cal1 iu slides(2ndsem09-10) chapter1 sv

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CALCULUS I

Dr. Nguyen Ngoc Hai

DEPARTMENT OF MATHEMATICS

INTERNATIONAL UNIVERSITY, VNU-HCM

April 19, 2010

Dr. Nguyen Ngoc Hai CALCULUS I

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References

Main textbook:

J. Steward, Calculus. Concepts and Contexts , 5th ed., ThomsonLearning, 2001.

Other textbooks:

[1.] R.N. Greenwell, N.P. Rithchey, M.L. Lial, Calculus withApplications for the Life Sciences , Pearson Education, 2003.

[2.] J. Rogawski, Calculus, Early Transcendentals , W. H. Freeman,2008.


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Chapter 1. FUNCTI NS, LIMITSAND CONTINUITY

Contents

[1.] What is Calculus?

[2.] Straight Lines. Equations of Lines

[3.] Functions and Graphs[4.] New Functions from Old Functions. Inverse Functions

[5.] Parametric Curves

[6.] Limits of Functions. One-sided Limits

[7.] Laws of Limits. Evaluating Limits

[8.] Continuity. The Intermediate Value Theorem

[9.] Limits Involving Infinity


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Ch 1 FUNCTI NS LIMITS



Chapter 1 FUNCTI NS, LIMITSAND CONTINUITY

1.1 WHAT IS CALCULUS?

Questions:

1. Why do the planets move in elliptical orbits around the sun?

2. How do radio waves propagate through space?

3. How can one predict the effects of interest rate changes oneconomies and stock markets?

4. Why does an epidemic spread faster and faster and then slow

down?

These and many other questions of interest and importance in ourworld relate directly to our ability to analyze motion and howquantities change with respect to time or each other.


Ch 1 FUNCTI NS LIMITS

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• Kepler described how the solar system worked. He didn’t knowwhy.

• Calculus and Newton’s laws explained why it worked that way.

• Algebra and geometry are useful tools for describing relationshipsamong static quantities, but they do not involve conceptsappropriate for describing how a quantity changes.

• Calculus provides the tools for describing motion quantitatively.It introduces two new operations called differentiation andintegration.


Ch t 1 FUNCTI NS LIMITS

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• Differential calculus dealt with the problem of calculatingrates of change.

• Integral calculus dealt with the problem of determining afunction from information about its rate of change.

•Calculus is the mathematics of motion and change.

• John von Neumann (1903-1957) wrote: “The calculus was thefirst achievement of modern mathematics”.


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HOW TO LEARN CALCULUS?

• Calculus introduces so many new concepts and computational

operations.

What should you do to learn?

1. Read the text carefully. Read the relevant passages in the

textbook and work through the examples step by step. Readand search for detail in a step by step logical fashion. It takesattention, patience, and practice.

2. Complete the homework exercises, keeping the following

principles in mind.a) Sketch a diagram whenever possible.b) Write your solution in a connected step-by-step logical fashion,

as if you were explaining to someone else.


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1.2 STRAIGHT LINES. EQUATIONS OF LINES1.2.1 STRAIGHT LINES

Linear functions are the simplest of all functions and their graphs(lines) are the simplest of all curves.

However, linear functions and lines play an enormously importantrole in calculus.


1 2 STRAIGHT LINES EQUATIONS OF LINES

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Slopes of Nonvertical Lines

If a particle moves from (x 1, y 1) to (x 2, y 2), the increments in itscoordinates are

∆x = x 2 − x 1 and ∆y = y 2 − y 1.

Let L be a nonvertical line in the plane. Let P 1(x 1, y 1) andP 2(x 2, y 2) be points on L.

Definition 2.1

The slope of a nonvertical line is

m =∆y

∆x =

y 2 − y 1x 2 − x 1



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• The slope of a horizontal line is zero since ∆y = 0.

• The slope of a vertical line is undefined.



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1.2 STRAIGHT LINES. EQUATIONS OF LINES1.2.2 EQUATIONS OF LINES

Definition 2.2

The equation

y − y 1 = m(x − x 1)

is the point-slope equation of the line that passes throughthe point (x 1, y 1) with slope m.

Example 2.1 Write an equation for the line that passes throughthe point (2, 3) with slope −3/2.

Example 2.2 Write an equation for the line through (−2,−1)and (3, 4).



