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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