CC30M - Topic 2.1 Exponent

7/27/2019 CC30M - Topic 2.1 Exponent

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CC30M Copyright Pinnacle Learning Centre

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


8/23CC30M Copyright Pinnacle Learning Centre

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



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



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



Solve the following :

Quick test

1222 3 xx

01292212

xx

125

1255 13 xx



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



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



2 2

Solution

Finally, sketch the graph:

x

y

4

2

f(x) = 2x

Exponent Graph



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



2 2


Solution

Finally, sketch the graph:

x

y

4

2

f(x) = (1/2)x

Exponent Graph



2 2


Solution

Note the symmetry between the two functions:

x

y

4

2

f(x) = (1/2)x

f(x) = 2x

Exponent Graph



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



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



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




3 1 1 3

5

3

1


Solution

Sketching the graph:

x

yf(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



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3 1 1 3

5

3

1


Solution

Sketching the graph:

x

y

f(x) = ex


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CC30M - Topic 2.1 Exponent