MATH1720 Mathematics Fundamentals PDF

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Department of Mathematics and Statistics

M AT H 1 7 2 0 M AT H E M AT I C S F U N D A M E N TA L S

2

Acknowledgements: The following members of the Department of Mathematics and Statistics, UWA, have contributed in one way or another for the production of these Lecture Notes: G. Coates, D. Hill, R. Honeybul, M. Matthews.

Contents

Formula Sheet

5

1. Numbers and operations 2. Algebra

7

19

3. Simultaneous equations 4. Quadratics 5. Indices

27

31 41

6. Re-arranging formulae

51

7. Exponentials and logarithms

55

8. Equations involving logarithms and/or exponentials 9. Quadratic equations 10. Functions and graphs

71 79

11. Quadratic functions and Parabolae

93

61

4

12. Calculus

99

Appendix 1: Answers to exercises Appendix 2: Solutions to problem sets Index

181

109 125

Formula Sheet The laws of indices km = km−n kn

km × kn = km+n 0

1

k =1

k =k

k

1 kn

1

kn =

√ n

−1

 − 1 j k = k j

1 = k

 n jn j = n k k

( jk )n = j n × k n k−n =

(k m )n = k mn

1

k

k2 =

√

k

The laws of logarithms

  x logb = logb x − logb y y

logb ( xy ) = logb x + logb y logb ( xr ) = r logb x logb b = 1

logb 1 = 0

e and the natural logarithm

es = e s−t et

e s e t = e s+t

(e s )t = e st

e0 = 1 ln(st) = ln s + ln t

e1 = e ln

ln 1 = 0

 s t

ln (sr ) = r ln s

= ln s − ln t ln e = 1

The quadratic formula

If

2

ax + bx + c = 0

e−s =

then

x=

−b ±

√

b2 − 4ac 2a

1 es

1. Numbers and operations Types of Numbers The natural numbers are the usual numbers we use for counting: 1, 2, 3, 4, · · · We also have the very special number zero 0 and the negative numbers

· · · − 4, −3, −2, −1 which are used to indicate deficits (that is, debts) or to indicate opposite directions. These numbers collectively form the integers

· · · − 4, −3, −2, −1, 0, 1, 2, 3, 4, · · ·

The basic operations of arithmetic Hopefully, the everyday operations +, −, × and ÷ are familiar: (i) 3 + 4 = 7

(iii) 4 × 9 = 36

(ii) 11 − 6 = 5

(iv) 24 ÷ 6 = 4

Arithmetic involving negative numbers When negative numbers are involved we need to be careful, and must use brackets if necessary. Multiplication examples: (i) 4 × (−7) = −28

(iii) (−3) × (−6) = 18

(ii) (−5) × 6 = −30

(iv) 3 × 5 = 15

Other examples: (i) 13 + (−8) = 5

(iv) −16 − 13 = −29

(ii) 21 − (−3) = 24

(v) (−12) ÷ 4 = −3

(iii) −16 + 13 = −3

(vi)

−12 =3 −4

positive × negative = negative negative × positive = negative negative × negative = positive positive × positive = positive

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math1720 mathematics fundamentals

BIDMAS. The order of operations Consider the simple question: What is 2 + 3 × 4?

Which operation (+ or ×) do we perform first?

Mathematical expressions are not simply calculated from left to right. They are performed in the following order: B:

Expressions within Brackets (. . .)

I:

Indices (powers, square roots, etc)

DM: AS:

We will consider Indices in detail later

Divisions (÷) and Multiplications (×) Additions (+) and Subtractions (−)

A couple of things to note: • Division and multiplication have the same precedence. • Addition and subtraction have the same precedence. • Operations with the same precedence are performed left to right. Some BIDMAS examples (i) 2 + 3 × 4 = 2 + 12 = 14 (ii) (4 + 6) × (12 − 4) = 10 × 8 = 80 (iii) 4 + 7 × (6 − 2) = 4 + 7 × 4 = 4 + 28 = 32 (iv) 3 × (4 + 8) − 2 × (10 − 6) = 3 × 12 − 2 × 4 = 36 − 8 = 28

When brackets are involved we can omit the multiplication sign. So instead of 3 × (4 + 8) − 2 × (10 − 6) we could write 3(4 + 8) − 2(10 − 6)

More examples

4÷2×3÷3

= 2×3÷3

= 6÷3

=2

− [2 − 3(4 − 6 ÷ 2 − (−3))] = −[2 − 3(4 − 3 + 3)] = −[2 − 3 × (1 + 3)] = −(2 − 3 × 4) = −(2 − 12) = −(−10)

