Electrophilic Aromatic Substitution and Nucleophilic Aromatic Substitution
Formulas 5 5.1Sequences 5.2Introduction to Functions 5.3Simple Algebraic Fractions Chapter Summary...
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Transcript of Formulas 5 5.1Sequences 5.2Introduction to Functions 5.3Simple Algebraic Fractions Chapter Summary...
Formulas5
5.1 Sequences
5.2 Introduction to Functions
5.3 Simple Algebraic Fractions
Chapter Summary
Case Study
5.4 Formulas and Substitution
5.5 Change of Subject
P. 2
Case Study
The following table shows the range of the BMI and the corresponding body condition.
Body Mass Index (BMI) is frequently used to check whether a person’s body weight and body height are in an appropriate proportion. It can be calculated by using the following formula:
)m(in heightBody
kg)(in t Body weigh (BMI)Index MassBody
22
For example, if a person’s weight and height are 50 kg and 1.6 m respectively, then the BMI 50 1.62 19.5.
BMI ConditionLess than 18.5 Underweight
18.5 – 22.9 Ideal23 – 24.9 Overweight25 – 29.9 Obese30 or over Severely obese
Let us check your body mass index and see whether you are normal or not.
Is my weight normal?
P. 3
We call such a list of numbers a sequence.
A. Introduction to Sequences
John recorded the weights of 10 classmates. He listed their weights (in kg) in the order of their class numbers.
Each number in a sequence is called a term.
We usually denote the first term as T1, the second term as T2, and so on.
In the above sequence, T1 46, T2 42, T3 50, ... , etc.
5.1 Sequences
P. 4
5.1 Sequences
B. Some Common Sequences
Some sequences have certain patterns, but some do not.
1. Consider the sequence 1, 5, 9, 13, ... .
When the difference between any 2 consecutive terms is a constant, such a sequence is called an arithmetic sequence, and the difference is called a common difference.
We can guess the subsequent terms: 17, 21, 25, ...
2. Consider the sequence 8, 16, 32, 64, ... .
When the ratio of each term (except the first term) to the preceding term is a constant, such a sequence is called a geometric sequence, and the ratio is called a common ratio.
We can guess the subsequent terms: 128, 256, 512, ...
+4 +4
2
2
P. 5
B. Some Common Sequences
5.1 Sequences
3. We can arrange some dots to form some squares.
4. We can arrange some dots to form some triangles.
The number of dots used in each squareis called a square number.
The number of dots used in each triangleis called a triangular number.
5. Consider the sequence 1, 1, 2, 3, 5, 8, ... .In this sequence, starting from the third term, each term is the sum of the 2 preceding terms. This sequence is called the Fibonacci sequence.
P. 6
5.1 Sequences
C. General Terms
For a sequence that shows a certain pattern, we can use Tn to represent the nth term.
It is a common practice to write the general term of a sequence as an algebraic expression in terms of n.
Tn is called the general term of the sequence.
For example, since the sequence 2, 4, 6, 8, 10, ... has consecutive even numbers, we can deduce that the general term of this sequenceis Tn 2n.Once the general term of a sequence is obtained, we can use it to describe any term in a sequence.
P. 7
(a) The sequence 7, 14, 21, 28, ... can be written as7(1), 7(2), 7(3), 7(4), ...
5.1 Sequences
C. General Terms
Example 5.1T
... ,16
1 ,
8
1 ,
4
1 ,
2
1
Find the general terms of the following sequences.(a) 7, 14, 21, 28, ... (b) 1, 2, 4, ...
(c) (d) 15, 14, 13, 12, ...
Solution:
(b) The sequence 1, 2, 4, ... can be written as21 1, 22 1, 23 1 ...
∴ The general term of the sequence is 7n.
∴ The general term of the sequence is 2n 1.
P. 8
5.1 Sequences
C. General Terms
Example 5.1T
Solution:
... ,16
1 ,
8
1 ,
4
1 ,
2
1
Find the general terms of the following sequences.(a) 7, 14, 21, 28, ... (b) 1, 2, 4, ...
(c) (d) 15, 14, 13, 12, ...
... ,21 ,
21 ,
21 ,
21 4321
(c) The sequence can be written as... ,16
1 ,
8
1 ,
4
1 ,
2
1
n
2
1∴ The general term of the sequence is .
(d) The sequence 15, 14, 13, 12, ... can be written as16 1, 16 2, 16 3, 16 4, ...
∴ The general term of the sequence is 16 n.
P. 9
5.1 Sequences
C. General Terms
Example 5.2TFind the general term and the 9th term of each of the following sequences.(a) 1, 2, 5, 8, ... (b) 1, 4, 9, 16, ...
Solution:Rewrite the terms of the sequence as expressions in which the order of the terms can be observed.
