALCULUSFor Business, Economics, and the Social and life Sciences Hoffmann, L.D. & Bradley, G.L.

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TA information

Mr Zhu Zhibin

Room E409 Tel: 3620630

zhibinzhu@uic.edu.hk

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Web-page for this class

Watch for announcements about this class and

download lecture notes from http://www.uic.edu.hk/~kentsang/calcu2012/c

alcu.htm Or from this page:

http://www.uic.edu.hk/~kentsang/

Or from Ispace

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Tutorials

One hour each week Time & place to be announced later (we need

your input)

More explanations More examples More exercises

Quizzes 20% Mid-term exam 20% Assignments 10% Final Examination 50%

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How is my final grade determined?

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UIC Score System

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Grade Distribution Guidelines

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What can you do to maximize your chances for success? Work hard, more importantly, work smart:

1. Understand, don't memorize.

2. Ask why, not how.

3. See every problem as a challenge.

4. Learn techniques, not results.

5. Make sure you understand each topic before going on to the next.

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More info about this Course

Assignments must be handed in before the deadline. There will be about 3 to 4 quizzes. We will tell you your scores for the mid-term test

and quizzes so that you know your progress. However, for the final examination, we cannot tell you the score before the AR release the official results.

Mathematics is about numbers, space, structures, …

Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.

Most important, it teaches us how to analysis problem in an abstract form, with logical thinking.

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Mathematics? Why?Mathematics? Why?

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They invented Calculus!

Sir Isaac Newton (1642-1727)

Gottfriend Wilhelm von Leibniz

(1646-1716)

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What is Calculus all about? Calculus is the study of changing quantities, or more

precisely, the rate of changes: e.g. velocities, interest rate, return on an asset.

The two key areas of Calculus are Differential Calculus and Integral Calculus.

The big surprise is that these two seemingly unrelated areas are actually connected via the Fundamental Theorem of Calculus.

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Calculus has practical applications, such as understanding the true meaning of the infinitesimals. (Image concept by Dr. Lachowska.)

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Isaac Newton (4 January 1643  – 31 March 1727)

English physicist, mathematician, astronomer, natural philosopher and theologian, one of the most influential men in human history.

Newton in a 1702 portrait by Godfrey Kneller

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Newton’s contributions

Newton described universal gravitation and the three laws of motion, laying the groundwork for classical mechanics, which dominated the scientific view of the physical Universe for the next three centuries and is the basis for modern engineering.Newton showed that the motions of objects on Earth and of celestial bodies are governed by the same set of natural laws.

17Cosmos1

Newton's own first edition copy of his Philosophiae Naturalis Principia Mathematica with his handwritten corrections for the second edition.The book can be seen in the Wren Library of Trinity College, Cambridge.

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Newton's 2nd law of motionNewton's Second Law states that an applied force, on an object equals the rate of change of its momentum, with time.

For a system with constant mass, the equation can be written in the iconic form: F= ma,where a is the acceleration of an object.Acceleration is the rate of change in velocity.This can be rewritten as a differential equation.

Most laws of nature can be expressed as differential equations or partial differential equations (PDE).

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If you are a finance major

Finance is a quantitative discipline How to calculate the return of your investment? Asset valuation Portfolio theory Derivatives Risk management

A simple example in asset valuation Suppose we have a riskless asset

r is the constant rate of return

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If your major is finance, you will know this: Fischer Black and Myron Scholes first articulated

the Black-Scholes formula in their 1973 paper, "The Pricing of Options and Corporate Liabilities."

Robert C. Merton was the first to publish a paper expanding the mathematical understanding of the options pricing model and coined the term "Black-Scholes" options pricing model.

Merton and Scholes received the 1997 Prize in Economic Sciences in Memory of Alfred Nobel for this and related work.

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The Black-Scholes model

In the Black-Scholes model, we assume that the underlying security (typically the stock) follows a geometric Brownian motion. That is,

where S is the price of the stock at time t,

μ is the drift rate of S, annualized,

σ is the volatility of the stock,

the dW term here stands in for any and all sources of uncertainty in the price history of a stock, modeled by a Brownian motion.

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If you are a science major

Science is Quantitative Logical

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Ecology: Population dynamics The basic accounting relation for population

dynamics is:N1 = N0 + B − D + I − E

where N1 is the number of individuals at time 1, N0 is the number of individuals at time 0, B is the number of individuals born, D the number that died, I the number that immigrated, and E the number that emigrated between time 0 and time 1.

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The Lotka–Volterra (predator–prey) equationsare a pair of first-order, non-linear, differential

equations frequently used to describe the dynamics of biological systems in which two species interact, one a predator and one its prey.

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where

* y is the number of some predator (for example, wolves);

* x is the number of its prey (for example, rabbits);

* dy/dt and dx/dt represents the growth of the two populations against time;

* t represents the time; and

* α, β, γ and δ are parameters representing the interaction of the two species.

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Suppose there are two species of animals, a baboon (prey) and a cheetah (predator). If the initial conditions are 80 baboons and 40 cheetahs, one can plot the progression of the two species over time.

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OK, any question?

That’s all for introduction. Let’s begin the real thing!

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Chapter 1Functions, Graphs and Limits

In this Chapter, we will encounter some

important concepts.

Functions ( 函数 )

Limits （极限）

One-sided Limits （单边极限） and Continuity （连续）

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Section 1.1 Functions （函数） A function is a rule that assigns to each object in a set A exactly one object in a set B.

