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FUNCTIONAND
LIMIT
2
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2.1 FUNCTION
• BACKGROUND
The term of “function” was first used by Leibniz in 1673 to denote the dependence of one quantity on another
Example :
The area of a circle depends on its radius r by the equation A = r2;
We say that “ A is a function of r “
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Notation
• Leonard Euler introduced the using of a letter of alphabet such as f to denote a function or relationship.
Example :
y = f(x)
is read “y equals f of x”, that is the value of y depends on the value of x
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DEFINITION
• A function is a rule that assigns to each element of set A one and only one element of set B
• The set A is called domain of the function
• The set of all possible value of f(x) as x varies over the domain is called the range of f
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DEFINITION
y = f(x)
• y is called dependent variable
• x is called independent variable
• The graph of a function f is the graph
of the equation y = f(x)
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Example
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Example
• f(x)=2x-1
• g(x)=x^2
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Which is a function?
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2.2 OPERATION ON FUNCTION;
CLASSIFYING FUNCTIONS
• Given function f and g, their sum f+g, difference f-g, product f.g and quotient f/g are defined by
(f+g)(x)=f(x)+g(x)
(f-g)(x)=f(x)-g(x)
(f.g)(x)=f(x).g(x)
(f/g)(x)=f(x)/g(x)
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• For the function f+g, f-g, and f.g the
domain is defined to be the
intersection of the domains of f and g
and for f/g the domain is this
intersection with the points where
g(x) = 0 excluded
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• If f is a function and k is a real
number, then the function kf is
defined by
(kf)(x)=k.f(x)
and the domain of kf is the same as
the domain of f
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• Given two function f and g, the composition of f with g, denoted by f o g, is the function defined by
(fog)(x)=f(g(x))
where the domain of f o g consists of all x in the domain of g for which g(x) is in the domain of f.
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Classification of Functions
• Constant function, f(x)=c, c is a constant value
• Monomial in x, f(x)=cxn, c is a constant value, n is a nonnegative
• Polynomial in x, f(x)=a0+a1x+a2x2+…+anx
n
• Linear, f(x)=a0+a1x
• Quadratic, f(x)=a0+a1x+a2x2
• Cubic f(x)=a0+a1x+a2x2+a3x
3
• Rational function, ratio of two polynomial
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2.3 INTRODUCTION TO CALCULUS :
TANGENTS AND VELOCITY
• TWO FUNDAMENTAL PROBLEM OF CALCULUS :
1. The tangent problem (differential calculus)
2. The area problem
(integral calculus)
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The tangent problem
• Given a function f
and a point P(x0,y0)
on its graph, find
the equation of the
tangent to the
graph at P (figure
2.3.1)
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The area problem
Given a function f,
find the area
between the graph
of f and an interval
[a,b] on the x-axis
(figure 2.3.2)
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• Secant line is the line
through P and Q where
Q is any point on the
curve different from P.
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• If we move the point Q along the curve toward P, the secant line will rotate toward “limiting” position. The line T occupying this limiting position is called the tangent line.
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• If P(x0,y0) and Q(x1,y1) lie on the graph f so
that f(x0)=y0 and f(x1)=y1, then the slope of the
secant line through P and Q is :
msec =slope of PQ =y1-y0 =
f(x1)-f(x0)
x1-x0 x1-x0
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• If x1-x0=h so that x1=x0+h then we can write :
• As Q approaches P along the graph of f, or
equivalently as h=x1-x0 gets closer and
closer to zero, the secant line through P and
Q approaches the tangent line at P.
• Thus the slope of the secant line msec
approaches the slope of the tangent line
mtan.
msec=f(x0+h)-f(x0)
h
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mtan=limiting value as h approaches zero of
f(x0+h)-f(x0)
h
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Velocity
• The average velocity of an object moving in one direction along a line is :
Average velocity =Distance traveled
Time elapsed
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• If over the time interval from t0 to t1 the
distance traveled is
s1-s0
and the time elapsed is
t1-t0
so the average velocity during the interval is
given by
Average velocity =s1-s0
t1-t0
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Geometric Interpretation of
Average Velocity
• For a particle moving in one direction on a
straight line, the average velocity between
time t0 and t1 is represented geometrically
by the slope of the secant line connecting
(t0, s0) and (t1,s1 ) on the position versus
time curve.
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Geometric Interpretation of
Instantaneous Velocity
• For a particle moving in one direction on a
straight line, the instantaneous velocity at
time t0 is represented geometrically by the
slope of the tangent line at (t0,s0) on the
position versus time curve.
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Geometric Interpretation of Average
and Instantaneous Velocity
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2.4 LIMIT (AN INTUITIVE
INTRODUCTION)
• In the last section we saw that the concepts of tangent and instantaneous velocity ultimately rest on the notion of a "limit" or "value approached by" a function. In this section as well as the next few we will investigate the notion of limit in more detail. Our development of limits in this text proceeds in three stages:
1. First we discuss limits intuitively.
2. Then we discuss methods for computing limits.
3. finally, we give a precise mathematical discussion of limits
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• Limits are used to describe how a function
behaves as the independent variable moves
toward a certain value. To illustrate, consider
the function
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• If the value of f(x) approaches the
number L1 as x approaches x0 from the
right side, we write
1)(lim0
Lxfxx
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• If the value of f(x) approaches the
number L1 as x approaches x0 from the
left side, we write
2)(lim0
Lxfxx
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Lxfxfxxxx
)(lim)(lim
00
• If limit from the left side is the same as
the limit from the right side, say
Then we write
Lxfxx
)(lim0
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2.5 LIMITS (COMPUTATIONAL
TECHNIQUES)
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THEOREM
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EXERCISE