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Slope-Intercept Equations

Definition 2.3

The equationy = mx + b

is the slope-intercept equation of the line with slope m andy -intercept b .

Example 2.3 The standard equation for converting Celsiustemperature to Fahrenheit temperature is a slope-interceptequation. If 0◦C corresponds to 32◦F (the freezing point of water)and 100◦C corresponds to 212◦F (the boiling point of water at seelevel), represent Fahrenheit temperature F as a function of Celsiustemperature C .



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Parallel and Perpendicular Lines

Parallel lines have equal angles of inclination. Thus,

Two lines are parallel if and only if they have the same slope,or if they are both vertical.

Example 2.4 Find the equation of the line that passes throughthe point (3, 5) and is parallel to the line 2x + 5y = 4.

Two lines are perpendicular if and only if the product of their

slopes is −1 or, if one is vertical and the other horizontal.

Example 2.5 Find the slope of any line L perpendicular to theline having the equation 5x − y = 4.


1 3 FUNCTIONS AND GRAPHS

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1.3 FUNCTIONS AND GRAPHS1.3.1 FUNCTIONS

In many practical situations, the value of one quantity may dependon the value of a second.

• For example, the area A of a circle depends on the radius r of the circle. The rule that connects A and r is given by A = πr 2.

• The human population of the world P depends on the time t .For instance,

P (1980) = 4.45 billions,

P (1990) = 5.28 billions,

P (2000) = 6.070 billions.

Such relationships can often be represented mathematically asfunctions .



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Definition 3.1

A function from a set A to a set B is a rule that assigns toeach element in A a single element of B .

The set A is called the domain of the function.

The set of all possible values of the function is called the range.

• We usually consider functions for which the sets A and B aresets of real numbers .

•To denote that y is a function of x we write

y = f (x )

The number f (x ) is the value of f at x .



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• A symbol that represents an arbitrary number in the domain of afunction f is called an independent variable.

• A symbol that represents a number in the range of f is called a

dependent variable.

So

The set of all possible values of the independent variable in afunction is its domain, and the resulting set of all possible

values of the dependent variable is the range.



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Example 3.1 Which of the following are functions?

(a)

(b) The key x 2 on a calculator.

(c) The set of order pairs with first elements children and secondelements their birth mothers.

(d) The set of order pairs with first elements mothers and secondelements their children.



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Representations of functions

There are four possible ways to represent a function

• verbally (by a description in words)• numerically (by a table of values)

•visually (by a graph)

• algebraically (by an explicit formula)

For instance, the most useful representation of the area of a circleas a function of its radius is probably the algebraic formulaA = πr 2.

• In most cases in this course, a function is expressed as anequation, such as C (x ) = 5x − 2 +

√ x 2 − 1.

• When an equation is given for a function, we say that theequation defines the function.



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Example 3.2 Let g (x ) = −x 2 + 4x − 5.Find each of the following

(a) g (3)

(b) g (a)

(c) g (x + h)

(d) g ( 2r

)

(e) Find all values of x such that g (x ) = −2.



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Agreement on Domains

When a function f is defined without specifying its domain,we assume that the domain consist of all real numbers x forwhich the value f (x ) of the function is a real number.

Example 3.3 Find the domain and range for each of thefunctions defined as follows.

(a) f (x ) = √ x 2 − 5x + 6 (b) g (t ) = t t 2−1

.


1.3 FUNCTIONS AND GRAPHS

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Piecewise Defined Functions

Example 3.4 The absolute value function f (x ) = |x | is definedby

f (x ) =

|x

|= x if x ≥ 0

−x if x < 0

Example 3.5 The signum function is defined by

sgn(x ) =

1 if x > 0

0 if x = 0

−1 if x < 0



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1.3 FUNCTIONS AND GRAPHS1.3.2 GRAPHS OF EQUATIONS AND FUNCTIONS

Graphs of Equations

Each point in the plane corresponds to an ordered pair of numbers.

• The first member is called the first coordinate of the point, andthe second member is called the second coordinate. Together,

these are called the coordinates of the point.

• In the xy -plane, the vertical line is often called the y -axis, andthe horizontal line is often called the x -axis.

Definition 3.2

The graph of an equation is a drawing that represents all thesolutions of the equation.



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3 U C O S G S1.3.2 GRAPHS OF EQUATIONS AND FUNCTIONS

Graphs of Functions

Definition 3.3

If f is a function with domain A, then its graph is the set of all order pairs

x , f (x ) | x ∈ A

In other words, the graph of f consists of all points (x , y ) in thexy -plane such that y = f (x ) and x is in the domain of f .