The set A is called the domain （定义域） of the function, and the set of assigned objects in B is called the range. （值域）

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Function, or not?

fffA B

A B

fA B

f YES

NONO NONO

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To be convenient, we represent a functional relationship by an equation

In this context, x and y are called variables, furthermore, we refer to y as the dependent variable ( 因变量 ) and to x as the independent variable （自变量） . For instant, the function representation

Noted that x and y can be substituted by other letters. For example, the above function can be represented by

)(xfy

4)( 2 xxfy

42 ts

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Function that describes tabular data

Academic Year Tuition and

Ending in Period n Fees

1973 1 $1,898

1978 2 $2,700

1983 3 $4,639

1988 4 $7,048

1993 5 $10,448

1998 6 $13,785

2003 7 $18,273

Table 1.1 Average Tuition and Fees for 4-Year Private CollegesTable 1.1 Average Tuition and Fees for 4-Year Private Colleges

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We can describe this data as a function f defined by the rule

Thus,

Noted that the domain of f is the set of integers

periodyear -5th theof beginning

at the fees and tuition average)(

nnf

273,18)7(,.......,700,2)2(,898,1)1( fff

}7,....,2,1{A

Solution:Solution:

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Piecewise-defined function （分段函数）

A piecewise-defined function is such a function that is often defined using more than one formula, where each individual formula describes the function on a subset of the domain.

Here is an example of such a function

1 xif 13

1 xif 1

1)(

2xxxf

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Find f(-1/2), f(1), and f(2), where the piecewise-defined function f(x) is given at the above slide.

Since satisfies x<1, use the top part of the formula to find

However, x=1 and x=2 satisfy x≥1, so f(1) and f(2) are both found by using the bottom part of the formula:

and

Example 1Example 1

Solution:Solution:

2

1x

3

2

2/3

1

12/1

1

2

1

f

41)1(3)1( 2 f 131)2(3)2( 2 f

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Domain Convention We assume the domain of f to be the set of all

numbers for which f(x) is defined (as a real number).

We refer to this as the natural domain of f. In general, there are two situations where a number is not in the domain of a function:1) division by 0 2) The even number root of a negative number

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Example 2

21

1)(

xxf

Find the domain and range of each of these functionsFind the domain and range of each of these functions

a. b. a. b. 4 2)( uug

Solution:Solution:

a.a. Since division by any number other than Since division by any number other than 00 is possible, the is possible, the domain of domain of ff is the set of all numbers except is the set of all numbers except -1-1 and and 11. The range . The range of of ff is the set of all numbers is the set of all numbers yy except except 00..

b.b. Since negative numbers do not have real fourth roots, so the Since negative numbers do not have real fourth roots, so the domain of domain of gg is the set of all numbers is the set of all numbers uu such as such as u≥-2u≥-2. The range . The range ofof gg is the set of all nonnegative numbers. is the set of all nonnegative numbers.

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Functions Used in Economics A demand function ( 需求函数 ) p=D(x) is a function that

relates the unit price p for a particular commodity to the number of units x demanded by consumers at that price.

The total revenue （总收入） is given by the product R(x)=(number of items sold)(price per item) =xp=xD(x)

If C(x) is the total cost （总成本） of producing the x units, then the profit （利润） is given by the function P(x)=R(x)-C(x)=xD(x)-C(x)

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Example 3

Market research indicates that consumers will buy Market research indicates that consumers will buy xx thousand units of a particular kind of coffee maker when thousand units of a particular kind of coffee maker when the unit price is dollars. The cost of the unit price is dollars. The cost of producing the producing the xx thousand units is thousand units is

5127.0 xp

855.323.2)( 2 xxxC

thousand dollarsthousand dollars

a. What are the revenue and profit functions, a. What are the revenue and profit functions, R(x)R(x) and and P(x),P(x), for this production process? for this production process?

b. For what values of b. For what values of xx is production of the coffee is production of the coffee makers profitable?makers profitable?

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Solution:Solution:

a.a. The demand function is , so the revenue is The demand function is , so the revenue is

thousand dollars, and the profit is (thousand dollars) thousand dollars, and the profit is (thousand dollars)

5127.0)( xxD

xxxxDxR 5127.0)()( 2

855.475.2

)855.323.2(5127.0

)()()(

2

22

xx

xxxx

xCxRxP

b. Production is profitable when b. Production is profitable when P(x)>0P(x)>0. We find that. We find that

0)17)(2(5.2

)3419(5.2

855.475.2)(2

2

xx

xx

xxxP

Thus, production is profitable for Thus, production is profitable for 2<x<172<x<17..

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Composition of Functions （复合函数）

Composition of Functions: Given functions f(u) and g(x), the composition f(g(x)) is the function of x formed by substituting u=g(x) for u in the formula for f(u).

Example 4

Find the composition function Find the composition function f(g(x)),f(g(x)), where and where and 1)( xxg

1)( 3 uuf

Solution:Solution:

ReplaceReplace u u by by x+1x+1 in the formula for in the formula for f(u)f(u) to get to get

2331)1())(( 233 xxxxxgf

Question: How about Question: How about g(f(x))g(f(x))??

Note: In general, Note: In general, f(g(x))f(g(x)) and and g(f(x))g(f(x)) will not be the same. will not be the same.

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Example 5

An environmental study of a certain community suggests An environmental study of a certain community suggests that the average daily level of carbon monoxide in the air that the average daily level of carbon monoxide in the air will be parts per million when the population will be parts per million when the population is p thousand. It is estimated that t years from now the is p thousand. It is estimated that t years from now the population of the community will be population of the community will be thousand. thousand.

15.0)( ppc

21.010)( ttp

a.a. Express the level of carbon monoxide in the air as a Express the level of carbon monoxide in the air as a function of time.function of time.

b.b. When will the carbon monoxide level reach When will the carbon monoxide level reach 6.86.8 parts parts per million?per million?

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Solution:Solution:

a. Since the level of carbon monoxide is related to the variablea. Since the level of carbon monoxide is related to the variable p p by the equation , and the variable by the equation , and the variable pp is related to is related to the variable the variable tt by the equation by the equation

15.0)( ppc21.010)( ttp

It follows that the composite function It follows that the composite function 222 05.061)1.010(5.0)1.010())(( tttctpc

expresses the level of carbon monoxide in the air as a function of expresses the level of carbon monoxide in the air as a function of the variablethe variable t t. .

b. Set b. Set c(p(t))c(p(t)) equal to equal to 6.8 6.8 and solve for and solve for t t to get to get

solution. natural anot is 4 4

16

8.005.0

8.605.06

2

2

2

tt

t

t

t

That is,That is, 4 4 years from now the level of carbon monoxide will be years from now the level of carbon monoxide will be 6.8 6.8 parts per million.parts per million.

