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About the Instructor Instructor: Dr. Jianli Xie Office hours: Mon. Thu. afternoon, or by appointment...
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Transcript of About the Instructor Instructor: Dr. Jianli Xie Office hours: Mon. Thu. afternoon, or by appointment...
![Page 1: About the Instructor Instructor: Dr. Jianli Xie Office hours: Mon. Thu. afternoon, or by appointment Contact: Email: xjl@sjtu.edu.cnxjl@sjtu.edu.cn Office:](https://reader036.fdocuments.net/reader036/viewer/2022062422/56649dde5503460f94ad79c9/html5/thumbnails/1.jpg)
About the Instructor Instructor: Dr. Jianli Xie Office hours: Mon. Thu. afternoon,
or by appointment Contact: Email: [email protected]
Office: Math Building Rm.1211
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About the TAs Xie Jun: [email protected] Jiang Chen: [email protected] Liu Li: [email protected] Wang Chengsheng: klaus19890602@hotma
il.com
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About the Course
Course homepageSAKAI http://202.120.46.185:8080/portal Grading policy
30%(HW)+35%(Midterm)+35%(Final) Important date
Midterm (Oct. 21), Final exam (Dec. 10)
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To The Student
Attend to every lecture Ask questions during lectures Do not fall behind Do homework on time Presentation is critical
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Ch.1 Functions and Models 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}
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Variable independence A function is independent of what variable is used
Ex. Find f if
Sol. Since
we have f(x)=x2-2.
Q: What is the domain of the above function f ?
A: D(f)=R(x+1/x)=(-1,-2][[2,+1)
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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
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Representation of a function
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.
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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.
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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.
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Properties of functions 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)
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Example
Ex. Given , is it even, odd, or
neither?
Sol.
Therefore, f is an odd function.
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Example
Ex. Given an increasing function f, let
What is the relationship between A and B?
Sol.
{ ( ) }, { ( ( )) }.A x f x x B x f f x x
.A B
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Essential functions I Polynomials (linear, quadratic, cubic……)
Power functions
Rational (P(x)/Q(x) with P,Q polynomials) Algebraic (algebraic operations of polynom
ials)
11 1 0( ) n n
n np x a x a x a x a
ay x
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Essential functions II Trigonometric (sine, cosine, tangent……) Inverse trigonometric (arcsin,arccos,arctan
……) Exponential functions ( ) Logarithmic functions ( ) Transcendental functions (non-algebraic)
xy a
logay x
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New functions from old functions 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
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Example
Ex. Find if f(x)=x/(x+1), g(x)=x10, and
h(x)=x+3.
Sol.
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Inverse functions A function f is called a one-to-one function if
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
for any y in B.
f(x1) f(x2) whenever x1 x2
f -1(y)=x , f(x)=y
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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
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Laws of exponential and logarithm
Laws of exponential
Laws of logarithm
Relationship
, ( ) , ( )x y x y x y xy x x xa a a a a a b ab
log log log ( ), log logba a a a ax y xy x b x
log ba x b x a
loglog
logc
ac
bb
a
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ex and lnx Natural exponential function ex
constant e¼2.71828 Natural logarithmic function lnx lnx=logex
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Graph of essential functions1/n ny x y x
logxay a y x
sin arcsiny x y x
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Homework 1 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