Example 3.6 Sketch the graph of

(a) x 2 + y 2 = 4 (b) y = x 2 (c) x = y 2.



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1.3.2 GRAPHS OF EQUATIONS AND FUNCTIONS

• The graph of a function is a curve in the xy -plane.

The Vertical Line Test

Vertical Line Test A curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects the curve more than one .



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1.3.3 EVEN AND ODD FUNCTIONS. SYMMETRY AND REFLECTIONS

Definition 3.4

Suppose that −x belongs to the domain of f whenever x does. We say that f is an even function if

f (−x ) = f (x ) for every x in the domain of f .

We say that f is an odd function if

f (−x ) = −f (x ) for every x in the domain of f .

• The graph of an even function is symmetric about the y axis .

• The graph of an odd function is symmetric about the origin.If an odd function f is defined at x = 0, then f (0) = 0.



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Example 3.7 Determine whether the function is even, odd, or

neither.(a) f (x ) = x 6 (b) g (x ) = 1

x (c) h(x ) = x 3 + x 2.



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Increasing and Decreasing Functions

Definition 3.5

A function f is called increasing on an interval I if

f (x 1) < f (x 2) for all x 1, x 2 ∈ I such that x 1 < x 2.

It is called decreasing on I if

f (x 1) > f (x 2) for all x 1, x 2 ∈ I such that x 1 < x 2.


1.4 NEW FUNCTIONS FROM OLD FUNCTIONSINVERSE FUNCTIONS

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INVERSE FUNCTIONS1.4.1 TRANSFORMATIONS OF FUNCTIONS

Two important ways of modifying a graph are translation (orshifting) and scaling.

Vertical and horizontal translation

Suppose c > 0. To obtain the graph of y = f (x ) + c , shift the graph of y = f (x ) a distance c unitsupward;

y = f (x ) − c , shift the graph of y = f (x ) a distance c units

downward;y = f (x − c ), shift the graph of y = f (x ) a distance c units to theright;

y = f (x + c ), shift the graph of y = f (x ) a distance c units to the

left. Dr. Nguyen Ngoc Hai CALCULUS I


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Example 4.1 Given the graph of

y = f (x ) =1

x 2 + 1,

use transformation to graph

y =x 2 + 2

x 2 + 1, y =

−2x 2 − 1

x 2 + 1, y =

1

(x + 1)2 + 1.

Example 4.2 Sketch the graph of the functionf (x ) = x 2 + 4x − 5 using the graph of y = x 2.



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Given a line L and a point P not on L, we call a point Q thereflection of P in L if L is the right bisector of the line segment

PQ .

The reflection of any graph G in L is the graph consisting of thereflections of all of the points of G.



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Vertical and Horizontal Stretching and Reflecting

Suppose that c > 1. To obtain the graph of

y = cf (x ), stretch the graph of y = f (x ) vertically by a factorof c ;

y = 1c f (x ), compress the graph of y = f (x ) vertically by a

factor of c ;

y = f (cx ), compress the graph of y = f (x ) horizontally by afactor of c ;

y = f ( x c ), stretch the graph of y = f (x ) horizontally by afactor of c ;

y = −f (x ), reflect the graph of y = f (x ) about the x -axis;

y = f (

−x ), reflect the graph of y = f (x ) about the y -axis;



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Example 4.3 Sketch the graph of the following functions(a) y = sin 5x (b) y = 3− sin2x (c) y = | ln x |.



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INVERSE FUNCTIONS1.4.2 SUMS, DIFFERENCES, PRODUCTS QUOTIENTS, AND MULTIPLES

Definition 2.1If f and g are functions, then for every x that belongs to thedomains of both f and g we define functions f + g , f − g , fg ,f /g by the formulas:

(f + g )(x ) = f (x ) + g (x )

(f − g )(x ) = f (x ) − g (x )

(fg )(x ) = f (x )g (x )

f

g

(x ) =

f (x )

g (x ) , where g (x ) = 0.

In particular, if c is a real number, then the function cf isdefined for all x in the domain of f by (cf )(x ) = c · f (x ).