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Section 1.2 The Graph of a FunctionSection 1.2 The Graph of a Function The graph of a function f consists of all points (x,y) where x

is in the domain of f and y=f(x), that is, all points of the form (x,f(x)).

Rectangular coordinate system （平面直角坐标系） , Horizontal axis （横坐标） , vertical axis （纵坐标） .

The below example shows that the function can be sketched by plotting a few points.

2)( 2 xxxfx -3 -2 -1 0 1 2 3 4

f(x) -10 -4 0 2 2 0 -4 -10

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Intercepts

x intercepts: The points where a graph crosses the x axis. A y intercept: A point where the graph crosses the y axis. How to find the x and y intercepts: The only possible y

intercept for a function is , to find any x intercept of y=f(x), set y=0 and solve for x.

Note: Sometimes finding x intercepts may be difficult. Following above example, the y intercept is f(0)=2. To

find the x intercepts, solve the equation f(x)=0, we have x=-1 and 2. Thus, the x intercepts are (-1,0) and (2,0).

)0(0 fy

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Parabolas （抛物线） Parabolas: The graph of as long as A≠0. All parabolas have a “U shape” and the parabola opens up if

A>0 and down if A<0. The “peak” or “valley” of the parabola is called its vertex

（顶点） , and it always occurs where

CBxAxy 2

A

Bx

2

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Example 6

A manufacturer determines that when x hundred units of a particular A manufacturer determines that when x hundred units of a particular commodity are produced, they can all be sold for a unit price given by commodity are produced, they can all be sold for a unit price given by the demand function the demand function p=60-xp=60-x dollars. At what level of production is dollars. At what level of production is revenue maximized? What is the maximum revenue?revenue maximized? What is the maximum revenue?

Solution:Solution:

The revenue function The revenue function R(x)=x(60-x)R(x)=x(60-x) hundred dollars. Note that hundred dollars. Note that R(x) ≥0R(x) ≥0 only for only for 00≤x≤60≤x≤60. . The revenue function can be rewritten as The revenue function can be rewritten as

xxxR 60)( 2 which is a parabola that opens downward (Since which is a parabola that opens downward (Since A=-1<0A=-1<0) and has its ) and has its high point (vertex) at high point (vertex) at 30

)1(2

60

2

A

Bx

Thus, revenue is maximized when Thus, revenue is maximized when x=30x=30 hundred units are produced, hundred units are produced, and the corresponding maximum revenue is and the corresponding maximum revenue is R(30)=900R(30)=900 hundred hundred dollars. dollars.

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Intersections of Graphs Sometimes it is necessary to determine when two

functions are equal.

For example, an For example, an economist may wish to economist may wish to compute the market compute the market price at which the price at which the consumer demand for consumer demand for a commodity will be a commodity will be equal to supply.equal to supply.

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Power Functions, Polynomials, and Rational Functions A Power Function （幂函数） : A function of the

form , where n is a real number. A Polynomial Function （多项式） : A function of the form

where n is a nonnegative integer and are constants. If , the integer n is called the degree （阶） of the polynomial.

A Rational Function （有理函数） : A quotient of two polynomials p(x) and q(x).

nxxf )(

011

1)( axaxaxaxp nn

nn

naaa ,,, 10

0na

)(

)(

xq

xp

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The Vertical Line Test The Vertical Line TestThe Vertical Line Test: A curve is the graph of a : A curve is the graph of a function if and only if no vertical line intersects the function if and only if no vertical line intersects the curve more than once. curve more than once.

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Section 1.3 Linear FunctionsA linear functionA linear function （线性函数）（线性函数） is a function that is a function that changes at a constant rate with respect to its changes at a constant rate with respect to its independent variable.independent variable.

The graph of a linear function is a straight line.The graph of a linear function is a straight line.

The equation of a linear function can be written in the The equation of a linear function can be written in the formform

bmxy

where where mm and and bb are constants. are constants.

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The Slope of a Line （斜率） The Slope of a Line: The slope of the non-vertical line

passing through the points and is given by the formula

),( 11 yx ),( 22 yx

12

12

in x change

yin changeSlope

xx

yy

x

y

Sign and magnitude of slopeSign and magnitude of slope

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Forms of the equation of a line The Slope-Intercept Form: The equation is the

equation of a line whose slope is m and whose y intercept is (0,b).

The Point-Slope Form: The equation is an equation of the line that passes through the point

and that has slope equal to m.

bmxy

)( 00 xxmyy

),( 00 yx

The slope-Intercept form is The slope-Intercept form is 3

1

)05.1(

)5.00(

m

2

1

3

1 xy

The point-slope form that passes The point-slope form that passes through the point through the point (-1.5,0)(-1.5,0) is is

)5.1(3

10 xy

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Example 7

Table 1.2 lists the percentage of the labour force that was Table 1.2 lists the percentage of the labour force that was unemployed during the decade unemployed during the decade 1991-20001991-2000. Plot a graph with the time . Plot a graph with the time (years after 1991) on the (years after 1991) on the xx axis and percentage of unemployment on axis and percentage of unemployment on thethe y y axis. Do the points follow a clear pattern? Based on these data, axis. Do the points follow a clear pattern? Based on these data, what would you expect the percentage of unemployment to be in the what would you expect the percentage of unemployment to be in the year year 20052005??

Number of Years Percentage of

Year from 1991 Unemployed

1991 0 6.8

1992 1 7.5

1993 2 6.9

1994 3 6.1

1995 4 5.6

1996 5 5.4

1997 6 4.9

1998 7 4.5

1999 8 4.2

2000 9 4.0

Table 1.2 Percentage of Civilian Unemployment

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Solution:Solution:

The pattern does suggest that we may be able to get useful The pattern does suggest that we may be able to get useful information by finding a line that “best fits” the data in information by finding a line that “best fits” the data in some meaningful way. One such procedure, called “some meaningful way. One such procedure, called “least-least-squares approximationsquares approximation”, require the approximating line to ”, require the approximating line to be positioned so that the sum of squares of vertical distances be positioned so that the sum of squares of vertical distances from the data points to the line is minimized. from the data points to the line is minimized.

