Transcript EXPONENTIAL FUNCTION

PROGETTO CLIL
2006 – 2007
EXPONENTIAL AND
LOGARITHMIC FUNCTIONS
Prof.ssa Barbara di Majo
in collaborazione con
Prof.ssa Cinzia Calella
Istituto Tecnico Commerciale “G. R. Carli” Trieste
e
Prof.ssa Patrizia Torelli Prof.ssa Barbara Fasano
Istituto Tecnico Nautico “T. di Savoia” Trieste
EXPONENTIALS AND
LOGARITHMS
• WHO: IV CLASS
• WHEN: I TERM
• TIME: 8-10 HOURS
2
AIMS
Students should :
• Improve their abilities with numbers and
symbols
• Understand some practical applications of
maths
3
OBJECTIVES
• To work with powers, roots, logarithms
• To revise and use some topics studied in
the I and II class
• To draw exponential and logarithmic
functions
4
CONTENTS
•
•
•
•
•
Powers (revise)
Exponential and logarithmic functions
Logarithms
Tasks
Practical application
5
PREREQUISITES
Students should know:
• Powers of numbers and their properties
• Coordinates
6
LESSON 1
Objective: to encourage students to read
and speak
Time: 1 h (task included)
7
VOCABULARY
potenza
base
a alla n
esponente
per
più
meno
diviso
power
base
a to the n
exponent/index
times
plus
minus
divided
8
radice
radice quadrata
radice cubica
alla seconda
alla terza
elevare
inversa
logaritmo
log in base 10
log naturale
root
square root
cube root
squared
cubed
to raise to
inverse
logarithm
common logarithm
natural/neperian log
9
Task: singular work
Every student should create a
sentence using one of the words
of the glossary (except for the last
3) and report to the whole class
10
LESSON 2
Objective: to revise the powers
(contents studied in the I and II class)
Time: 1 h (task included)
11
WHAT ARE POWERS?
There are many particular multiplications in which all the
factors are all the same
For example: 2·2∙2·2·2
Not to write in a such long way, it has been introduced a
new mathematical operation:
the power
So 2·2·2·2·2 is written as 25
12
PROPERTIES
a 1
0
a a
1
0 ?
0
13
PROPERTIES
an · am ≡ an+m
am : an ≡ am-n
(am)n ≡ am·n
14
PROPERTIES
a
n
m
n
1
 n
a
a  a
n
m
15
Task: individual work
Each student should create an
example of the previous
properties and report to the whole
class
16
Indicate if every raltionship is true or false:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
53 = 15
24 = 16
5455 520
(53)7 = 521
00 =1
83 : 8 3 = 0
(73 : 72)0 = 1
(142 : 72) = (14 : 7)2
(153 : 33) = 125
(24)3 : (24)2 = 16
(34)2  3 = 39
23  25 : 22 : 28 = 2-2
T
T
T
T
T
T
T
T
T
T
T
T
F
F
F
F
F
F
F
F
F
F
F
F
Calculate the value of these expressions using powers properties:
1.





:8


8:
8

8

8
:
8

8
2.



:10



:

:10
10

8
8
3.












5

5

5
:
5

5

5

5
:
5
:
5

5
:
5
4.
















3

3

3
:
3

3

3
:
3

3
:
3

4

2

8
:
2

2
22 2
3
4
7
2
3
5
4
4
2
2
33
5
3
2
710
3
215
13
3
2
2
4
5
62
8
0
4
6
24
4
2
3
4
LESSON 3
Objective:
1. To draw “a raised to the power of x”
2. To understand the characteristics of
these functions
Time: 2h
19
First task (individual or pair work)
Calculate the values of the function
y2
x
giving to x positive and negative values
20
Remember!
Properties of the power of the numbers:
a
x
1
 
a
x
21
Second task (individual work)
Calculate the values of the function
1
y 
2
x
giving to x positive and negative values
22
Third task (small groups)
• Draw the two functions on the same
Cartesian Plane
• Compare your results
• Describe their properties
23
Plenary lesson
The teamleaders of the groups report their
conclusions to the whole class
24
Blue:a>1; Red: 0<a<1
8,00
7,00
6,00
5,00
4,00
3,00
2,00
x
1,00
0,5
x
2
0,00
-4,0000 -3,0000 -2,0000 -1,0000 0,0000 1,0000 2,0000 3,0000 4,0000
esponenziale base 2
esponenziale base 0,5
25
Properties
1.
2.
3.
4.
Increasing
Continue
Asymptote: x axis
Positive
26
LESSON 4
Objective: to work with logarithms
Time: 2 h (task included)
27
We call logarithm of b in base a logarithms
the exponent x
we give to the base a to obtain the number b
x = loga b
ax = b
base
Definition:
loga b  x  a  b
x
log2 8  3because2  8
3
30
PROPERTIES:
logab  log a  log b
a
log  log a  log b
b
log a b  b log a
l
31
Changing the base of a logarithm
logac = x → c ≡ ax
so
logbc ≡ logbax ≡ x·logba
Therefore
logb c
x
logb a
or
log
bc
log
a c
log
ba
Task: (individual work):
Calculate (remember the properties!)
1. log (3xy)
6. log 2a a  b
2. log (a2bc3)
3x 2 y
7. log
xy
3. log
3x y 
a 2  b2
4. log
5. log
6
x
3
x
x
8. log
x
2
 y2

