functions & graphs (1.2) what is a function? review domain & range boundedness open &...
TRANSCRIPT
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Functions & Graphs (1.2)What is a function?Review Domain & RangeBoundednessOpen & Closed IntervalsDistance from a point to a line
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Even & Odd Functions...
Ex: Identify the domain, range, (use interval notation) and whether the function is odd or even or neither.
y = x2
y = √(1-x2)
y = √x
y = 1/x
y = 2x/(x-1)
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Functions Defined in Pieces
While some functions are defined by single formulas, others are defined by applying different formulas to different parts of their domain.
These are called piecewise functions.
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Examples:
-x ; x < 0 y = x2 ; 0 < x < 1 1 ; x > 1
-x ; 0 < x < 1y =
2x – 2 ; 1< x < 2
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The Absolute Value Function
The absolute value function is defined piecewise:
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Composite Functions
Suppose that some of the outputs of a function can be used as inputs of
a function . We can then link and to form a new function whose inputs
are inputs of and whose outputs are the numbers
g
f g f
x g ( )( )( )( ) ( )
.
We say that the function read of of is
. The usual standard notation for the composite is ,
which is read " of ."
f g x
f g x f g x
f g
f g
the composite
of and og f
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Examples
f(x) = x2 + 1 g(x) = x- 7
Find:g(f(2))
f(g(2))
g(g(3))
f(f(x))
g(f(x))
g(g(x))
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Trig ReviewComplete Packet (will be part of
HW #6) on your ownSeek help either during seminar
or at next week’s review session
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1.3Exponential Functions
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Slide 1- 11
Exponential GrowthExponential DecayApplicationsThe Number e
…and why
Exponential functions model many growth patterns.
What you’ll learn about…
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Slide 1- 12
Exponential Function
Let be a positive real number other than 1. The function
( )
is the .
x
a
f x a
a
=
exponential function with base
The domain of f(x) = ax is (-∞, ∞) and the range is (0, ∞). Compound interest investment and population growth are examples of exponential growth.
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Slide 1- 13
Exponential Growth
If 1 the graph of looks like the graph
of 2 in Figure 1.22ax
a f
y=
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Slide 1- 14
Exponential Decay
If 0 1 the graph of looks like the graph
of 2 in Figure 1.22b.x
a f
y -=
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Slide 1- 15
Exponential Growth and Exponential Decay
The function , 0, is a model for
if 1, and a model for if 0 1.
xy k a k
a a
exponential growth
exponential decay
= × >
> < <
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Graphing Exponential Functions
Graph y = 2x
◦x-intercept:_______
◦y -intercept:_______
◦Domain: _______◦Range: _______◦Type:
_______Slide 1- 16
Graph y = 2-x
x-intercept:_______
y -intercept:_______
Domain: _______ Range: _______ Type:
_______
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Slide 1- 17
Rules for Exponents
See page 21 to review these!
Half-life
Exponential functions can also model phenomena that produce decrease over time, such as happens with radioactive decay. The half-life of a radioactive substance is the amount of time it takes for half of the substance to change from its original radioactive state to a non-radioactive state by emitting energy in the form of radiation.
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Use the Law of Exponents to expand or condense
1. ax ay
2. (ax)y
3. ax bx
4. (a/b)y
Slide 1- 18
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Slide 1- 19
Example Exponential Functions
( )Use a grapher to find the zero's of 4 3.xf x = -
( ) 4 3xf x = -
[-5, 5], [-10,10]
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Rewrite the exponential expression to have the indicated base
(9)2x , base 3(1/8) 2x , base 2
Slide 1- 20
Applications The Population of Knoxville is 500,000 and is increasing at the rate
of 3.75% annually. Approximately when will the population reach 1 million?
Suppose the half-life of a certain radioactive substance is 20 days and that there are 5 g present initially. When will there only be 1 g of the substance left?
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Interest ProblemsSimple Interest FormulaCompound Interest FormulaInterest compounded
continuously◦ How much would you get if P = $1, r =
100% and the principal were compounded continuously (every second of each day for 365 days) for one year?
Slide 1- 21
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Slide 1- 22
The Number e
Many natural, physical and economic phenomena are best modeled
by an exponential function whose base is the famous number , which is
2.718281828 to nine decimal places.