It produces the “best-fitting line” . It produces the “best-fitting line” . Based on this formula, we can attempt a prediction of the Based on this formula, we can attempt a prediction of the unemployment rate in the year unemployment rate in the year 20052005: :

338.7389.0 xy

892.1338.7)14(389.0)14( yNote: Note: Care must be taken when making predictions by extrapolating Care must be taken when making predictions by extrapolating from known data, especially when the data set is as small as the one from known data, especially when the data set is as small as the one in this example. in this example.

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Parallel （平行） and Perpendicular （垂直） Lines Let and be the slope of the non-vertical lines

and . Then

and are parallel if and only if

and are perpendicular if and only if

1m2L

2m 1L

1L2L 21 mm

1L2L

12

1

mm

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Example 8

Let Let LL be the line be the line 4x+3y=34x+3y=3

a.a. Find the equation of a line parallel to Find the equation of a line parallel to LL through through P(-1,4).P(-1,4).

b.b. Find the equation of a line perpendicular toFind the equation of a line perpendicular to L L through through Q(2,-3).Q(2,-3).

1L

2L

Solution:Solution:

By rewriting the equation By rewriting the equation 4x+3y=34x+3y=3 in the slope-intercept form in the slope-intercept form

, we see that , we see that LL has slope has slope

a.a. Any line parallel to Any line parallel to LL must also have slope must also have slope -4/3-4/3. The required line. The required line

contains contains P(-1,4),P(-1,4), we have we have

b.b. A line perpendicular to A line perpendicular to LL must have slope must have slope m=3/4m=3/4.Since the .Since the required line contains required line contains Q(2,-3),Q(2,-3), we have we have

13

4 xy

3

4Lm

1L3

8

3

4)1(

3

44 xyxy

2

9

4

3

)2(4

33

xy

xy2L

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Section 1.4 Functional Models To analyze a real world problem, a common procedure is to

make assumptions about the problem that simplify it enough to allow a mathematical description. This process is called mathematical modelling and the modified problem based on the simplifying assumptions is called a mathematical model.

Real-world problem

Testing

Interpretation

Mathematicalmodel

adjustments

PredictionAnalysis

Formulation

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Elimination of Variables In next example, the quantity you are seeking is expressed

most naturally in term of two variables. We will have to eliminate one of these variables before you can write the quantity as a function of a single variable.

Example 9

The highway department is planning to build a picnic area The highway department is planning to build a picnic area for motorists along a major highway. It is to be rectangular for motorists along a major highway. It is to be rectangular with an area of with an area of 5,0005,000 square yards and is to be fenced off square yards and is to be fenced off on the three sides not adjacent to the highway. Express the on the three sides not adjacent to the highway. Express the number of yards of fencing required as a function of the number of yards of fencing required as a function of the length of the unfenced side.length of the unfenced side.

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Solution:Solution:

We denote We denote xx and and y y as the lengths of the sides of the picnic area. as the lengths of the sides of the picnic area. Expressing the number of yards Expressing the number of yards FF of required fencing in terms of of required fencing in terms of these two variables, we get . Using the fact that the area these two variables, we get . Using the fact that the area is to be is to be 5,0005,000 square yards that is square yards that is

yxF 2x

yxy5000

000,5

and substitute the resulting expression for and substitute the resulting expression for yy into the formula for into the formula for F F to to

getget x

xx

xxF100005000

2)(

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Modelling in Business and Economics Example 10

A manufacturer can produce blank videotapes at a cost of $2 per cassette. The cassettes have been selling for $5 a piece. Consumers have been buying 4000 cassettes a month. The manufacturer is planning to raise the price of the cassettes and estimates that for each $1 increase in the price, 400 fewer cassettes will be sold each month.

a: Express the manufacturer’s monthly profit as a function of the price at which the cassettes are sold.

b: Sketch the graph of the profit function. What price corresponds to maximum profit? What is the maximum profit?

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Solution:Solution:

a.a. As we know, As we know, Profit=(number of cassettes sold)(profit Profit=(number of cassettes sold)(profit per cassette)per cassette)

Let Let p p denote the price at which each cassette will be sold denote the price at which each cassette will be sold and let and let P(p)P(p) be the corresponding monthly profit. be the corresponding monthly profit.

Number of cassettes soldNumber of cassettes sold

==4000-400(number of $1 increases)4000-400(number of $1 increases)

==4000-400(p-5)=6000-400p4000-400(p-5)=6000-400p

Profit per cassette=p-2Profit per cassette=p-2

The total profit is The total profit is 120006800400

)2)(4006000()(2

pp

pppP

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b.b. The graph of The graph of P(p)P(p) is the downward opening parabola is the downward opening parabola shown in the bottom figure. Profit is maximized at the shown in the bottom figure. Profit is maximized at the value of value of pp that corresponds to the vertex of the parabola. that corresponds to the vertex of the parabola. We know We know

Thus, profit is maximized when the manufacturer charges Thus, profit is maximized when the manufacturer charges $8.50$8.50 for each cassette, and the maximum monthly profit for each cassette, and the maximum monthly profit is is

5.8)400(2

6800

2

A

Bp

16900$12000)5.8(6800)5.8(400)5.8( 2max PP

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Market EquilibriumThe law of supply and demandThe law of supply and demand: In a competitive market : In a competitive market environment, supply tends to equal demand, and when this environment, supply tends to equal demand, and when this occurs, the market is said to be in equilibrium. occurs, the market is said to be in equilibrium.