= 10

When brackets are nested we usually use different sorts, e.g. (· · · ) [· · · ] {· · · }

1. numbers and operations

9

Word problems We are sometimes asked quesions in written or spoken conversation that we will be able to use mathematics to help us solve. Our first task is to turn the ‘word problem’ into a mathematics problem. Example: What is three times the difference between 12 and 4? Answer: 3 × (12 − 4)

= 3×8

= 24

Another example: Joy takes the product of the sum of two and six, and the difference between nineteen and nine. She then result the answer by four. What number does she arrive at? Answer: {(2 + 6) × (19 − 9)} ÷ 4

= {8 × 10} ÷ 4 = 80 ÷ 4 = 20

The number line The integers sit nicely on a number line: ✛ -5

✲ -4

-3

-2

-1

0

1

2

3

4

5

Fractions These are numbers that sit between the integers. For example, the 1 number “one-half”, which we write “ ”, is exactly halfway between 2 0 and 1. 1 2 ✛ ✲ s -5 -4 -3 -2 -1 0 1 2 3 4 5

Fractions are ratios of integers, that is, p q where the numerator p and denominator q are integers. When the numerator (top line) is smaller than the denominator (bottom line) we call this a proper fraction.

Note that p or q or both can be negative

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math1720 mathematics fundamentals

2 The number “four and two-fifths” is written “4 ” and is between 5 4 and 5. To pinpoint this number, imagine splitting up the interval between 4 and 5 into fifths (five equal pieces). Now, starting at 4, move right along two of these fifths. ✛ -5

✲

s -4

-3

-2

-1

0

1

2

3

4

5

2 We call 4 a mixed numeral, as it is the sum of an integer and a 5 proper fraction.

Converting mixed numerals to improper fractions An improper fraction is one in which the numerator is larger than the denominator. 1 Example: We wish to convert the mixed numeral 3 into an 2 improper fraction. We have three wholes and one half:

or three lots of two halves (making six) plus one half is seven halves:

so

3

1 7 = 2 2

Arithmetically, we write 3

7 6+1 3×2+1 1 = = = 2 2 2 2

Some examples 4

13 12 + 1 4×3+1 1 = = = 3 3 3 3 2

1

1 9 8+1 2×4+1 = = = 4 4 4 4

5 11 6+5 1×6+5 = = = 6 6 6 6

3 43 40 + 3 8×5+3 = = = 5 5 5 5 Note: Improper fractions turn out to be used more often in maths, mainly because mixed numbers are confusing in algebra. 8

1. numbers and operations

11

Multiplication of fractions To multiply two fractions simply multiply their numerators and denominators separately. For example, 1 5 × 2 6 1×5 2×6 5 12

= =

Think of this as finding “one-half of five-sixths”

Another example is 3 2 × 5 7

3×2 5×7

=

=

6 35

This is referred to as inline notation

Division of fractions 2 The notation 2 ÷ 3 means the same as , that is “two-thirds”. Now 3 think of 2 and 3 as fractions, that is 2÷3 =

2 3 ÷ 1 1

We can also use fraction multiplication to say that 2 1 2 = × 1 3 3 Comparing these last two we have 2 1 2 3 ÷ = × 1 3 1 1 That is, dividing by a fraction is equivalent to multiplying by its reciprocal, that is, the fraction having been inverted. For example, 2 ÷ 7 2 × 7 18 35

= =

5 9 9 5

Examples 5 1 2 × ÷ 7 6 3 1 4

× 3 5

2

7 8

7 32 3 5

=

1 2 5 ÷ ×3 7 4 3

=

=

5 2 ÷ 6 21

=

=

3 7 ÷ 32 5

13 2 22 ÷ × 7 3 4

=

6 2 × 5 21

=

=

5 7 × 3 32

22 13 3 × × 7 2 4

12 105

=

=

35 96 858 56

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math1720 mathematics fundamentals

Factorization of numbers This is the process of expressing a given number as a product of two (or more) smaller numbers. For example, 28 = 4 × 7

= 2 × 54 = 2×6×9

108

64

= 2 × 32 = 2 × 2 × 16 ··· = 2×2×2×2×2×2

Later we’ll write this as 64 = 26

Cancellation of numbers The process of simplifying fractions by looking for factors which are common to both the numerator and the denominator and then eliminating (cancelling) them. For example, 18 20

=

✁2 × 9 ✁2 × 10

=

9 10

100 250

=

10 × 10 ✚ 25 × ✚ 10 ✚

=

10 25

=

5✁ × 2 5✁ × 5

=

2 5

24 108

=

4✁ × 6 ✁4 × 27

=

6 27

=

3✁ × 2 ✁3 × 9

=

2 9

Adding and subtracting fractions Consider the sum

1 2 + 3 5 We can’t add these fractions yet because thirds are different to fifths. However, we can make use of reverse cancellation to make the denominators the same: 1 2 + 5 3