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INVERSE FUNCTIONS1.4.2 SUMS, DIFFERENCES, PRODUCTS QUOTIENTS, AND MULTIPLES

Example 4.4 If f (x ) = √ x and g (x ) =

√ 4 − x

2

, find thefunctions 6f , f + g , f − g , fg , and f /g , and specify the domainsof each of these functions.



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INVERSE FUNCTIONS1.4.3 COMPOSITE FUNCTIONS

Definition 4.2

Given two functions f and g , the composite function f ◦ g (also called the composition of f and g ) is given by

f ◦ g

(x ) = f

g (x )

.

• The domain of f ◦ g is the set of all x in the domain of g forwhich g (x ) is in the domain of f .

If the range of g is contained in the domain of f then the domainof f ◦ g is just the domain of g .



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INVERSE FUNCTIONS1.4.3 COMPOSITE FUNCTIONS

Example 4.5 If f (x ) = x + 1 and g (x ) =√

4 − x 2, calculate thefour composite functions f ◦ g , f ◦ f , g ◦ g , and g ◦ f , and specifythe domain of each.

Note In general f ◦ g = g ◦ f .

Example 4.6 Given F (x ) =

2 + cos(x 2 + 1), find functionsf , g and h such that F = f ◦ g ◦ h.



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INVERSE FUNCTIONS1.4.4 THE BASIC CLASSES OF FUNCTIONS

Polynomials

Definition 4.3

• For any real number α, the function f (x ) = x α is called thepower function with exponent α.

•A function P is called a polynomial if

P (x ) = anx n + an−1x n−1 + · · ·+ a1x + a0

where n is a nonnegative integer number and the numbersan, an−1,..., a0 are constants.

The numbers an, an−1,..., a0 are called coefficients.

The degree of P is n (assuming that an = 0).

The coefficient an is called the leading coefficient.

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• A polynomial of degree 1 is of the form f (x ) = ax + b and soit is a linear function.

• A polynomial of degree 2 is called a quadratic function. Itsgraph is always a parabola.

• A polynomial of degree 3 is of the form

P (x ) = ax 3

+ bx 2

+ cx + d (a = 0)

and is called a cubic function.



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Rational Functions

Definition 4.4

A rational function is a quotient of two polynomials

f (x ) =P (x )

Q (x ).

Every polynomial is also a rational function (with Q (x ) = 1).

• The domain of a rational function P (x )Q (x ) is the set {x | Q (x ) = 0}.



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Algebraic Functions

Definition 4.5

A function is called an algebraic function if it can beconstructed using algebraic operations (such as addition,

substraction, multiplications, division, and taking roots)starting with polynomials.

For example,

f (x ) =

x 2 − 5x + 6, g (x ) =x 7 − x 2 + 3

x 4 −√ x 2 − 1+ (x − 1) 6

√ 2x + 1

are algebraic functions.



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INVERSE FUNCTIONS1.4.5 INVERSE FUNCTIONS

Definition 4.7

A function f is called a one-to-one function if f (x 1) = f (x 2)whenever x 1 and x 2 belong to the domain of f and x 1 = x 2.

In other words, a function is one-to-one if it never takes on thesame values twice, that is, fore every value c , the equationf (x ) = c has at most one solution for x .

An equivalent statement is that

f is one-to-one if

f (x 1) = f (x 2) =⇒ x 1 = x 2

.



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INVERSE FUNCTIONS1.4.5 INVERSE FUNCTIONS

The Horizontal Line Test

Horizontal Line Test A function is one-to-one if and only

if no horizontal line intersects its graph more than one.

Example 4.7 Which of the following functions is one-to-one?

(a) f (x ) = x 2 (b) g : [0,∞) → IR , g (x ) = x 2



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1.4.5 INVERSE FUNCTIONS

Definition 4.8Let f be a one-to-one function with domain D and range E .Then its inverse function f −1 has domain E and range D and defined by

f −1(y ) = x ⇐⇒ f (x ) = y

for any y in E .

Note that

domain of f −1 = range of f range of f −1 = domain of f



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Example 4.8 Find the inverse of f (x ) =√

2x + 1.

How to find the inverse function of f

1. Solve the equation y = f (x ) for x in terms of y (if possible).

2. Interchange x and y . The resulting equation will be

y = f −1(x ).



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Notef −1

f (x )

= x for every x in D

f f −1(x ) = x for every x in R

The graph of f −1 is obtained by reflecting the graph of f

about the line y = x.


