cc30m - topic 2.1 exponent
TRANSCRIPT
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TOPIC 2 ALGEBRA
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TOPICS
2.1 Exponent
2.2 Logarithm
2.3 Surds
2.4 Inequalities
2.5 Quadratics Equation
2.6 Partial Fraction
2.7 Polynomials
2.8 Simultaneous Equations
2.9 Binomial Theorem2.10 Iteration
2.11 Function
2.12 Complex Numbers
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2.1 EXPONENT
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2.1 EXPONENT
-The Law Of Exponent
- Exponent with Natural Base e- Exponent Manipulating
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2.1 EXPONENT
The Law Of Exponent / Indices
The Laws Of Indices :
1. IndicesMultiplication
Example :
2. IndicesDivision
Example :
nmnm pxpp
cbcbaxcba
babxaba2352272
134532
3065
nm
n
m
pp
p
26)5(7)4(21
543
724
222
8
cabcbacba
cba
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2.1 EXPONENT
The Law Of Exponent / Indices
3. IndicesPowers
Example :
4. IndicesRoots and Reciprocals
Example :
mnmxnnm ppp
636332 328424 babaab
nn pp
1
m
mp
p
1and
3/73/43/17437
4
babaa
b
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5
4
1
3
12
1
34
1
yxyx
2
22
2442
4
84
yx
yxyx
Solve the following :
Quick test
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Like and i, e denotes a number.
Called The Euler Number after Leonhard Euler (1707-1783)
It can be defined by:
e= lim(1+1/n)
n
As n gets larger (n), gets closer and closer to
2.718281828459.
(e is irrational number)
2.1 EXPONENT
Exponent With Natural Base e
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e3 e4 =
e7
10e3 =
5e2
2e3-2 =
2e
(3e-4x)2
9e(-4x)2
9e
-8x
9
e8x
2.1 EXPONENT
Exponent With Natural Base e
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2.1 EXPONENT
Manipulating
Indices / exponent Manipulating :
634 2 xx
122 2793 xxx
21
2
2
22
,2421123
3
2
2
3232
xxxx
x
x
xx
3
364
3331234
x
xxx
xxx
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Solve the following :
Quick test
1222 3 xx
01292212
xx
125
1255 13 xx
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Sketch the graph of the exponential function
f(x) = 2x.
Solution
First, recall that the domain of this function is
the set of real numbers.
Next, puttingx= 0 gives y= 20 = 1, which is the
y-intercept.
(There is nox-intercept, since there is no value
ofxfor which y= 0)
Exponent Graph
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Solution
Now, consider a few values forx:
Note that 2xapproaches zero asxdecreases without
bound:
There is a horizontal asymptote at y= 0.
Furthermore, 2xincreases without bound whenx
increases without bound.
Thus, the rangeoffis the interval(0, ).
x 5 4 3 2 1 0 1 2 3 4 5
y 1/32 1/16 1/8 1/4 1/2 1 2 4 8 16 32
Exponent Graph
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2 2
Solution
Finally, sketch the graph:
x
y
4
2
f(x) = 2x
Exponent Graph
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Sketch the graph of the exponential function f(x) = (1/2)x.
Solution
Now, consider a few values forx:
Note that (1/2)xincreaseswithout bound whenxdecreases
without bound. Furthermore, (1/2)xapproaches zero asxincreaseswithout
bound: there is a horizontal asymptote at y= 0.
As before, the range offis the interval (0,).
x
5
4
3
2
1 0 1 2 3 4 5
y 32 16 8 4 2 1 1/2 1/4 1/8 1/16 1/32
Exponent Graph
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2 2
Sketch the graph of the exponential function f(x) = (1/2)x.
Solution
Finally, sketch the graph:
x
y
4
2
f(x) = (1/2)x
Exponent Graph
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2 2
Sketch the graph of the exponential function f(x) = (1/2)x.
Solution
Note the symmetry between the two functions:
x
y
4
2
f(x) = (1/2)x
f(x) = 2x
Exponent Graph
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The exponential function y= bx (b > 0, b
1) has the following properties:
1. Its domain is (, ).2. Its range is (0, ).
3. Its graph passes through the point (0, 1)
4. It is continuous on (, ).
5. It is increasing on (, ) if b > 1 and
decreasing on (, ) if b < 1.
Properties of Exponential Graph
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Exponential functions to the basee, where e is
an irrational number whose value is
2.7182818, play an important role in both
theoretical and applied problems.
It can be shown that
1
lim 1
m
me m
Exponent Function Base e
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Sketch the graph of the exponential function f(x) = ex.
Solution
Since ex> 0 it follows that the graph ofy= exis similar to the
graph ofy= 2x.
Consider a few values forx:
x 3 2 1 0 1 2 3
y 0.05 0.14 0.37 1 2.72 7.39 20.09
Exponent Function Base e
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3 1 1 3
5
3
1
Sketch the graph of the exponential function f(x) = ex.
Solution
Sketching the graph:
x
yf(x) = ex
Exponent Function Base e
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Sketch the graph of the exponential function f(x) = ex.
Solution
Since ex> 0 it follows that 0 < 1/e < 1 and so
f(x) = ex= 1/ex= (1/e)x is an exponential function with base
less than 1.
Therefore, it has a graph similar to that of y= (1/2)x.
Consider a few values forx:
x
3
2
1 0 1 2 3
y 20.09 7.39 2.72 1 0.37 0.14 0.05
Exponent Function Base e
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3 1 1 3
5
3
1
Sketch the graph of the exponential function f(x) = ex.
Solution
Sketching the graph:
x
y
f(x) = ex
Exponent Function Base e