= =

2×3 1×5 + 5×3 3×5 6 5 + 15 15

Now we just have a total of 5 + 6 = 11 fifteenths, that is 1 2 + 3 5

= =

5+6 15 11 15

1. numbers and operations

9 2 − . Following 20 5 the above pattern the denominator would be 20 × 5 = 100, but we can simplify the numbers a little as follows. Since 5 divides into 20 we could just do this instead: The same applies to subtraction. For example,

2 9 − 5 20

= = = =

9 − 20 9 − 20 9−8 20 1 20

2×4 5×4 8 20

Some examples

(a )

1 4 + 7 4

=

( b)

16 7 + 28 28

2 1 5 + − 54 9 6

=

1×7 4×4 + 4×7 7×4

=

=

21 5 − 54 54

23 28

=

5 9 12 − + 54 54 54 16 54

=

Decimals Decimals are a convenient and useful way of writing fractions with denominators 10, 100, 1000, etc. 2 is written as 0.2 10 35 is written as 0.35 100 The number 237.46 is shorthand for

4 is written as 0.04 100 612 is written as 0.612 1000

6 4 + 10 100 The decimal point separates the whole numbers from the fractions. To the right of the decimal point, we read the names of the digits individually. For example, 237.46 is read as ‘two hundred and thirty-seven point four six’. The places after the decimal point are called the decimal places. We say that 25.617 has 3 decimal places. 2 × 100 + 3 × 10 + 7 × 1 +

which if required can be simplified: ✁2 × 8 = 8 = 27 2 ✁ × 27

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math1720 mathematics fundamentals

Some fractions expressed as decimals We can write 3 10

as

0.3

−

47 100

as

0.47

as

931 1000 6 100

−

23 100

4 10

as

−0.4

2 12 = 1+ 10 10

as

1.2

−0.23

27 327 = 3+ 100 100

as

3.27

as

0.931

28 1000

as

0.028

as

0.06

5 1000

as

0.005

Common fractions expressed as decimals All fractions can be written as decimals. However, the decimal sequence may go on forever. Examples 1 = 0.1 10

1 = 0.01 100

1 = 0.5 2 3 = 0.75 4 1 = 0.3333 · · · 3 1 = 0.142857142857 · · · 7

1 = 0.25 4 1 = 0.2 5 2 = 0.6666 · · · 3

1 = 0.001 1000 1 = 0.125 8 2 = 0.4 5 2 − = −0.4 5 1 22 = 3 + = 3.1428 · · · 7 7

The · · · indicate that the sequence goes on forever

Fixing the number of decimal places When working with decimals we usually limit ourselves to a certain number of decimal places. Examples To three decimal places 1 = 0.333 3

1 = 0.143 7

1 = 0.125 8

2 = 0.400 5

Multiplying and dividing decimal numbers by 10 To multiply a decimal number by 10 simply move the decimal point to the left, and to divide a decimal number by 10 move the decimal point to the right Examples 0.47 × 10 = 4.7

−0.12 × 10 = −1.2

0.06 × 10 = 0.6 0.000234 × 10 = 0.00234

Note that we have ‘rounded up’ the 2 to 3 because the 8 after the 2 in 1 = 0.14285714 · · · is larger than 5 7 (out of 10) Note that we can ‘pad out’ with zeros 2 to the left in the decimal for 5

1. numbers and operations

0.000234 × 100 = 0.0234 0.378 ÷ 10 = 0.0378 2.571 ÷ 10 = 0.2571

0.56 × 100 = 56.0

− 0.04 ÷ 10 = −0.004 4.63 ÷ 100 = 0.0463

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math1720 mathematics fundamentals

Exercises 1 1.1 Evaluate the following (i) 3(4 + 2 × 6) + 12 ÷ 4

(ii) 6 + 2[25 − 3(2 + 5)]

1.2 What is the difference between the product of 6 and 7, and the sum of 3 and 8? 1.3 Convert to improper fractions

(a ) 1

1 4

( b) 2

3 8

1.4 Evaluate (that is, calculate)

(a )

3 1 × 5 4

( b)

2 1 ÷ 3 5

1.5 What fraction is two-thirds of four-fifths? 1.6 Evaluate (i)

3 2 + 5 4

(ii)

2 2+4 − 5−1 3

1.7 Reduce to simplest form by using cancellation: 78 102 1.8 What is one-quarter of the sum of one-third and two-sevenths?