(a) T1 1T2 2T3 5T4 8∴ Tn 3n 4
3 4 6 4 9 4 12 4
(b) T1 1T2 4T3 9T4
16 ∴ Tn n2
12 22
32
42
T9 3(9) 4 23
T9 92
81
+3 +3 +3
3(1) 4 3(2) 4 3(3) 4 3(4) 4
P. 10
5.2 Introduction to Functions
Consider a sequence with the general term Tn 5n 2.
The above figure shows an ‘input-process-output’ relationship, which is called a function.
In this example, we call Tn a function of n.
In the previous section, we learnt how to find the values of the terms in a sequence from the general term by substituting different values of n in the general term.
For each input value of n, there is exactly one output value of Tn.
P. 11
For every value of x, there is only one corresponding value of y.
Suppose that each can of cola costs $5.
Let x be the number of cans of cola, and $y be the corresponding total cost.
Since the total cost of x cans of cola is $5x, the equation y 5x represents the relationship between x and y.The following table shows some values of x and the correspondingvalues of y.
5.2 Introduction to Functions
The idea of function is common in our daily lives.
x 1 2 3 4 5
y 5 10 15 20 25
We say that y is a function of x.
P. 12
5.2 Introduction to Functions
Example 5.3T
5
2
If p is a function of q such that p 4q 5, find the values of p for the following values of q.(a) 6 (b) 5 (c)
Solution:
Substitute q 6 into the expression 4q 5.
(a) When q 6, p 4(6) 5
(b) When q 5, p 4(5) 5 25
(c) When q , p 5
25
5
24
55
8
5
17
19
P. 13
we call such an expression an algebraic fraction.
,)2)(1(
6or
12
5 ,
4
3 2
xx
x
b
b
b
A. Simplification
5.3 Simple Algebraic Fractions
b
aWe learnt at primary level that numbers in the form , where
a and b are integers and b 0, are called fractions.
When both the numerator and the denominator of a fractional expression are polynomials, where the denominator is not a constant, such as:
Note that is
not an algebraic fraction because the denominator is a constant.
5
43 yx
P. 14
For example:
3
2312
212
36
24
For algebraic fractions, we can simplify them in a similar way, when the common factors are numbers, variables or polynomials.
2
23
3
2)3(2
)2(2
6
4
a
aa
a
a
a
A. Simplification
5.3 Simple Algebraic Fractions
We can simplify a numerical fraction, whose numerator and denominator both have common factors, by cancelling the common factors.
For example:
P. 15
ucua
ucua
42
105 (a)
y
yy
21
714 (b)
2
A. Simplification
5.3 Simple Algebraic Fractions
Example 5.4T
ucua
ucua
42
105
y
yy
21
714 2
Simplify the following algebraic fractions.
(a) (b)
First factorize the numerator and the denominator. Then cancel out the common factors.
Solution:
2
5
)2(2
)2(5
cau
cau
3
12 y)3(7
)12(7
y
yy
3
21or
y
We cannot cancel common factors from the terms 7y and 21y only, i.e., the fraction cannot be simplified as
.3
14
21
714 22 y
y
yy
3
P. 16
hk
knkmnhmh
2
422 (a)
2)26(
3 (b)
x
x
A. Simplification
5.3 Simple Algebraic Fractions
Example 5.5T
hk
knkmnhmh
2
422Simplify the following algebraic fractions.
(a) (b)2)26(
3
x
x
Solution:
nm 2
hk
nmknmh
2
)2(2)2(
hk
nmkh
2
)2)(2(
2)]3(2[
3
x
x
)3(4
1
x
2)3(4
3
x
x
First factorize the numerator. You may check if 2k + h (the denominator) is a factor of the numerator.
P. 17
B. Multiplication and Division
5.3 Simple Algebraic Fractions
For example:
We can perform the multiplication or division of algebraic fractions in a similar way.
When multiplying or dividing a fraction, we usually try to cancel out all common factors before multiplying the numerator and the denominator separately to get the final result.
For example:
6
5)3(3
)5(5
)5(2
3
9
25
10
3
a
b
aa
bb
b
a
a
b
b
a
6
5
)3(3
)5(5
)5(2
3
9
25
10
3
2
2
2
3
P. 18
32
2 8
864
2 (a)
k
t
tt
k
205
24
4
6 (b)
2
n
q
n
q
B. Multiplication and Division
5.3 Simple Algebraic Fractions
Example 5.6T
Solution:
32
2 8
864
2
k
t
tt
k
Simplify the following algebraic fractions.
(a) (b)205
24
4
6 2
n
q
n
q
3
2 8
)8(8
2
k
t
tt
k
tk4
1
224
)4(5
4
6
q
n
n
q
q4
5
P. 19
a
aaa
9
79
52
9
5
9
2
a
aaa
18
718
)2(2)3(1
9
2
6
1
C. Addition and Subtraction
5.3 Simple Algebraic Fractions
For example:
The method used for the addition and subtraction of algebraicfractions is similar to that for numerical fractions.