SOME IMPORTANT INVERSE FUNCTIONS

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Exponential and Logarithmic Functions

An exponential function is a function of the form f (x ) = ax ,where a > 0 and a = 1. The number a is called the base.

• Exponential functions are positive: ax > 0 for all x .

• f (x ) = ax is increasing if a > 1 and decreasing if a < 1.

• The domain of an exponential function is IR = (−∞,∞) andthe range is (0, ∞).



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Definition 4.9

If a > 0 and a = 1, then the logarithm to the base a,denoted loga x , is the inverse of f (x ) = ax .

y = loga x ⇐⇒ x = ay

• the domain of loga x is (0,∞).

•the range of loga x is the set of all real number IR .

• f (x ) = loga x is increasing if a > 1 and decreasing if a < 1.



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Inverse Trigonometric Functions

• The function f (x ) = sin x is one-to-one on [−π/2, π/2].

• Its inverse is called the inverse sine function or the arcsinefunction and denoted sin−1 x or arcsin x :

y =sin−1 x is the unique angle in [−π2

, π2

] such that sin y = x

y = sin−1 x ⇐⇒ sin y = x and − π

2≤ x ≤ π

2

• The domain of sin−1 x is [−1, 1] and the range is [−π/2, π/2].



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• The cosine function is one-to-one on [0, π].

• Its inverse is called the inverse cosine function or the arccosfunction and denoted cos−1 x or arccos x :

y =cos−1

x is the unique angle in [0, π] such that cos y = x y = cos−1 x ⇐⇒ cos y = x and 0 ≤ x ≤ π

•The domain of cos−1 x is [

−1, 1] and the range is [0, π].



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• The tangent function is one-to-one on the interval (−π/2, π/2).

• The inverse is called the inverse tangent function and isdenoted tan−1 x or arctan x :

y =tan−1 x is the unique angle in−π2 , π2

such that tan y = x

y = tan−1 x ⇐⇒ tan y = x and − π

2< x <

π

2



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Similarly,

y =cot−1 x is the unique angle in (0, π) such that cot y = x

y = cot−1 x ⇐⇒ cot y = x and 0 < x < π

• tan−1 x and cot−1 x have domain IR .

• The range of tan−1 x is (−π/2, π/2).

• The range of cot−1

x is (0, π).


ELEMENTARY FUNCTIONS

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• New functions may be produced using the operations of addition,multiplication, division, as well as composition, extraction of roots,and taking inverses.

• It is convenient to refer to a function constructed in this wayfrom the basic functions listed above as an elementary function.


1.5 PARAMETRIC CURVES

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Definition 5.1

If x and y are given as functions

x = f (t ), y = g (t )

over an interval I of t −values, then the set of points(x , y ) = f (t ), g (t ) defined by these equations is a curve in

the coordinate plane. The equations are parametricequations for the curve. The variable t is a parameter forthe curve and its domain I is the parameter interval. If I is aclosed interval, a ≤ t ≤ b , the point

f (a), g (a)

is the initial

point of the curve and f (b ), g (b ) is the terminal point of

the curve. When we give parametric equations and aparameter interval for a curve in the plane, we say that wehave parametrized the curve. The equations and intervalconstitute a parametrization of the curve.



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• In many applications t denotes time, but it might instead denotean angle or the distance a particle has traveled along its path from

its starting point.

• We could use a letter other than t for the parameter.

Example 5.1 The Unit Circle x 2 + y 2 = 1. What curve isrepresented by the parametric equations

x = cos t , y = sin t , 0≤

t ≤

2π?

Since the curve starts and ends at the same point, it is called aclosed curve.



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Example 5.2 A parametrization of the Ellipse x 2

a2 + y 2

b 2= 1.

Describe the motion of a particle whose position P (x , y ) at time t

is given by

x = a cos t , y = b sin t , 0 ≤ t ≤ 2π.

Example 5.3 A parametrization of the Circle x 2 + y 2 = R 2.The equations and parameter interval

x = R cos t , y = R sin t , 0≤

t ≤

2π

obtained by taking b = a = R in the previous example, describethe circle x 2 + y 2 = R 2.



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Example 5.4 Cycloids. A wheel of radius a rolls (withoutslipping) along a horizontal straight line. Find parametricequations for the path traced by a point P on the wheel’scircumference. The path is called a cycloid.