a 2  2ab
9. log
xy  y 2
10. log
3a
23
b2  c2

34
Semplify:
1. log x – log y – log z
2. 2log a + 3log b – 5log c
3. 1 log x + 5 log y - 2 log z
3
2
3
5. log 16 – 3log 2 + log 4
4. 2log a –log a3
7. 1 log x 2  1  logx  1
2
6. 1 log 27 – log 5 + 1 log 3
3
2
8. 1
3
log a  2 log b   3log a  log b 
35
LESSON 5
Objective:
1. to apply the definition of logarithm
2. to draw logarithmic functions
3. to understand their carachteristics
Time: 2 h
36
Inverse functions:
y  log2 x
y  log1 x
2
37
First task (in pairs)
Calculate the values of the functions and draw them
on the same cartesian plane
y  log2 x
y  log1 x
2
giving to x positive
values only
38
Second task (small groups)
• Draw the two functions on the same
cartesian plane
• Compare your results
• Describe their properties
39
Red:a>1, Blue: 0<a<1
4,00
3,00
2,00
1,00
0,00
0,0000 1,0000 2,0000 3,0000 4,0000 5,0000 6,0000 7,0000 8,0000
-1,00
-2,00
-3,00
-4,00
40
Exponential equations
• You are dealing with an exponential
equation when the unknown is at the
exponent.
So, in general, an exponential equation is
presented in the form:
ax=b
41
How to solve it?
As you certainly remember, to solve an equation, you pass through
inverse operations (in both member of the equation), in order to
semplify it.
• For example:
 x+3=5
 5x=15
 x 3
x=5-3
x=15:5
x=32
- is the inverse operation of +
: is the inverse operation of •
the power is the inverse operation
of
, the unknown is the base
42
• So, if the unknown is at the exponent…try to
guess how to solve it: which is the inverse
operation of the power when the unknown is at
the exponent?
• ax=b
so
logax=logb
and, using the logarithmic properties:
xloga=logb
so
logb
x
log a
43
First task
• Try to solve the following exponential equations using the
logarithms and their properties:
x
5
 10
2 x  3x  15
1
3x
 27
3
3 2 9
x
x
25
2
3x
2
x
1

4
4x
 2x
16
44
Can you guess how to solve simplier
if you have just powers of the same
base?
And can you try to solve equations
like these?
4  23 0
2x
x
2  42  3  0
2x
x
45
LESSON 6
Objective: to connect maths with other
disciplines
Time: 1 h
46
Applications of exponential equations in
financial mathematics
•
•
•
•
Glossary
Linear capitalization law
Exponential capitalization law
Searching for the time in the capitalization law
47
Glossary
-Matematica finanziaria = financial mathematics
-Interest rate = tasso di interesse
-Montante = total amount
-Valore attuale = current value
48
Linear capitalization law
•
•
•
•
•
I = i C t where:
I = interest
i = interest rate
C = nominal value of the capital
t = time
• So M=C+Cit=C(1+It)
Where M = final amount
total amount with the linear capitalization
5000
Total amount
• In the linear
capitalization law,
interests are always
proportional to the time
and the nominal value
of the capital
4000
3000
2000
1000
0
0
2
4
6
8
10
12
time
49
14
Exponential capitalization
•
In the esponential capitalization the total amount
is calculated on the total amount of the previous
year
0
M=c
1
montante
total amount
7000
2
M1=C+Ci M2=M1+M1i
M1=C(1+i)
M2=M1(1+i)
M2=c(1+i)(1+i)
M2=C(1+i)2
Totaldel
amount
with the
exponential
law
Calcolo
montante
in regime
di interesse
composto
6000
8000
5000
6000
4000
3000
4000
2000
2000
1000
00
M1
00
2
54
6
108
10
1512
14
tempotime
M  C  (1  i)t
50
Searching for the time in the capitalization law
• Since in the capitalization law M=C(1+i)t,
if you are searching for the time you have to solve
an exponential equation:
M=C(1+i)t
M
t
 (1  i)
C
M
log
 t  log(1  i )
C
51