The demand functionThe demand function: : p=D(x)p=D(x)

The supply functionThe supply function: : p=S(x)p=S(x)

The equilibrium priceThe equilibrium price: :

Shortage: Shortage: D(x)>S(x)D(x)>S(x)

Surplus: Surplus: S(x)>D(x)S(x)>D(x)

)()( eee xSxDp

66

Example 11

Market research indicates that manufacturers will supply Market research indicates that manufacturers will supply x x units of a particular commodity to the marketplace when the units of a particular commodity to the marketplace when the price is price is p=S(x)p=S(x) dollars per unit and that the same number of dollars per unit and that the same number of units will be demanded by consumers when the price is units will be demanded by consumers when the price is p=D(x)p=D(x) dollars per unit, where the supply and demand dollars per unit, where the supply and demand functions are given byfunctions are given by

xxDxxS 6174)( 14)( 2

a.a. At what level of production At what level of production x x and unit price and unit price pp is market is market equilibrium achieved? equilibrium achieved?

b.b. Sketch the supply and demand curves, Sketch the supply and demand curves, p=S(x)p=S(x) and and p=D(x),p=D(x), on the same graph and interpret. on the same graph and interpret.

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Solution:Solution:

a.a. Market equilibrium occurs when Market equilibrium occurs when S(x)=D(x)S(x)=D(x), we have, we have

16or 10

0)16)(10(

6174142

x

xx

xx

Only positive values are meaningful, Only positive values are meaningful, 114)10(6174)10( Dpe

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Break-Even AnalysisAt low levels of production, the manufacturer suffers At low levels of production, the manufacturer suffers a loss. At higher levels of production, however, the total a loss. At higher levels of production, however, the total revenue curve is the higher one and the manufacturer revenue curve is the higher one and the manufacturer realizes a profit. realizes a profit.

Break-even point Break-even point : The total revenue equals total cost.: The total revenue equals total cost.

69

Example 12

A manufacturer can sell a certain product for A manufacturer can sell a certain product for $110$110 per unit. per unit. Total cost consists of a fixed overhead of Total cost consists of a fixed overhead of $7500$7500 plus plus production costs of production costs of $60$60 per unit. per unit.

a.a. How many units must the manufacturer sell to break even?How many units must the manufacturer sell to break even?

b.b.What is the manufacturer’s profit or loss if What is the manufacturer’s profit or loss if 100100 units are units are sold?sold?

c.c.How many units must be sold for the manufacturer to How many units must be sold for the manufacturer to realize a profit of realize a profit of $1250$1250?? Solution:Solution:

If If xx is the number of units manufactured and sold, the total is the number of units manufactured and sold, the total revenue is given by revenue is given by R(x)=110xR(x)=110x and the total cost by and the total cost by C(x)=7500+60xC(x)=7500+60x

70

a.a. To find the break-even point, set To find the break-even point, set R(x)R(x) equal to equal to C(x)C(x) and solve and solve 110x=7500+60x110x=7500+60x, so that, so that x=150 x=150..

It follows that the manufacturer will have to sell It follows that the manufacturer will have to sell 150150 units to break units to break even. even.

b.b. The profit The profit P(x) P(x) is revenue minus cost. Hence,is revenue minus cost. Hence,

P(x)=R(x)-C(x)=110x-(7500+60x)=50x-7500P(x)=R(x)-C(x)=110x-(7500+60x)=50x-7500

The profit from the sale of The profit from the sale of 100100 units is units is P(100)=-2500P(100)=-2500

It follows that the manufacturer will lose It follows that the manufacturer will lose $2500 $2500 if if 100 100 units are units are sold.sold.

c.c. We set the formula for profit We set the formula for profit P(x) P(x) equal to equal to 1250 1250 and solve for and solve for x, x, we have we have P(x)=1250, x=175. P(x)=1250, x=175. That is That is 175175 units must be sold to units must be sold to generate the desired profit.generate the desired profit.

71

Example 13

A certain car rental agency charges A certain car rental agency charges $25$25 plus plus 6060 cents per cents per mile. A second agency charge mile. A second agency charge $30$30 plus plus 5050 cents per mile. cents per mile. Which agency offers the better deal? Which agency offers the better deal?

Solution:Solution:

Suppose a car is to be driven Suppose a car is to be driven xx miles, then the first agency miles, then the first agency will charge dollars and the second will charge will charge dollars and the second will charge

. So that . So that x=50x=50. .

For shorter distances, the first agency offers the better deal, For shorter distances, the first agency offers the better deal, and for longer distances, the second agency is better. and for longer distances, the second agency is better.

xxC 60.025)(1

xxC 50.030)(2

72

Section 1.5 Limits （极限） Roughly speaking, the limit process involves examining the

behavior of a function f(x) as x approaches a number c that may or may not be in the domain of f.

To illustrate the limit process, consider a manager who determines that when x percent of her company’s plant capacity is being used, the total cost is

96068

3206368)(

2

2

xx

xxxC

hundred thousand dollars. The company has a policy of hundred thousand dollars. The company has a policy of rotating maintenance in such a way that no more than rotating maintenance in such a way that no more than 80% 80% of capacity is ever in use at any one time. What cost of capacity is ever in use at any one time. What cost should the manager expect when the plant is operating at should the manager expect when the plant is operating at full permissible capacity?full permissible capacity?

73

It may seem that we can answer this question by simply It may seem that we can answer this question by simply evaluating evaluating C(80), C(80), but attempting this evaluation results in but attempting this evaluation results in the meaningless fraction the meaningless fraction 0/0. 0/0. However, it is still possible However, it is still possible to evaluate to evaluate C(x) C(x) for values of for values of x x that that approachapproach 80 80 from the from the left (left (x<80x<80) and the right () and the right (x>80x>80), as indicated in this table:), as indicated in this table:

x approaches 80 from the left → ←x approaches 80 from the right

x 79.8 79.99 79.999 80 80.0001 80.001 80.04

C(x) 6.99782 6.99989 6.99999 7.000001 7.00001 7.00043

The values of The values of C(x) C(x) displayed on the lower line of this table displayed on the lower line of this table suggest that suggest that C(x) C(x) approaches the number approaches the number 77 as as xx gets closer gets closer and closer to and closer to 80. 80. The functional behavior in this example The functional behavior in this example can be describe by can be describe by 7)(lim

80

xC

x

74

LimitLimit: If : If f(x)f(x) gets closer and closer to a number gets closer and closer to a number LL as as xx gets gets closer and closer to closer and closer to cc from both sides, then from both sides, then LL is the limit of is the limit of f(x)f(x) as as xx approaches approaches cc. The behavior is expressed by writing. The behavior is expressed by writing

Lxf

cx

)(lim

75

Example 14

Solution:Solution:

Use a table to estimate the limit Use a table to estimate the limit

1

1)(

x

xxfLet and compute Let and compute f(x) f(x) for a succession of valuesfor a succession of values

of of x x approaching approaching 11 from the left and from the right. from the left and from the right.