Note The graph of a function y = f (x ) can always beparametrized as

x = t

y = f (t )


1.6 LIMITS1.6.1 LIMITS OF FUNCTIONS

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Calculus is based on the fundamental concept of the limit of afunction.

It is this idea of limit that distinguishes calculus from algebra,geometry, and trigonometry, which are useful for describing staticsituations.



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Example 6.1 Describe the behaviour of the function

f (x ) =x 2

−1

x − 1

near x = 1.



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Definition 6.1

We writelimx →a

f (x ) = L

and say “the limit of f (x ) as x approaches a equals L” if we can make the values of f (x ) arbitrarily close to L (as close

to L as we like) by taking x to be sufficiently close to a (oneither side of a) but not equal to a. We also say that f (x )approaches L or converges to L as x approaches a.

An alternative notation for limx →a f (x ) = L is

f (x ) → L as x → a

which is usually read “f (x ) approaches L as x approaches a”.



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Example 6.2

(a) limx →a x = a

(b) limx →a c = c (where c is a constant).

Example 6.3 Investigate

limx →0

sinπ

x .


1.6 LIMITS1.6.2 ONE-SIDED LIMITS

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The limit we have discussed so far are two-sided.

In some instances, f (x ) may approach L from one side of awithout necessarily approaching it from the other side, or f (x ) maybe defined on only one side of a.

Example 6.4 The Heaviside function H is defined by

H (t ) =

0 if t < 0

1 if t ≥ 0

As t approaches 0 from the left, H (t ) approaches 0. As t approaches 0 from the right, H (t ) approaches 1. There is no singlenumber that H (t ) approaches as t approaches 0. Therefore,limt →0 H (t ) does not exist.



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Definition 6.2

We write

limx →a− f (x ) = L

and say “the left-hand limit of f (x ) as x approaches a (orthe limit of f (x ) as x approaches a from the left) equalsL” if we can make the values of f (x ) as close to L as we want

by taking x to be sufficiently close to a and x less than a. Wealso say that f (x ) has left limit L at x = a.

Similarly, if we require that x be greater than a, we get “the

right-hand limit of f (x ) as x approaches a is equal to L” (orf (x ) has right limit L at x = a), and we write

limx →a+

f (x ) = L.



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Theorem 6.1

A function f (x ) has limit L at x = a if and only if it has bothleft and right limits there and these one-sided limits are both

equal to L:

limx →a

f (x ) = L ⇐⇒ limx →a−

f (x ) = limx →a+

f (x ) = L.



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Example 6.5 If

f (x ) =|x − 2|

x 2 + x − 6,

find limx →2+ f (x ), limx →2− f (x ), and limx →2 f (x ).


1.7 LAWS OF LIMITS. EVALUATING LIMITS1.7.1 LAWS OF LIMITS

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Theorem 7.1Suppose that c is a constant and the limits limx →a f (x ) and limx →a g (x ) exist. Then

1. limx →a f (x ) + g (x ) = limx →a f (x ) + limx →a g (x )

2. limx →a

f (x ) − g (x )

= limx →a f (x ) − limx →a g (x )

3. limx →a

cf (x )

= c limx →a f (x )

4. limx →a f (x )g (x )

= limx →a f (x ) · limx →a g (x )

5. limx →a f (x )g (x ) = limx →a f (x )limx →a g (x ) if limx →a g (x ) = 0.


1.7 LAWS OF LIMITS. EVALUATING LIMITS1.7.1 LAWS OF LIMITS

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6. limx →a f (x )]n = limx →a f (x )n

,

where n is a positive integer.

7. limx →an

f (x ) = n

limx →a f (x ),where n is a positive integer.

The Limit Laws also hold for one-sided limits.

Example 7.1 Find

limt →0

√ t 2 + 9 − 3

t 2.


1.7 LAWS OF LIMITS. EVALUATING LIMITS1.7.2 THE SQUEEZE THEOREM

Theorem 7 2

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Theorem 7.2

If f (x ) ≤ g (x ) when x is near a (except possibly at a) and the

limits of f and g both exist as x approaches a, then

limx →a

f (x ) ≤ limx →a

g (x ).

Theorem 7.3 (The Squeeze Theorem)If f (x ) ≤ g (x ) ≤ h(x ) when x is near a (except possibly at a)and limx →a f (x ) = limx →a h(x ) = L, then

limx →a

g (x ) = L.

Similar statements hold for left and right limits.

The Squeeze Theorem is sometimes called the SandwichTheorem or the Pinching Theorem.


