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About the Instructor
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Instructor: Dr. Jianli Xie
Office hours: Mon. Thu. afternoon,
or by appointment
Contact: Email: xjl@sjtu.edu.cn
Office: Math Building Rm.1211
About the TAs
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Xie Jun: beiwei3_4803@hotmail.com
Jiang Chen: bert@sjtu.edu.cn
Liu Li: terriclisa@hotmail.com
Wang Chengsheng:
klaus19890602@hotmail.com
About the Course
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Course homepage
SAKAI http://202.120.46.185:8080/portal
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Grading policy
30%(HW)+35%(Midterm)+35%(Final)
Important date
Midterm (Oct. 21), Final exam (Dec. 10)
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To The Student
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Attend to every lecture
Ask questions during lectures
Do not fall behind
Do homework on time
Presentation is critical
Ch.1 Functions and Models
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Functions are the fundamental objects that we
deal with in Calculus
A function f is a rule that assigns to each element x
in a set A exactly one element, called f(x), in a set B
f: x2 A! y=f(x)2 B
x is independent variable, y is dependent variable
A is domain of f, range of f is defined by {f(x)|x2 A}
Variable independence
A function is independent of what variable is used
Ex. Find f if
Sol. Since
we have f(x)=x2-2.
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Q: What is the domain of the above function f ?
A: D(f)=R(x+1/x)=(-1,-2][[2,+1)
Example
Ex. Find f if f(x)+2f(1-x)=x2.
Sol. Replacing x by 1-x, we obtain
f(1-x)+2f(x)=(1-x)2.
From these two equations, we have
Representation of a function
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Description in words (verbally)
Table of values (numerically)
Graph (visually)
Algebraic expression (algebraically)
The 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 once.
Example
Ex. Find the domain and range of
Sol. 4-x2¸0) –2· x·2
So the domain is
.
Since 0·4-x2·4,
the range is
.
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Piecewise defined functions
Ex. A function f is defined by
Evaluate f(0), f(1) and f(2) and sketch the graph.
Sol. Since 0·1, we have f(0)=1-0=1.
Since 1·1, we have f(1)=1-1=0.
Since 2>1, we have f(2)=22=4.
Piecewise defined functions
The graph is as the following. Note that we use the
open dot to indicate (1,1) is excluded from the graph.
Properties of functions
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Symmetry
even function: f(-x)=f(x)
odd function: f(-x)=-f(x)
Monotony
increasing function: x1<x2) f(x1)<f(x2)
decreasing function: x1<x2) f(x1)>f(x2)
Periodic function: f(x+T)=f(x)
Example
Ex. Given
neither?
Sol.
, is it even, odd, or
Therefore, f is an odd function.
Example
Ex. Given an increasing function f, let
A  {x f ( x)  x},
B {x f ( f ( x))  x}.
What is the relationship between A and B?
Sol. A  B.
Essential functions I
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Polynomials (linear, quadratic, cubic……)
p( x)  an x n  an 1 x n 1 
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Power functions
yx
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 a1 x  a0
a
Rational (P(x)/Q(x) with P,Q polynomials)
Algebraic (algebraic operations of
polynomials)
Essential functions II
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Trigonometric (sine, cosine, tangent……)
Inverse trigonometric
(arcsin,arccos,arctan……)
x
y

a
Exponential functions (
)
Logarithmic functions ( y  log a x )
Transcendental functions (non-algebraic)
New functions from old functions
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Transformations of functions
f(x)+c, f(x+c), cf(x), f(cx)
Combinations of functions
(f+g)(x)=f(x)+g(x), (fg)(x)=f(x)g(x)
Composition of functions
Example
Ex. Find
if f(x)=x/(x+1), g(x)=x10, and
h(x)=x+3.
Sol.
Inverse functions
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A function f is called a one-to-one function if
f(x1) f(x2)
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whenever x1 x2
Let f be a one-to-one function with domain A and
range B. Then its inverse function f -1 has domain
B and range A and is defined by
f -1(y)=x
for any y in B.
,
f(x)=y
Example
Ex. Find the inverse function of f(x)=x3+2.
Sol. Solving y=x3+2 for x, we get
Therefore, the inverse function is
Laws of exponential and logarithm
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Laws of exponential
a a  a
x
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y
x y
,
(a )  a ,
x y
xy
a  b  (ab)
x
x
Laws of logarithm
log a x  log a y  log a ( xy ),
log a xb  b log a x
log c b
log a b 
log c a
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x
b
log
x
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b
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x

a
Relationship
a
ex and lnx
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Natural exponential function ex
constant e¼2.71828
Natural logarithmic function lnx
lnx=logex
Graph of essential functions
y  xn
y  sin x
y  ax
y  x1/ n
y  arcsin x
y  log a x
Homework 1
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Section 1.1: 24,27,36,66
Section 1.2: 3,4
Section 1.3: 37,44,52
Section 1.6: 18,20,28,51,68,71,72