[email protected] mth55_lec-54_sec_8-5a_polynom_inequal.ppt 1 bruce mayer, pe chabot college...
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[email protected] • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§9.1a§9.1aExponential FcnsExponential Fcns
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Bruce Mayer, PE Chabot College Mathematics
Review §Review §
Any QUESTIONS About• §8.5 → Rational InEqualities
Any QUESTIONS About HomeWork• §8.5 → HW-41
8.5 MTH 55
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Bruce Mayer, PE Chabot College Mathematics
Exponential FunctionExponential Function
A function, f(x), of the form
is called an EXPONENTIAL function with BASE a.
The domain of the exponential function is (−∞, ∞); i.e., ALL Real Numbers
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Bruce Mayer, PE Chabot College Mathematics
Recall Rules of ExponentsRecall Rules of Exponents
Let a, b, x, and y be real numbers with a > 0 and b > 0. Then
ax ay axy ,
ax
ay ax y ,
ab x axbx ,
ax yaxy ,
a0 1,
a x 1
ax 1
a
x
.
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Bruce Mayer, PE Chabot College Mathematics
Evaluate Exponential FunctionsEvaluate Exponential Functions
Example
Solution
Example
Solution
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Bruce Mayer, PE Chabot College Mathematics
Evaluate Exponential FunctionsEvaluate Exponential Functions
Example
Solution
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Bruce Mayer, PE Chabot College Mathematics
Example Example Graph Graph yy = = ff((xx) =3) =3xx
Graph the exponential fcn: ( ) 3 .xf x
Make T-Table,& Connect Dots
x y0
1
–1
2
–2
3
1
3
1/3
9
1/9
27
x
y
-5 -4 -3 -2 -1 1 2 3 4 5
4
3
6
2
5
1
-1
-2
78
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Bruce Mayer, PE Chabot College Mathematics
Example Example Graph Exponential Graph Exponential
Graph the exponential fcn:
Make T-Table,& Connect Dots
x y
0
1
–1
2
–2
–3
1
1/3
3
1/9
9
27• This fcn is a
REFLECTION of y = 3x
3xy
x
y
-5 -4 -3 -2 -1 1 2 3 4 5
4
3
6
2
5
1
-1
-2
78
1( )
3
xy f x
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Bruce Mayer, PE Chabot College Mathematics
Example Example Graph Exponential Graph Exponential
Graph the exponential fcn:
Construct SideWays T-Table
x −3 −2 −1 0 1 2 3
y = (1/2)x 8 4 2 1 1/2 1/4 1/8
Plot Points and Connect Dots with Smooth Curve
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Bruce Mayer, PE Chabot College Mathematics
Example Example Graph Exponential Graph Exponential
As x increases in the positive direction, y decreases towards 0
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Bruce Mayer, PE Chabot College Mathematics
Exponential Fcn PropertiesExponential Fcn Properties
Let f(x) = ax, a > 0, a ≠ 1. Then
A. The domain of f(x) = ax is (−∞, ∞).
B. The range of f(x) = ax is (0, ∞); thus, the entire graph lies above the x-axis.
C. For a > 1 (e.g., 7)i. f is an INcreasing function; thus, the graph
is RISING as we move from left to right
ii. As x→∞, y = ax increases indefinitely and VERY rapidly
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Bruce Mayer, PE Chabot College Mathematics
Exponential Fcn PropertiesExponential Fcn Properties
Let f(x) = ax, a > 1, a ≠ 1. Then iii. As x→−∞, the values of y = ax get
closer and closer to 0.
D. For 0 < a < 1 (e.g., 1/5)i. f is a DEcreasing function; thus, the graph
is falling as we scan from left to right.
ii. As x→−∞, y = ax increases indefinitely and VERY rapidly
iii. As x→ ∞, the values of y = ax get closer and closer to 0
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Bruce Mayer, PE Chabot College Mathematics
Exponential Fcn PropertiesExponential Fcn Properties
Let f(x) = ax, a > 0, a ≠ 1. Then
E. Each exponential function f is one-to-one; i.e., each value of x has exactly ONE target. Thus:
i. 2121 xxaa xx
ii. f has an inverse
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Bruce Mayer, PE Chabot College Mathematics
Exponential Fcn PropertiesExponential Fcn Properties
Let f(x) = ax, a > 0, a ≠ 1. Then
F. The graph f(x) = ax has no x-intercepts • In other words, the graph of f(x) = ax
never crosses the x-axis. Put another way, there is no value of x that will cause f(x) = ax to equal 0
G. The x-axis is a horizontal asymptote for every exponential function of the form f(x) = ax.
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Bruce Mayer, PE Chabot College Mathematics
Translate Exponential GraphsTranslate Exponential Graphs
Translation Equation Effect on Equation
HorizontalShift
y = ax+b
= f (x + b)Shift the graph of y = ax, b units(i) Left if b > 0.(ii) Right if b < 0.