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1.8 CONTINUITYTHE INTERMEDIATE VALUE THEOREM

1.8.1 CONTINUITY



Definition 8.1

A function f is continuous at a number a if

limx →a

f (x ) = f (a).

If f is not continuous at a, we say that f is discontinuous ata, or f has a discontinuity at a.

Notice that if f is continuous at a, then:

1. f (a) is defined, that is, a is in the domain of f ;

2. limx →a f (x ) exists;

3. limx →a f (x ) = f (a).



1.8.1 CONTINUITY

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Definition 8.2

A function f is continuous from the right at a number a if

limx →a+

f (x ) = f (a)

and f is continuous from the left at a if

limx →a−

f (x ) = f (a).

For example, the Heaviside function

H (x ) =

0 if x < 01 if x ≥ 0

is continuous at every number x except 0. It is right continuousat 0 but is not left continuous or continuous there.



1.8.1 CONTINUITY

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Definition 8.3A function f is continuous on an interval if it is continuousat every number in the interval. If f is continuous at all pointsin its domain, then f is simply called continuous.

Here, if f is defined only on one side of an endpoint of the interval,we understand continuous at the endpoint to mean “continuousfrom the right” or “continuous from the left.”

Example 8.1 Show that the function f (x ) =√

1 − x 2 iscontinuous.



1.8.1 CONTINUITY

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Theorem 8.1If f and g are continuous at a, and c is a constant, then the following functions are also continuous at a:(a) f ± g (b) cf (c) fg (d) f /g if g (a)

= 0.

Corollary 8.1

(a) Any polynomial is continuous everywhere, that is, it is

continuous on IR = (−∞,∞).(b) Any rational function is continuous wherever it is defined-that is, it is continuous on its domain.


1.8 CONTINUITYTHE INTERMEDIATE VALUE THEOREM1.8.1 CONTINUITY

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Example 8.2 Where are each of the following functions

discontinuous?

(a) f (x ) =

x 2−x −2x −2 if x = 2

1 if x = 2

(b) g (x ) = 1

x 2if x

= 0

1 if x = 0

(c) h(x ) =

x 2 if x ≤ 0

x + 1 if x > 0

• The kind of discontinuity illustrated in part (a) is calledremovable.

• The discontinuity in part (b) is called an infinite discontinuity.

•The discontinuities in part (c) are called jump discontinuities.



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Theorem 8.2

The following types of functions are continuous at every number in their domains: Polynomials, Rational functions,Root functions, Trigonometric functions.

Example 8.3 Where is the function

f (x ) =

ln x + tan−1 x

x 2 − 1

continuous?



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Theorem 8.3

If f is continuous at b and limx →a g (x ) = b, thenlimx →a f

g (x )

= f (b ). In other words,

limx →a

f

g (x )

= f

limx →a

g (x )

.

Theorem 8.4

If g is continuous at a and f is continuous at g (a), then the composite function f ◦ g given by (f ◦ g )(x ) = f

g (x )

is

continuous at a.

Theorem 8.5

If f is continuous on an interval I with range J and if the inverse f −1 exists, then f −1 is continuo u s o n the d omai n J .


1.8 CONTINUITYTHE INTERMEDIATE VALUE THEOREM1.8.1 THE INTERMEDIATE VALUE THEOREM

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Theorem 8.6 (The Intermediate Value Theorem)

A function f that is continuous on a closed interval [a, b ]takes on every value between f (a) and f (b ).

A point c where f (c ) = 0 is called a zero or root of f .

Corollary 8.2 (Existence of Zeros)

If f is continuous on [a, b ] and if f (a) and f (b ) have opposite signs, that is, f (a)f (b ) < 0, then f has a zero in (a, b ).

Example 8.4 Show that the equation x 3 − x − 1 = 0 has asolution in the interval [1, 2].


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1.9 LIMITS INVOLVING INFINITY1.9.1 INFINITE LIMITS



• “x → a−” means that we consider only values of x that are lessthan a.

•“x → a+

” means that we consider only values of x that aregreater than a.

• Keep in mind that ∞ and −∞ are not numbers.


