calculus 2 function and limit.pdf
TRANSCRIPT
FUNCTIONAND
LIMIT
2
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 “
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
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
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)
Example
Example
• f(x)=2x-1
• g(x)=x^2
Which is a function?
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)
• 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
• 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
• 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.
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
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)
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)
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)
• Secant line is the line
through P and Q where
Q is any point on the
curve different from P.
• 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.
• 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
• 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
mtan=limiting value as h approaches zero of
f(x0+h)-f(x0)
h
Velocity
• The average velocity of an object moving in one direction along a line is :
Average velocity =Distance traveled
Time elapsed
• 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
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.
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.
Geometric Interpretation of Average
and Instantaneous Velocity
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
• Limits are used to describe how a function
behaves as the independent variable moves
toward a certain value. To illustrate, consider
the function
• If the value of f(x) approaches the
number L1 as x approaches x0 from the
right side, we write
1)(lim0
Lxfxx
• If the value of f(x) approaches the
number L1 as x approaches x0 from the
left side, we write
2)(lim0
Lxfxx
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
2.5 LIMITS (COMPUTATIONAL
TECHNIQUES)
THEOREM
EXERCISE