1

1lim

1

x

xx

x→ 1 ← x x 0.99 0.999 0.9999 1 1.00001 1.0001 1.001

f(x) 0.50126 0.50013 0.50001 0.499999 0.49999 0.49988

The numbers on the bottom line of the table suggest that The numbers on the bottom line of the table suggest that f(x)f(x) approaches approaches 0.50.5 as as xx approaches approaches 11. That is . That is

5.01

1lim

1

x

xx

76

Three functions for which Three functions for which 4)(lim3

xfx

It is important to remember that limits describe the behavior It is important to remember that limits describe the behavior of a function of a function nearnear a particular point, not a particular point, not necessarilynecessarily at the at the point itself. point itself.

77

The figure below shows that the graph of two functions The figure below shows that the graph of two functions that do not have a limit as that do not have a limit as xx approaches approaches 22. .

Figure Figure (a)(a): The limit does not exist; Figure : The limit does not exist; Figure (b)(b): The : The function has no finite limit as function has no finite limit as xx approaches approaches 22. Such so-. Such so-called called infinite limitsinfinite limits will be discussed later. will be discussed later.

78

Properties of Limits thenexist, )(limand)(lim If xgxf

cxcx

)(lim)(lim)]()([lim xgxfxgxfcxcxcx

)(lim)(lim)]()([lim xgxfxgxfcxcxcx

constant any for )(lim)(lim kxfkxkfcxcx

)](lim)][(lim[)]()([lim xgxfxgxfcxcxcx

0)(lim if )(lim

)(lim]

)(

)([lim

xg

xg

xf

xg

xfcx

cx

cx

cx

exists )](lim[ if )](lim[)]([lim p

cx

p

cx

p

cxxfxfxf

79

For any constant For any constant k, k,

That is, the limit of a constant is the constant itself, and That is, the limit of a constant is the constant itself, and the limit of the limit of f(x)=xf(x)=x as as xx approaches approaches cc is is cc. .

cxkkcx

cx

lim and lim

80

Computation of Limits

)843(lim 3

1

xx

x 2

83lim

3

0

x

xx

Example 15

Find Find (a) (b) (a) (b) Solution:Solution:

a.a. Apply the properties of limits to obtainApply the properties of limits to obtain

98)1(4)1(38limlim4lim3)843(lim 3

11

3

1

3

1

xxxxxxxx

b.b. Since , you can use the quotient rule for Since , you can use the quotient rule for limits to get limits to get

0)2(lim0

xx

420

80

2limlim

8limlim3

2

83lim

00

0

3

03

0

xx

xx

x x

x

x

x

81

Limits of Polynomials and Rational FunctionsLimits of Polynomials and Rational Functions: If : If p(x)p(x) and and q(x)q(x) are polynomials, then are polynomials, then

andand

)()(lim cpxpcx

0)( if )(

)(

)(

)(lim

cqcq

cp

xq

xpcx

Example 16

Find Find 2

1lim

2

x

xx

Solution:Solution:

The quotient rule for limits does not apply in this case The quotient rule for limits does not apply in this case since the limit of the denominator is since the limit of the denominator is 00 and the limit of the and the limit of the numerator is numerator is 33. So the limit of the quotient does not exist. . So the limit of the quotient does not exist.

82

If and , then is said to be If and , then is said to be indeterminateindeterminate. The term . The term indeterminateindeterminate is used since the is used since the limit may or may not exist. limit may or may not exist.

Indeterminate Form （不定形）

)(

)(lim

xg

xfcx

0)(lim

xfcx

23

1lim

2

2

1

xx

xx

0)(lim

xgcx

Example 17

(a)(a) Find Find (b) (b) Find Find 1

1lim

1

x

xx

Solution:Solution:

a.a. 21

2

2

1lim

)2)(1(

)1)(1(lim

23

1lim

112

2

1

x

x

xx

xx

xx

xxxx

b. b. 2

1

1

1lim

1)1(

1lim

1)1(

11lim

1

1lim

1111

xxx

x

xx

xx

x

xxxxx

83

Limits Involving Infinity

Lxfx

)(lim

Limits at InfinityLimits at Infinity: If the value of the function : If the value of the function f(x)f(x) approach the approach the number number LL as as xx increases without bound, we write increases without bound, we write

Similarly, we write Similarly, we write

when the functional values when the functional values f(x)f(x) approach the number approach the number MM as as xx decreases without bound. decreases without bound.

Mxfx

)(lim

84

Reciprocal Power RulesReciprocal Power Rules: For constants : For constants AA and and kk, with , with k>0k>0

0lim and 0lim kxkx x

A

x

A

Example 18

FindFind 2

2

21lim

xx

xx

Solution:Solution:

5.0200

1

2lim/1lim/1lim

1lim

/2//1

/lim

21lim

22222

22

2

2

xxx

x

xx xxxxxxx

xx

xx

x

85

Procedure for Evaluating a Limit at Infinity of f(x)=p(x)/q(x)

Step 1. Divide each term in f(x) by the highest power xk that appears in the denominator polynomial q(x).

Step 2. Compute or using algebraic properties of limits and the reciprocal rules.