1.9 LIMITS INVOLVING INFINITY1.9.1 INFINITE LIMITS

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The line x = a is called a vertical asymptote of the curvey = f (x ) if at least one of the following statements is true.

limx →a f (x ) = ∞ limx →a− f (x ) = ∞ limx →a+ f (x ) = ∞limx →a f (x ) =

−∞limx →a− f (x ) =

−∞limx →a+ f (x ) =

−∞

Example 9.1 Find the vertical asymptotes of

(a) f (x ) = ln x (b) g (x ) = tan x .


1.9 LIMITS INVOLVING INFINITY1.9.2 FINITE LIMITS AT INFINITY

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Definition 9.3

Let f be a function defined on some interval (a,∞). Then,

limx →∞

f (x ) = L

means that the values of f (x ) can be made arbitrarily close toL by taking x sufficiently large.

Another notation for limx →∞ f (x ) = L is

f (x ) → L as x →∞.



Definition 9.4

L f b f i d fi d i l ( ) Th

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Let f be a function defined on some interval (−∞, a). Then,

limx →−∞

f (x ) = L

means that the values of f (x ) can be made arbitrarily close toL by taking x sufficiently large negative.

Definition 9.5

The line y = L is called a horizontal asymptote of the curvey = f (x ) if either

limx →∞ f (x ) = L or limx →−∞ f (x ) = L.

• Most of the Limit Laws given in Section 1.7 also hold for limitsat infinity.

Dr Nguyen Ngoc Hai CALCULUS I


Example 9.2 If n is a positive integer, then

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limx →±∞

1

x n = 0.

Example 9.3 Evaluate

limx →±∞

20x 2 − 3x 3x 5 − 4x 2 + 5

.

Example 9.4 Find the horizontal and vertical asymptotes of the

graph of the function

f (x ) =

√ 2x 2 + 1

3x − 5.



Theorem 9.1

Th li it

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The limits

limx →−∞

1 + 1

x

x and lim

x →∞

1 + 1

x

x

exist and equal. This value is called the number e.

Thus we havelimt →0

(1 + t )1t = e

limt →

0

ln(1 + t )

t

= 1

and

limu →0

e u − 1

u = 1


1.9 LIMITS INVOLVING INFINITY1.9.3 INFINITE LIMITS AT INFINITY

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The notation limx →∞

f (x ) = ∞is used to indicate that the values of f (x ) become large as x becomes large.

Similar meanings are attached to the following symbols:

limx →∞

f (x ) = −∞ limx →−∞

f (x ) = ∞ limx →−∞

f (x ) = −∞



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Example 9.5 If a > 1 then

limx →−∞

ax = 0 and limx →∞

ax = ∞

Example 9.6 Calculate

(a) limx →±∞11x +2x 3−1

(b) limx →∞−4x 3+7x

2x 2−3x −10

(c) limx →−∞ −4x 3+7x 2x 2−3x −10



Asymptotic Behavior of a Rational Function The

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Asymptotic Behavior of a Rational Function The

asymptotic behavior of a rational function depends only onthe leading terms of its numerator and denominator. Supposean, b m = 0 and

L = limx →±∞

anx n + an−1x n−1 + · · · + a0

b mx m

+ b m−1x m−1

+ · · · + b 0

.

• If m > n, then L = 0.

•If m = n, then L = an/b m.

• If m < n, then L = ±∞, depending on the signs of numerator and denominator.


Exercises and Assignments

Text book: J. Stewart, Calculus. Concepts and Contexts , 2nd,Thomson Learning 2001

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Thomson Learning, 2001.

Pages Exercises Assignments

22–24 5, 6, 26, 35 , 40 7, 8, 18, 20, 27, 28 3943, 48, 51

35–38 1, 10, 13 2, 4, 6, 9, 12, 14

46–49 3, 7, 24, 32, 37 5, 23, 38, 40, 44, 53, 55

73–75 5, 20, 26, 11 6, 7, 22, 25, 28, 33

86-87 19, 23, 24


Exercises and Assignments

Text book: J. Stewart, Calculus. Concepts and Contexts , 2nd,Thomson Learning, 2001.

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g,

Pages Exercises Assignments

108-110 3, 6 5, 10

117-119 2, 15, 18, 26, 32, 36 7, 10, 20, 27, 28,33, 35, 38, 43

128–130 3, 13, 24, 29, 37 4, 6, 7, 15, 16, 23, 3031, 33, 38, 40, 46

139–142 2, 4, 24, 39, 41 3, 11, 19, 20, 23, 31, 3437, 42


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cal1 iu slides(2ndsem09-10) chapter1 sv

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