We can define to be the numbe
e
e ( ) 1r that the function 1
approaches as approaches infinity.
x
f xx
x
æ ö÷ç= + ÷ç ÷çè ø
f(x) = (1 + 1/x)x
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Slide 1- 23
The Number e
The exponential functions and are frequently used as models
of exponential growth or decay.
Interest compounded continuously uses the model , where is the
initial investment, is t
x x
r t
y e y e
y P e P
r
-= =
= ×
he interest rate as a decimal and is the time in years.t
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Slide 1- 24
Example The Number e( ) 0.03
The approximate number of fruit flies in an experimental population after
hours is given by 20 , 0.
a. Find the initial number of fruit flies in the population.
b. How large is the populat
tt Q t e t= ³
ion of fruit flies after 72 hours?
c. Use a grapher to graph the function .Q
[0,100] by [0,120] in 10’s
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Slide 1- 25
( )
( ) ( ) ( )
( ) ( )
0.03 0
0.03 2.72 16
0
a. To find the initial population, evaluate at 0.
20 20 20 1 20 flies.
b. After 72 hours, the population size is
20 2
0
0 173 flies.
c.
72
Q t t
Q e e
Q e e
=
= = = =
= = »
( ) 0.0320 , 0tQ t e t= ³
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Slide 1- 26
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Slide 1- 27
Quick Quiz Sections 1.1 – 1.3
( )
( )
( )
( )( )
You may use a graphing calculator to solve the following problems.
1. Which of the following gives an equation for the line through 3, 1
and parallel to the line: 2 1?
1 7A
2 21 5
B2 2
C 2 5
D 2
y x
y x
y x
y x
y x
-
=- +
= +
= -
=- +
=-
( )
7
E 2 1y x
-
=- +
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Slide 1- 28
Quick Quiz Sections 1.1 – 1.3
( ) ( )( )( )
( )( )( )( )( )
22. If 1 and 2 1, which of the
following gives 2 ?
A 2
B 5
C 9
D 10
E 15
f x x g x x
f g
= + = -
o
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Slide 1- 30
Warm-Up
( ) ( )( )
( )
In Exercises 1 3, write an equation for the line.
1. the line through the points 1, 8 and 4, 3
2. the horizontal line through the point 3, 4
3. the vertical line through the point 2, 3
In Exercises 4 - 6
-
-
-
2 2 2
2
, find the - and -intercepts of the graph of the relation.
4. 1 5. 19 16 16 9
6. 2 1
x y
x y x y
y x
+ = - =
= +
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1.4Parametric Equations
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Slide 1- 32
RelationsLines and Other Curves
What you’ll learn about…
…and why
Parametric equations can be used to obtain graphs of relations and functions.
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Slide 1- 33
Relations
A relation is a set of ordered pairs (x, y) of real numbers.
The graph of a relation is the set of points in a
plane that correspond to the ordered pairs of the relation.
If x and y are functions of a third variable t, called a parameter, then the equations that define x and y are parametric equations.
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Slide 1- 34
Parametric Curve, Parametric Equations
( ) ( )
( ) ( ) ( )( )
If and are given as functions
,
over an interval of -values, then the set of points , ,
defined by these equations is a . The equations are
of th
x y
x f t y g t
t x y f t g t
= =
=
parametric curve
parametric equations e curve.
Lines, line segments and many other curves can be defined parametrically.
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General parametric equations involving angular measure:x = v0 cosθt and y = -16t2 + v0 sinθt + s
Ex. 1: Consider the path followed by an object that is propelled into the air as an angle of 45 degrees with an initial velocity of 48 ft/sec. The object will follow a parabolic path.
Write a Cartesian equation and a set of parametric equations to model this example.
Slide 1- 35
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Graph each set of parametric equations, then find the Cartesian equation relating the variables (eliminate the parameter):
x = 2t + 1
y = 2 – t
Cartesian Equation:Slide 1- 36
t 0 1 2
x
y
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Slide 1- 37
x = r2 – 3r + 1y = r + 1
Cartesian Equation:
r
x
y
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x = sin ry = cos r
Cartesian Equation:
Slide 1- 38
r
x
y
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x = t3
y = t2/2
Slide 1- 39
t
x
y