)(lim xfx

)(lim xfx

Exercise15

283lim

4

24

x

xxxx

Example 19

If a crop is planted in soil where the nitrogen level is If a crop is planted in soil where the nitrogen level is N, N, then the crop yield then the crop yield Y Y can be modeled by the can be modeled by the Michaelis-Michaelis-MentenMenten function function 0 )(

N

NB

ANNY

where where AA and and BB are positive constants. What happens to are positive constants. What happens to crop yield as the nitrogen level is increased indefinitely? crop yield as the nitrogen level is increased indefinitely?

x

xx

)sin(lim

((Optional Optional

Question!Question!))

86

Solution:Solution:

We wish to computeWe wish to compute

A

A

NB

ANNNB

NANNB

ANNY

N

N

NN

101/lim

//

/lim

lim)(lim

Thus, the crop yield tends toward the constant value Thus, the crop yield tends toward the constant value AA as as the nitrogen level the nitrogen level NN increases indefinitely. For this reason, increases indefinitely. For this reason, AA is called the is called the maximum attainable yieldmaximum attainable yield. .

87

Infinite Limits Infinite Limits （无穷极限）（无穷极限） : If : If f(x) f(x) increases or increases or decreases without bound asdecreases without bound as x x→→c, c, we have we have

)(limor )(lim xf xfcxcx

For example For example 22 )2(lim

x

xx

From the figure, we can From the figure, we can guest that guest that

22 )2(

limx

xx

88

Section 1.6 One-sided Limits and Continuity

If f(x) approaches L as x tends toward c from the left (x<c), we write

Lxfcx

)(lim

where where LL is called the is called the limit from the leftlimit from the left (or left- (or left-hand limit) hand limit) （左极限）（左极限）Likewise if Likewise if f(x)f(x) approaches approaches MM as as xx tends tends toward toward cc from the right from the right (x>c),(x>c), then then Mxf

cx

)(lim

MM is called the is called the limit from the rightlimit from the right (or right-hand (or right-hand limit.) limit.) （右极限）（右极限）

89

Example 20

For the function For the function

2 if 12

2 if 1)(

2

xx

xxxf

evaluate the one-sided limits and evaluate the one-sided limits and )(lim2

xfx

)(lim2

xfx

Solution:Solution:

Since for Since for x<2, x<2, we have we have 21)( xxf

3)1(lim)(lim 2

22

xxf

xx

Similarly, Similarly, f(x)=2x+1f(x)=2x+1 if if xx≥≥22, so , so 5)12(lim)(lim

22

xxf

xx

90

Existence of a LimitExistence of a Limit: The two-sided limit : The two-sided limit exists if and only if the two one-sided limits exists if and only if the two one-sided limits and exist and are equal, and then and exist and are equal, and then

)(lim2

xfx

)(lim2

xfx

)(lim2

xfx

)(lim)(lim)(lim222

xfxfxfxxx

Notice that the limit of the piecewise-define Notice that the limit of the piecewise-define function function f(x)f(x) in example in example 2020 does not exist, that is does not exist, that is

does not exist, since does not exist, since

)(lim2

xfx

)(lim)(lim22

xfxfxx

91

At x=1: 1

lim 0x

f x

1

lim 1x

f x

1 1f

Left-hand limit

Right-hand limit

value of the function

does not does not exist! exist!

Since the left and right Since the left and right hand limits are not hand limits are not equal.equal.

)(lim1

xfx

92

At x=2: Left-hand limit

Right-hand limit

value of the function

2

lim 1x

f x

2

lim 1x

f x

2 2f

does exist!does exist!

Since the left and right Since the left and right hand limits are equal, hand limits are equal, However, the limit is However, the limit is not equal to the value not equal to the value of function. of function.

)(lim2

xfx

93

At x=3: Left-hand limit

Right-hand limit

value of the function

3

lim 2x

f x

3

lim 2x

f x

3 2f

does exist!does exist!

Since the left and right Since the left and right hand limits are equal, hand limits are equal, and the limit is equal to and the limit is equal to the value of function.the value of function.

)(lim3

xfx

94

Nonexistent One-sided LimitsNonexistent One-sided LimitsA simple example is provided by the function A simple example is provided by the function

)/1sin()( xxf

As As xx approaches approaches 0 0 from either the left or the right, from either the left or the right, f(x)f(x) oscillates oscillates between between -1-1 and and 11 infinitely often. Thus neither one-sided limit at infinitely often. Thus neither one-sided limit at 00

exists. exists.

95

Continuity Continuity （连续性）（连续性）A continuous function is one whose graph can be A continuous function is one whose graph can be drawn without the “pen” leaving the paper. (no holes or drawn without the “pen” leaving the paper. (no holes or

gaps )gaps )

96

A “hole “ at A “hole “ at x=cx=c

97

A “gap” at A “gap” at x=cx=c

98

So what properties will guarantee that So what properties will guarantee that f(x)f(x) does not does not have a “hole” or “gap” at have a “hole” or “gap” at x=cx=c??

ContinuityContinuity: A function : A function ff is continuous at is continuous at c c if all three of if all three of these conditions are satisfied:these conditions are satisfied:

a.a.

b.b.

c.c.

If If f(x)f(x) is not continuous at is not continuous at cc, it is said to have a , it is said to have a discontinuity there. discontinuity there.

exists )(lim xfcx

)()(lim cfxfcx

defined is )(cf

99

f(x)f(x) is continuous at is continuous at x=3x=3 because the left because the left and right hand limits and right hand limits exist and equal to exist and equal to f(3).f(3).

At x=1:At x=1:

At x=2:At x=2:

At x=3:At x=3:

)(lim)(lim11

xfxfxx

)2()(lim)(lim22

fxfxfxx

)3()(lim)(lim33

fxfxfxx

Discontinuous

Discontinuous

Continuous

100

Continuity Polynomials and Rational FunctionsContinuity Polynomials and Rational Functions

If p(x) and q(x) are polynomials, then

)()(lim cpxpcx

0)( if )(

)(

)(

)(lim

cqcq

cp

xq

xpcx

A polynomial or a rational function is continuous A polynomial or a rational function is continuous wherever it is definedwherever it is defined

101

Example 21

Show that the rational function is continuous at x=3. 2

1)(

x

xxf

Solution:Solution:

Note that f(3)=(3+1)/(3-2)=4, since , you will find that

0)2(lim3

xx

)3(41

4

)2(lim

)1(lim

2

1lim)(lim

3

3

33f

x

x

x

xxf

x

x

xx

as required for f(x) to be continuous at x=3, since the three criteria for continuity are satisfied.