VerticalShift
y = ax + b = f (x) + b
Shift the graph of y = ax, b units(i) Up if b > 0.(ii) Down if b < 0.
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Bruce Mayer, PE Chabot College Mathematics
Example Example Sketch Graph Sketch Graph
By TranslationMove DOWNy = 3x by 3 Units
Note• Domain: (−∞, ∞)
• Range: (−4, ∞)
• Horizontal Asymptote: y = −4
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Bruce Mayer, PE Chabot College Mathematics
Example Example Sketch Graph Sketch Graph
By TranslationMove LEFTy = 3x by 1 Unit
Note• Domain: (−∞, ∞)
• Range: (0, ∞)
• Horizontal Asymptote: y = 0
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Bruce Mayer, PE Chabot College Mathematics
Alternative Graph: Swap Alternative Graph: Swap xx & & yy
It will be helpful in later work to be able to graph an equation in which the x and y in y = ax are interchanged
-6 -4 -2 0 2 4 6 8 10 12
-6
-4
-2
0
2
4
6
8
10
12
yx 3.2
xy 3.2
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Bruce Mayer, PE Chabot College Mathematics
Example Example Graph Graph xx = 3 = 3yy
Graph the exponential fcn:
Make T-Table,& Connect Dots
x y
1
3
1/3
9
1/9
27
0
1
–1
2
–2
3
x
y
-3 -2 -1 1 2 3 4 5 6 7 8 9
4
3
6
2
5
1
-1-2
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Bruce Mayer, PE Chabot College Mathematics
Example Example Apply Exponential Apply Exponential
Example Bank Interest compounded annually.
The amount of money A that a principal P will be worth after t years at interest rate i, compounded annually, is given by the formula
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Bruce Mayer, PE Chabot College Mathematics
Example Example Compound Interest Compound Interest
Suppose that $60,000 is invested at 5% interest, compounded annually
a) Find a function for the amount in the account after t years
SOLUTION
a) = $60000(1 + 0.05 )t
= $60000(1.05)t
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Bruce Mayer, PE Chabot College Mathematics
Example Example Compound Interest Compound Interest
Suppose that $60,000 is invested at 5% interest, compounded annually
b) Find the amount of money amount in the account at t = 6.
SOLUTION
b) A(6) = $60000(1.05)6 $80,405.74
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Bruce Mayer, PE Chabot College Mathematics
Example Example Bacterial Growth Bacterial Growth
A technician to the Great French microbiologist Louis Pasteur noticed that a certain culture of bacteria in milk doubled every hour.
Assume that the bacteria count B(t) is modeled by the equation
• Where t is time in hours
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Bruce Mayer, PE Chabot College Mathematics
Example Example Bacterial Growth Bacterial Growth
Given Bacterial Growth Equation
Find:a) the initial number of bacteria,
b) the number of bacteria after 10 hours; and
c) the time when the number of bacteria will be 32,000.
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Bruce Mayer, PE Chabot College Mathematics
Example Example Bacterial Growth Bacterial Growth
a) INITIALLY time, t, is ZERO → Sub t = 0 into Growth Eqn:
B0 B 0 200020 20001 2000
b) At Ten Hours Sub t = 10 into Eqn:
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Bruce Mayer, PE Chabot College Mathematics
Example Example Bacterial Growth Bacterial Growth
c) Find t when B(t) = 32,000
Thus 4 hours after the starting time, the number of bacteria will be 32k
32000 20002t
16 2t
24 2t
4 t
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Bruce Mayer, PE Chabot College Mathematics
WhiteBoard WorkWhiteBoard Work
Problems From §9.1 Exercise Set• 36, 40, 54
USAPersonalSavingsRate
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Bruce Mayer, PE Chabot College Mathematics
All Done for TodayAll Done for Today
BacteriaGrowFAST!
• Note: 37 °C = 98.6 °F (Body Temperature)
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Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
AppendiAppendixx
–
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Bruce Mayer, PE Chabot College Mathematics
Irrational ExponentsIrrational Exponents
By The Properties of Exponents we Can Evaluate Bases Raised to Rational-Number Powers Such as
23
2
3
13 23
123
2
77777
What about expressions with IRrational exponents such as:
To attach meaning to this expression consider a rational approximation, r, for the Square Root of 2
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Bruce Mayer, PE Chabot College Mathematics
Irrational ExponentsIrrational Exponents
Approximate byITERATION on:
1.4 < r < 1.5
1.41 < r < 1.42
1.414 < r < 1.415
1.4 1.515.245 7 7 18.520p 1.41 1.4215.545 7 7 18.850p
1.414 1.41515.666 7 7 15.697p
closes in on 2r 2 closes in on 7p
277
2
rp
r
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Bruce Mayer, PE Chabot College Mathematics
Irrational ExponentsIrrational Exponents
Thus by Iteration
27 15.6728909 Any positive irrational exponent can be
interpreted in a similar way. Negative irrational exponents are then
defined using reciprocals.