cal1 lec 01 function tt1213

7/24/2019 Cal1 Lec 01 Function TT1213

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HUT DEPARTMENT OF MATH. APPLIED--------------------------------------------------------------------------------------------------------

CALCULUS 1

LEC 01: FUNCTION

(A CATALOG OF ESSENTIAL FUNCTIONS)

Instructor: Dr. Nguyen Quoc Lan (December, 2012)

Email: [email protected]


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CONTENTS-----------------------------------------------------------------------------------------------------------------------------------

2- BASIC FUNCTIONS.

3- ONE TO ONE FUNCTIONS.

4- INVERSE FUNCTIONS

1- FUNCTIONS: DOMAIN, RANGE, GRAPH.

5- INVERSE TRIGONOMETRIC FUNCTIONS

6- HYPERBOLIC FUNCTIONS


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FUNCTION--------------------------------------------------------------------------------------------------------------------------------------------

Domain Range

Highschool: Function y = f(x), x independent variable , y

dependent variable

General: Function f is a rule that for each object x

A we can

find exactly one object y

B. We write: f: A B. A is called

the DOMAIN of the function f; B is called the RANGE of f.


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EXAMPLE--------------------------------------------------------------------------------------------------------------------------------------------

( ) ( )xfxxf ofrangetheanddomainthefind,sinGiven:Example =

[ ]1;1RangeR,D:Answer ==

Find Range of y = f(x): A value y

Range of f(x)

The

equation y = f(x) has at least one root.

( ) ( )xfx

xf ofrangetheanddomainthefind,1

1Given:Example

2 =

1

1

equationheConsider t:Range.1:D:Answer 2 = xyx

( )Range

0

1

0

01

0

01

0

11 22

>

+

+

=

=

y

y

y

yy

y

y

yx

y

yx


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GRAPH OF FUNCTION--------------------------------------------------------------------------------------------------------------------------------------------

Graph: Curve M(x,f(x)) Vertical line test: One x

One y


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LINEAR FUNCTION--------------------------------------------------------------------------------------------------------------------------------------------

Linear function (1st degree of x): y = mx + b. Graph of linear

function: A line

Linear equation: ax + by + c = 0.

The slope m of one

line (d): y = mx + b

shows the direction:

m > 0: d is up

m < 0: d is down

Vertical line x = C

does not have slope!


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QUADRATIC FUNCTIONS--------------------------------------------------------------------------------------------------------------------------------------------

Quadratic function (2nd degree of x): y = ax2 + bx + c. Graph of

quadratic function: parabola. Distinguish 2 case: a > 0, a < 0.

Upward

:0>a Downward

:0


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BASIC FUNCTIONS--------------------------------------------------------------------------------------------------------------------------------------------

( )

.13:ExCubic.:3Quadratic,:2Linear;:1

.0,...:ndegreeofPolynomial

23

01

1

+====

+++=

xxynnn

aaxaxaxP nn

nn

n

( ) ( )

( ) 34

:Ex..:Polynomial

PolynomialfunctionRational

2

===

x

xxy

xQ

xPxR

==a

0lim,lim:1 =+=> +x

x

x

x aaa

down

Go:10


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LOGARITHMIC FUNCTIONS--------------------------------------------------------------------------------------------------------------------------------------------

( )0,1,0log:Definition >>== xaaaxxy ya


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ONE TO ONE FUNCTION---------------------------------------------------------------------------------------------------------------------------

What is the difference

between two functions f & g?

Answer: x y f(x) f(y)

f is one to one. But

g(3) = g(2)

g is not so!

A function f is called a one

to one if it never takes

on the same value twice: x

y

f(x)

f(y)


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HORIZONTAL LINE TEST---------------------------------------------------------------------------------------------------------------------------

Horizontal Line Test: A

function s one to one if

and only if no horizontal

line intersects its graph

more than once

Example: Is the function g(x) = x2 one

to one?

Answer 1: No as g(1) = g(1)

Answer 2: Use horizontal line test


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INVERSE FUNCTION--------------------------------------------------------------------------------------------------------------------------------

Definition: Let f be a one to one function from domain A

to (range) B, that means f(A) = B. Than its INVERSE

FUNCTION f1 : B A defined by: f1(y) = x

f(x) = y

Example: Logarithmic function yexRyx

xy=

>

=

,0

ln

Remark 1: f: A B

f1 : B A

Remark 2: Do not mistake the 1 in f1

for an exponent: f1(x) doesnt mean

( ) ( ) ( )xf

xfxf

11reciprocalThe.1 =


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RULE TO FIND INVERSE FUNCTION--------------------------------------------------------------------------------------------------------------------------------

( )73

54offunctioninversetheFind:Example

+==

x

xxfy

How to find the inverse function of a one to one f(x)

Step 1: Write y = f(x) (1)

Step 2: Solve (1) for x in terms of y (if possible)

Step 3: To express f1 as a function of x, interchange x

and y. The resulting function is y = f1

(x)

( )

( )43

57functionInverse

3

4,

43

575473

73

54Let:Answer

1

+==

+=+=

+=

x

xxfy

yy

yxxxy

x

xy


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INVERSE TRIGONOMETRIC FUNCTION-----------------------------------------------------------------------------------------------------------------------------------

Four basic inverse trigonometric functions:

yxyxxxy sin

22

,11,arcsinsin 1 ===

yxyxxxy cos0,11,arccoscos 1 ===

yxyRxxxy tan

22

,,arctantan 1 =


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EXAMPLE-----------------------------------------------------------------------------------------------------------------------------------

3

1arcsintanb/

2

1arcsina/Evaluate:Example

66sin

21sin&

2221arcsina/:Solution ====

22

1

cos

sintan

3

22

9

11cos0cos

&1cossinAs.3

1sin&

223

1arcsinb/ 22

====>

=+=


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HYPERBOLIC FUNCTIONS--------------------------------------------------------------------------------------------------------------------------------

,2

sinh

xx

eex

= xxxxxxeex

xx

sinhcoshcoth,

coshsinhtanh,

2cosh ==

+=

We get directly hyperbolic formulas from all familiar

trigonometric formulas by changing cosx coshx and the sign

of any product of two sinx. For example: sin2x sinh2x

( )

( )Recognize?coshsinh22sinh1,sinhcoshShowb/

coshyp&sinhyp:Calculatorcosh0sinh0,Evaluatea/:Example

22 xxxxx ==

1sinhequationtheSolve:Example =x

Question: How to find + 21 x

dx =

tax

xa

dxsin:

22


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HYPERBOLIC FORMULAS--------------------------------------------------------------------------------------------------------------------------------

1cossin 22 =+ xx 1shch 22 = xx

( ) yxyxyx sinsincoscoscos m= ( ) yxyxyx shshchchch =( ) xyyxyx cossincossinsin = ( ) xyyxyx chshchshsh =

( ) xxx 22 sin211cos22cos == ( ) xxx 22 sh211ch22ch +==

( ) xxx cossin22sin = ( ) xxx chsh22sh =

2cos

2cos2coscos yxyx

yx +

=+2

ch2

ch2chch yxyx

yx +

=+

2sin

2sin2coscos yxyx

yx +=2

sh2

sh2chch yxyx

yx +=

HyperbolicTrigonometric

cal1 lec 01 function tt1213

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