about the instructor instructor: dr. jianli xie office hours: mon. thu. afternoon, or by appointment...
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About the Instructor Instructor: Dr. Jianli Xie Office hours: Mon. Thu. afternoon,
or by appointment Contact: Email: [email protected]
Office: Math Building Rm.1211
About the TAs Xie Jun: [email protected] Jiang Chen: [email protected] Liu Li: [email protected] Wang Chengsheng: klaus19890602@hotma
il.com
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)
To The Student
Attend to every lecture Ask questions during lectures Do not fall behind Do homework on time Presentation is critical
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}
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)
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
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 .
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 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 , is it even, odd, or
neither?
Sol.
Therefore, f is an odd function.
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
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
Essential functions II Trigonometric (sine, cosine, tangent……) Inverse trigonometric (arcsin,arccos,arctan
……) Exponential functions ( ) Logarithmic functions ( ) Transcendental functions (non-algebraic)
xy a
logay x
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
Example
Ex. Find if f(x)=x/(x+1), g(x)=x10, and
h(x)=x+3.
Sol.
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
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
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
ex and lnx Natural exponential function ex
constant e¼2.71828 Natural logarithmic function lnx lnx=logex
Graph of essential functions1/n ny x y x
logxay a y x
sin arcsiny x y x
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