7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 1/90
CALCULUS I
Dr. Nguyen Ngoc Hai
DEPARTMENT OF MATHEMATICS
INTERNATIONAL UNIVERSITY, VNU-HCM
April 19, 2010
Dr. Nguyen Ngoc Hai CALCULUS I
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 2/90
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.
Dr. Nguyen Ngoc Hai CALCULUS I
S S
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 3/90
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
Dr. Nguyen Ngoc Hai CALCULUS I
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 4/90
Ch 1 FUNCTI NS LIMITS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 5/90
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.
Dr. Nguyen Ngoc Hai CALCULUS I
Ch 1 FUNCTI NS LIMITS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 6/90
Chapter 1 FUNCTI NS, LIMITSAND CONTINUITY
1.1 WHAT IS CALCULUS?
• 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.
Dr. Nguyen Ngoc Hai CALCULUS I
Ch t 1 FUNCTI NS LIMITS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 7/90
Chapter 1 FUNCTI NS, LIMITSAND CONTINUITY
1.1 WHAT IS CALCULUS?
• 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”.
Dr. Nguyen Ngoc Hai CALCULUS I
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 8/90
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.
Dr. Nguyen Ngoc Hai CALCULUS I
S G S Q O S O S
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 9/90
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1 2 STRAIGHT LINES EQUATIONS OF LINES
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 10/90
1.2 STRAIGHT LINES. EQUATIONS OF LINES1.2.1 STRAIGHT LINES
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
Dr. Nguyen Ngoc Hai CALCULUS I
1 2 STRAIGHT LINES EQUATIONS OF LINES
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 11/90
1.2 STRAIGHT LINES. EQUATIONS OF LINES1.2.1 STRAIGHT LINES
• The slope of a horizontal line is zero since ∆y = 0.
• The slope of a vertical line is undefined.
Dr. Nguyen Ngoc Hai CALCULUS I
1 2 STRAIGHT LINES EQUATIONS OF LINES
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 12/90
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).
Dr. Nguyen Ngoc Hai CALCULUS I
1 2 STRAIGHT LINES EQUATIONS OF LINES
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 13/90
1.2 STRAIGHT LINES. EQUATIONS OF LINES1.2.2 EQUATIONS OF LINES
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 .
Dr. Nguyen Ngoc Hai CALCULUS I
1 2 STRAIGHT LINES EQUATIONS OF LINES
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 14/90
1.2 STRAIGHT LINES. EQUATIONS OF LINES1.2.2 EQUATIONS OF LINES
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1 3 FUNCTIONS AND GRAPHS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 15/90
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 .
Dr. Nguyen Ngoc Hai CALCULUS I
1 3 FUNCTIONS AND GRAPHS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 16/90
1.3 FUNCTIONS AND GRAPHS1.3.1 FUNCTIONS
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 .
Dr. Nguyen Ngoc Hai CALCULUS I
1 3 FUNCTIONS AND GRAPHS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 17/90
1.3 FUNCTIONS AND GRAPHS1.3.1 FUNCTIONS
• 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.
Dr. Nguyen Ngoc Hai CALCULUS I
1 3 FUNCTIONS AND GRAPHS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 18/90
1.3 FUNCTIONS AND GRAPHS1.3.1 FUNCTIONS
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1 3 FUNCTIONS AND GRAPHS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 19/90
1.3 FUNCTIONS AND GRAPHS1.3.1 FUNCTIONS
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1 3 FUNCTIONS AND GRAPHS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 20/90
1.3 FUNCTIONS AND GRAPHS1.3.1 FUNCTIONS
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1 3 FUNCTIONS AND GRAPHS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 21/90
1.3 FUNCTIONS AND GRAPHS1.3.1 FUNCTIONS
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
.
Dr. Nguyen Ngoc Hai CALCULUS I
1.3 FUNCTIONS AND GRAPHS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 22/90
1.3 FUNCTIONS AND GRAPHS1.3.1 FUNCTIONS
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
Dr. Nguyen Ngoc Hai CALCULUS I
1.3 FUNCTIONS AND GRAPHS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 23/90
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.3 FUNCTIONS AND GRAPHS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 24/90
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.3 FUNCTIONS AND GRAPHS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 25/90
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 .