When the denominators of algebraic fractions are not equal, first we have to find the lowest common multiple (L.C.M.) of the denominators.
For example:
P. 20
hh 8
3
2
7
C. Addition and Subtraction
5.3 Simple Algebraic Fractions
Example 5.7T
hh 8
3
2
7 Simplify .
Solution:
hh 8
3
8
)4(7
h8
328
h8
25
P. 21
uvvu
3
3
62
5
C. Addition and Subtraction
5.3 Simple Algebraic Fractions
Example 5.8T
Solution:
Simplify .uvvu
3
3
62
5
)3(2
1
uv
)3(
3
)3(2
5
vuvu
)3(2
)2(35
vu
)3(2
1
vu
For any number x 0,
xxx
111
P. 22
)3(3
275
pp
p
C. Addition and Subtraction
5.3 Simple Algebraic Fractions
Example 5.9T
pp
3
93
4
Simplify .
Solution:
Since 3(p 3) and p have no common factors other than 1, the L.C.M. of 3(p 3) and p is 3p(p 3).
pp
3
93
4 pp
3
)3(3
4
)3(3
)3(334
pp
pp
)3(3
2794
pp
pp
P. 23
5.4 Formulas and Substitution
Consider the volume (V) of a cuboid:
V lwh where l is the length, w is the widthand h is the height.
If the values of the variables l, w and h are already known, we can find V by the method of substitution.
Similarly, if the values of V, l and w are known, we can find h.
Actually, in any given formula, we can find any one of the variables by the method of substitution if all the others are known.
P. 24
22h
5.4 Formulas and Substitution
Example 5.10T
4
32 hhrT Consider the formula . Find the value of h if T 121
and r 2.5.
Solution:
4
34
3
2
2
rhT
hhrT
4
35.2121 2h
4
3
2
5121
2
h
4
3
4
25121 h
2
11121
h
11
2121h
Factorize the expression.
P. 25
5.5 Change of Subject
2
bhA
Given the area (A) of a triangle:
where b is the base length and h is the height.
In the formula, A is the only variable on the left-hand side.
We call A the subject of the formula.
This formula can be used if we want to find A, when b and h are known.
b
Ah
2
In another case, if we need to find h when A and b are known, it is more
convenient to use another formula , with h being the subject.
The process of obtaining the formula from is called the change of subject. 2
bhA
b
Ah
2
We can use the method of solving equations to change the subject of a formula.
P. 26
ukuk
543543
5
43 ku
5.5 Change of Subject
Example 5.11TMake u the subject of the formula 3k 4 5u.
Solution:
First move the other terms to one side such that only the variable u remains on the other side.
435 ku Rewrite the formula such that only thevariable u is on the L.H.S.
4 is transposed to the L.H.S. to become 4.
P. 27
prm
rmp132
321
rp
prm
3
2
)3(2 rpmpr
5.5 Change of Subject
Example 5.12T
rmp
321 Make m the subject of the formula .
Solution:
pr
rp
m
32
prrpm 2)3(
Simplify the fractions.The L.C.M. of r and p is pr.
Multiply both sides by mpr.
P. 28
mhmkm
mhk
2)23(23
2
)1(2
3
k
hkm
5.5 Change of Subject
Example 5.13T
m
mhk
23
2
Make m the subject of the formula .
Solution:
First move all the terms that involve m to one side.
)1(23 kmhk
mhmkk 223 Remove the brackets
mkmhk 223 Take out the common factor m.
hkkm 3)1(2
P. 29
)24(9
2Tt 6
)27(9
2
∴ It takes 6 hours for the temperature of the food to become 3C.
5.5 Change of Subject
Example 5.14T
2
924
tT
A bag of food is put into a refrigerator. The temperature T (in C) of
the food after time t (in hours) is given by the formula .
(a) Make t the subject of the formula.(b) How long will it take for the temperature of the food to become
–3C?
Solution:
Tt
tT
242
92
924(a) (b) )]3(24[
9
2 t
P. 30
Chapter Summary
5.1 Sequences
A list of numbers arranged in an order is called a sequence.
Each number in a sequence is called a term.
For a sequence with a certain pattern, we can represent the sequence by its general term.
P. 31
5.2 Introduction to Functions
Chapter Summary
A function describes an ‘input-process-output’ relationship between 2 variables. Each input gives only one output.
P. 32
5.3 Simple Algebraic Fractions
Chapter Summary
The manipulations of algebraic fractions are similar to those of numerical fractions.
P. 33
5.4 Formulas and Substitution
Chapter Summary
A formula is any equation that describes the relationship of 2 or more variables.
By the method of substitution, we can find the value of a variable in a formula when the other variables are known.