102

Example 22

Determine where the function below is not continuous.Determine where the function below is not continuous.

Solution:Solution:

Rational functions are continuous everywhere except Rational functions are continuous everywhere except where we have division by zero.where we have division by zero.

The function given will not be continuous at The function given will not be continuous at t=-3t=-3 and and t=5t=5..

103

Exercise

Discuss the continuity of each of the following functions

1 if 2

1 if 1)( .

1

1)( .

1)( .

2

xx

xxxhc

x

xxgb

xxfa

104

Example 23

For what value of the constant A is the following function For what value of the constant A is the following function continuous for all real x?continuous for all real x?

1 xif 43

1 if 5)( 2 xx

xAxxf

Solution:Solution:

Since Since Ax+5Ax+5 and are both polynomials, it follows and are both polynomials, it follows that that f(x)f(x) will be continuous everywhere except possibly at will be continuous everywhere except possibly at x=1x=1 . According to the three criteria for continuity, we . According to the three criteria for continuity, we have have

432 xx

This means that This means that ff is continuous for all is continuous for all xx only when only when A=-3A=-3

3)1(25)1()(lim)(lim11

AfAfxfxfxx

105

Example 24

Find numbers Find numbers aa and and b b so that the following function is so that the following function is continuous everywhere. continuous everywhere.

1 if

11 if

1 if

)( 2

xbx

xbax

-xax

xf

Solution:Solution:

Since the “parts” Since the “parts” ff are polynomials, we only need to are polynomials, we only need to choose choose aa and and b b so that so that ff is continuous at is continuous at x=-1x=-1 and and 1.1. At x=-1At x=-1 121)1()(lim)(lim

11

babaafxfxf

xx

At x=1At x=1 121)1()(lim)(lim11

babbafxfxfxx

We have We have a=-1/3 a=-1/3 and and b=1/3 b=1/3 for for ff is continuous everywhere is continuous everywhere

106

Continuity on an Interval A function A function f(x) f(x) is said to be continuous on an open interval is said to be continuous on an open interval

a<x<ba<x<b if it is continuous if it is continuous at each point x=c in that interval. ff is continuous on closed interval is continuous on closed interval a≤x≤ba≤x≤b, if it continuous , if it continuous

on the open interval on the open interval a<x<ba<x<b and and

is continuous on [-1,1]

)()(lim afxfax

)()(lim bfxfbx

21)( xxf

107

Example 25

Discuss the continuity of the function on the Discuss the continuity of the function on the open interval open interval -2<x<3-2<x<3 and on the closed interval and on the closed interval -2≤x≤3-2≤x≤3

3

2)(

x

xxf

Solution:Solution:

The rational function The rational function f(x)f(x)is continuous for all is continuous for all xx except except x=3x=3. Therefore, it is continuous on the open interval . Therefore, it is continuous on the open interval -2<x<3 -2<x<3 but not on the closed intervalbut not on the closed interval -2≤x≤ -2≤x≤3,since it is 3,since it is discontinuous at the endpoint 3 (where its denominator is discontinuous at the endpoint 3 (where its denominator is zero). The graph of zero). The graph of ff is shown in below Figure. is shown in below Figure.

108

The Intermediate Value Property

Suppose that Suppose that f(x)f(x) is continuous on the interval is continuous on the interval a≤x≤b a≤x≤b and and LL is a number between is a number between f(a)f(a) and and f(b), f(b), then there exists a then there exists a number number c c betweenbetween a a andand b, b, such that such that f(c)=Lf(c)=L. .

109

Corollary

If f is continuous on the closed interval [a,b], and f(a) and f(b) have opposite signs, then there exists a number c in (a,b) where f(c)=0.

110

Example 26

Show that the equation has a solution for Show that the equation has a solution for 1<x<2 1<x<2

1

112

xxx

Solution:Solution:

Let . Then Let . Then f(1)=-3/2f(1)=-3/2 and and f(2)=2/3f(2)=2/3. Since . Since f(x)f(x) is continuous for is continuous for 1≤x≤2, 1≤x≤2, it it follows from the intermediate follows from the intermediate value property that the graph value property that the graph must cross the x axis somewhere must cross the x axis somewhere between between x=1x=1 and and x=2x=2. .

1

11)( 2

x

xxxf

111

Summary Function:Function: domain and range of a function domain and range of a function composition of function composition of function f(g(x))f(g(x))

Graph of a functionGraph of a function: : xx and and yy intercepts, intercepts, Piecewise-defined function, power functionPiecewise-defined function, power function Polynomial, Rational function, Vertical line testPolynomial, Rational function, Vertical line test Linear functionLinear function: : Slope, Slope-intercept formula, point-slope formulaSlope, Slope-intercept formula, point-slope formula Parallel and perpendicular lines. Parallel and perpendicular lines.

112

Function Models:Function Models: Market equilibrium: law of supply and demandMarket equilibrium: law of supply and demand Shortage and surplus, Break-even analysisShortage and surplus, Break-even analysis

Limits:Limits: Calculation of limits: limits of polynomial and Calculation of limits: limits of polynomial and rational function, limits at infinity: calculation of rational function, limits at infinity: calculation of limits at the infinity (Reciprocal power Rules), limits at the infinity (Reciprocal power Rules),

Infinite limit, One sided limit, Existence of limitInfinite limit, One sided limit, Existence of limit Continuity Continuity of of f(x)f(x) at at x=cx=c:: Continuity on an interval, Continuity of Continuity on an interval, Continuity of polynomials polynomials and rational function, Intermediate value propertyand rational function, Intermediate value property

Lxfcx

)(lim