Dr. Nguyen Ngoc Hai CALCULUS I
1.3 FUNCTIONS AND GRAPHS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 26/90
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.3 FUNCTIONS AND GRAPHS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 27/90
1.3.3 EVEN AND ODD FUNCTIONS. SYMMETRY AND REFLECTIONS
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.3 FUNCTIONS AND GRAPHS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 28/90
1.3.3 EVEN AND ODD FUNCTIONS. SYMMETRY AND REFLECTIONS
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.4 NEW FUNCTIONS FROM OLD FUNCTIONSINVERSE FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 29/90
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
1.4 NEW FUNCTIONS FROM OLD FUNCTIONSINVERSE FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 30/90
INVERSE FUNCTIONS1.4.1 TRANSFORMATIONS OF FUNCTIONS
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.4 NEW FUNCTIONS FROM OLD FUNCTIONSINVERSE FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 31/90
INVERSE FUNCTIONS1.4.1 TRANSFORMATIONS OF FUNCTIONS
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.4 NEW FUNCTIONS FROM OLD FUNCTIONSINVERSE FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 32/90
INVERSE FUNCTIONS1.4.1 TRANSFORMATIONS OF FUNCTIONS
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;
Dr. Nguyen Ngoc Hai CALCULUS I
1.4 NEW FUNCTIONS FROM OLD FUNCTIONSINVERSE FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 33/90
INVERSE FUNCTIONS1.4.1 TRANSFORMATIONS OF FUNCTIONS
Example 4.3 Sketch the graph of the following functions(a) y = sin 5x (b) y = 3− sin2x (c) y = | ln x |.
Dr. Nguyen Ngoc Hai CALCULUS I
1.4 NEW FUNCTIONS FROM OLD FUNCTIONSINVERSE FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 34/90
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 ).
Dr. Nguyen Ngoc Hai CALCULUS I
1.4 NEW FUNCTIONS FROM OLD FUNCTIONSINVERSE FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 35/90
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.4 NEW FUNCTIONS FROM OLD FUNCTIONSINVERSE FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 36/90
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 .
Dr. Nguyen Ngoc Hai CALCULUS I
1.4 NEW FUNCTIONS FROM OLD FUNCTIONSINVERSE FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 37/90
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.4 NEW FUNCTIONS FROM OLD FUNCTIONSINVERSE FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 38/90
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.
The domain of a polynomial is IR .Dr. Nguyen Ngoc Hai CALCULUS I
1.4 NEW FUNCTIONS FROM OLD FUNCTIONSINVERSE FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 39/90
INVERSE FUNCTIONS1.4.4 THE BASIC CLASSES OF FUNCTIONS
• 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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.4 NEW FUNCTIONS FROM OLD FUNCTIONSINVERSE FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 40/90
INVERSE FUNCTIONS1.4.4 THE BASIC CLASSES OF FUNCTIONS
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}.
Dr. Nguyen Ngoc Hai CALCULUS I
1.4 NEW FUNCTIONS FROM OLD FUNCTIONSINVERSE FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 41/90
INVERSE FUNCTIONS1.4.4 THE BASIC CLASSES OF FUNCTIONS
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.4 NEW FUNCTIONS FROM OLD FUNCTIONSINVERSE FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 42/90
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
.
Dr. Nguyen Ngoc Hai CALCULUS I
1.4 NEW FUNCTIONS FROM OLD FUNCTIONSINVERSE FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 43/90
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
Dr. Nguyen Ngoc Hai CALCULUS I
1.4 NEW FUNCTIONS FROM OLD FUNCTIONSINVERSE FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 44/90
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
Dr. Nguyen Ngoc Hai CALCULUS I
1.4 NEW FUNCTIONS FROM OLD FUNCTIONSINVERSE FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 45/90
1.4.5 INVERSE FUNCTIONS
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 ).
Dr. Nguyen Ngoc Hai CALCULUS I
1.4 NEW FUNCTIONS FROM OLD FUNCTIONSINVERSE FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 46/90
1.4.5 INVERSE FUNCTIONS
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.
Dr. Nguyen Ngoc Hai CALCULUS I
SOME IMPORTANT INVERSE FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 47/90
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, ∞).
Dr. Nguyen Ngoc Hai CALCULUS I
SOME IMPORTANT INVERSE FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 48/90
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.
Dr. Nguyen Ngoc Hai CALCULUS I
SOME IMPORTANT INVERSE FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 49/90
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].
Dr. Nguyen Ngoc Hai CALCULUS I
SOME IMPORTANT INVERSE FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 50/90
• 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].
Dr. Nguyen Ngoc Hai CALCULUS I
SOME IMPORTANT INVERSE FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 51/90
• 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
Dr. Nguyen Ngoc Hai CALCULUS I
SOME IMPORTANT INVERSE FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 52/90
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).
Dr. Nguyen Ngoc Hai CALCULUS I
ELEMENTARY FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 53/90
• 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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.5 PARAMETRIC CURVES
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 54/90
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.5 PARAMETRIC CURVES
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 55/90
• 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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.5 PARAMETRIC CURVES
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 56/90
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.5 PARAMETRIC CURVES
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 57/90
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 )
Dr. Nguyen Ngoc Hai CALCULUS I
1.6 LIMITS1.6.1 LIMITS OF FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 58/90
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.6 LIMITS1.6.1 LIMITS OF FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 59/90
Example 6.1 Describe the behaviour of the function
f (x ) =x 2
−1
x − 1
near x = 1.
Dr. Nguyen Ngoc Hai CALCULUS I
1.6 LIMITS1.6.1 LIMITS OF FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 60/90
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”.
Dr. Nguyen Ngoc Hai CALCULUS I
1.6 LIMITS1.6.1 LIMITS OF FUNCTIONS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 61/90
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 .
