About the Instructor Instructor: Dr. Jianli Xie Office hours: Mon. Thu. afternoon, or by appointment Contact: Office: Math Building Rm.1211 TexPoint fonts used in EMF: A AA A AA

About the TAs Xie Jun: Jiang Chen: Liu Li: Wang Chengsheng: TexPoint fonts used in EMF: A AA A AA

About the Course Course homepage SAKAI 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: x 2 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)|x 2 A}

Variable independence  A function is independent of what variable is used Ex. Find f if Sol. Since we have f(x)=x 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)=x 2. 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-x 2 ¸ 0 ) –2 · x · 2 So the domain is. Since 0 · 4-x 2 · 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)=2 2 =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: x 1 <x 2 ) f(x 1 )<f(x 2 )  decreasing function: x 1 f(x 2 )  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.

Essential functions I Polynomials (linear, quadratic, cubic……) Power functions Rational (P(x)/Q(x) with P,Q polynomials) Algebraic (algebraic operations of polynomials)

Essential functions II Trigonometric (sine, cosine, tangent……) Inverse trigonometric (arcsin,arccos,arctan……) Exponential functions ( ) Logarithmic functions ( )  Transcendental functions (non-algebraic)

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)=x 10, 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(x 1 )  f(x 2 ) whenever x 1  x 2 f -1 (y)=x, f(x)=y

Example Ex. Find the inverse function of f(x)=x Sol. Solving y=x 3 +2 for x, we get Therefore, the inverse function is

Laws of exponential and logarithm Laws of exponential Laws of logarithm Relationship

e x and lnx  Natural exponential function e x  constant e ¼  Natural logarithmic function lnx  lnx=log e x

Graph of essential functions

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 TexPoint fonts used in EMF: A AA A AA