Dr. Nguyen Ngoc Hai CALCULUS I
1.6 LIMITS1.6.2 ONE-SIDED LIMITS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 62/90
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.6 LIMITS1.6.2 ONE-SIDED LIMITS
D fi iti 6 2
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 63/90
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.6 LIMITS1.6.2 ONE-SIDED LIMITS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 64/90
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.6 LIMITS1.6.2 ONE-SIDED LIMITS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 65/90
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 ).
Dr. Nguyen Ngoc Hai CALCULUS I
1.7 LAWS OF LIMITS. EVALUATING LIMITS1.7.1 LAWS OF LIMITS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 66/90
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.7 LAWS OF LIMITS. EVALUATING LIMITS1.7.1 LAWS OF LIMITS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 67/90
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.7 LAWS OF LIMITS. EVALUATING LIMITS1.7.2 THE SQUEEZE THEOREM
Theorem 7 2
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 68/90
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.
Dr. Nguyen Ngoc Hai CALCULUS I
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 69/90
1.8 CONTINUITYTHE INTERMEDIATE VALUE THEOREM
1.8.1 CONTINUITY
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 70/90
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).
Dr. Nguyen Ngoc Hai CALCULUS I
1.8 CONTINUITYTHE INTERMEDIATE VALUE THEOREM
1.8.1 CONTINUITY
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 71/90
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.8 CONTINUITYTHE INTERMEDIATE VALUE THEOREM
1.8.1 CONTINUITY
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 72/90
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.8 CONTINUITYTHE INTERMEDIATE VALUE THEOREM
1.8.1 CONTINUITY
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 73/90
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.8 CONTINUITYTHE INTERMEDIATE VALUE THEOREM1.8.1 CONTINUITY
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 74/90
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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.8 CONTINUITYTHE INTERMEDIATE VALUE THEOREM1.8.1 CONTINUITY
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 75/90
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?
Dr. Nguyen Ngoc Hai CALCULUS I
1.8 CONTINUITYTHE INTERMEDIATE VALUE THEOREM1.8.1 CONTINUITY
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 76/90
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 .
Dr. Nguyen Ngoc Hai CALCULUS I
1.8 CONTINUITYTHE INTERMEDIATE VALUE THEOREM1.8.1 THE INTERMEDIATE VALUE THEOREM
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 77/90
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].
Dr. Nguyen Ngoc Hai CALCULUS I
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 78/90
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 79/90
1.9 LIMITS INVOLVING INFINITY1.9.1 INFINITE LIMITS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 80/90
• “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.
Dr. Nguyen Ngoc Hai CALCULUS I
1.9 LIMITS INVOLVING INFINITY1.9.1 INFINITE LIMITS
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 81/90
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 .
Dr. Nguyen Ngoc Hai CALCULUS I
1.9 LIMITS INVOLVING INFINITY1.9.2 FINITE LIMITS AT INFINITY
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 82/90
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 →∞.
Dr. Nguyen Ngoc Hai CALCULUS I
1.9 LIMITS INVOLVING INFINITY1.9.2 FINITE LIMITS AT INFINITY
Definition 9.4
L f b f i d fi d i l ( ) Th
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 83/90
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
1.9 LIMITS INVOLVING INFINITY1.9.2 FINITE LIMITS AT INFINITY
Example 9.2 If n is a positive integer, then
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 84/90
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.
Dr Nguyen Ngoc Hai CALCULUS I
1.9 LIMITS INVOLVING INFINITY1.9.2 FINITE LIMITS AT INFINITY
Theorem 9.1
Th li it
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 85/90
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
Dr Nguyen Ngoc Hai CALCULUS I
1.9 LIMITS INVOLVING INFINITY1.9.3 INFINITE LIMITS AT INFINITY
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 86/90
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 ) = −∞
Dr Nguyen Ngoc Hai CALCULUS I
1.9 LIMITS INVOLVING INFINITY1.9.3 INFINITE LIMITS AT INFINITY
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 87/90
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
Dr Nguyen Ngoc Hai CALCULUS I
1.9 LIMITS INVOLVING INFINITY1.9.3 INFINITE LIMITS AT INFINITY
Asymptotic Behavior of a Rational Function The
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 88/90
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.
Dr Nguyen Ngoc Hai CALCULUS I
Exercises and Assignments
Text book: J. Stewart, Calculus. Concepts and Contexts , 2nd,Thomson Learning 2001
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 89/90
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
Dr Nguyen Ngoc Hai CALCULUS I
Exercises and Assignments
Text book: J. Stewart, Calculus. Concepts and Contexts , 2nd,Thomson Learning, 2001.
7/27/2019 Cal1 IU SLIDES(2ndSem09-10) Chapter1 SV
http://slidepdf.com/reader/full/cal1-iu-slides2ndsem09-10-chapter1-sv 90/90
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
Dr Nguyen Ngoc Hai CALCULUS I