Problem set 1 1.1 Evaluate the following. (a) 4 + 4 × 4 ÷ 4 − 4

(b) 4 + 4 × (4 ÷ 4 − 4)

(c) 2 × 4 − (8 − 1) (d) −14 ÷ 2 − (−20) ÷ (−5) (e) (−4 × 12) − [32 − (−4)] (f) [4 × (3 + 4) − 21] ÷ [2 × 14 ÷ 7 + 3] (g) 3 × 8 − [−16 − (−4)] (h) [−3 × (3 − 4) − 19] ÷ [8 ÷ 4 × 7 − 6] (i) (8 − 4) ÷ (4 + 4) − 8 (j) 4 + 4 × 4 ÷ (4 − 4)

1. numbers and operations

1.2 Compute the following, leaving your answer in the simplest fraction form. (a) (c) (e) (g)

1 5 3 + + 2 4 12 4 2 +1 3 24 4 3 × 5 7 1 5 7 ×1 ÷ 8 7 9

(b) (d) (f) (h)

3 3 4 − + 5 10 4 4 3 2 1 −2 +1 5 5 3 2 2 2 ÷4 9 3 3 34 2 ÷ 7 14

1.3 What is half of the difference between a third of 60 and a quarter of 80? 1.4 What is the sum of half of 45 and a third of the product of two 1 thirds and 1 ? 8 1.5 James and his wife Sweet Li have a 12 year old daughter called Lyn. James is 50 year old and Lyn is 12. Half of his age added to five thirds of Lyn’s age gives the age of Sweet Li. How old is Sweet Li? 1.6 Swee Khum put a third of her savings in the bank, a third in bonds, a quarter of the remainder in stocks and the rest in fixed deposit. If her total amount is $600,000 how much did she put in fixed deposit?

17

2. Algebra Algebra is a powerful tool for expressing fundamental relationships and solving real-world problems. An example: In the expression E = mc2 , the variables are the energy (E) of an object and its mass (m), c2 means c × c and c is a constant representing the inconveniently large value of the speed of light in a vacuum, i.e. 29,792,458 metres per second. The famous equation above summarises the relationship between the variables E and m We can also use algebra to express our rules for multiplying and dividing fractions:

Such constants appearing in relationships are called parameters An equation is something with an equals sign in it

a ad a d c ÷ = × = bc c b b d

a ac c × = bd b d

An example of algebra in action A simplified version of the formula for the length of time it takes a child to say “are we nearly there yet?” on a long car journey is T=

1+A C2

This formula was proposed by Prof. Dwight Barkley, Warwick University

where T = time to say “are we nearly there yet?” (in hours), A = number of activities provided and C =number of children. So, 2 children with 3 activities will take 1+3 22 4 = 4 = 1 hour to get restless

T=

Solving equations In many situations we are told the numerical values of all variables in an equation except one and asked to determine the value of that variable. That is, we are asked to solve for the unknown variable. Example. In the restlessness problem above we might know that there were 2 children in the back and it took them an hour and a half to complain. How many activities were they given? We have 1

1+A 1 = 2 22

that is

3 1+A = 2 22

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math1720 mathematics fundamentals

To answer questions like this we need to know how to re-arrange equations. Simple Equations Here are some typical equations with an unknown variable: x+4 = 7

2x + 7 =

3x +8 2

3 2 +4 = x x Our aim is to find out what value of the unknown variable x makes the equation work. That is, we wish to “solve for x”. Operations on equations Let’s look at x+4 = 7 To get x on its own, we can “remove” the other operations (in this case the “+ 4”) by performing operations that “undo” them. To maintain equality, operations must be applied to the entirety of both sides. Operations include: (1) Multiply (or divide) both sides of an equation by a non-zero number (2) Add (or subtract) the same number to (from) both sides of an equation Using these operations effectively is just a matter of technique and practice. Example:To solve x+4 = 7 subtract 4 from both sides: x + 4✁−4✁ = 7 − 4

⇒

x=3

The symbol ⇒ means ‘implies that’, and the diagonal bars indicate cancellation. It’s always a good idea to check that the value we have obtained is in fact correct. Clearly 3 + 4 = 7 so we are correct. Example: To solve 3x = 12 divide each side by 3, 12 ✁3 x = 3 ✁3 ⇒ x=4

⇒ Check: 3 × 4 = 12

X

2. algebra

21

Example: To solve y − 4 = 2y + 10 you need to get the two y terms on the same side,

⇒

⇒

y − 4−y = 2y + 10−y

⇒

−4 = y + 10 (since “two y’s” minus “one y” is “one y ”.)

−4 − 10 = y + 10 − 10

⇒

−14 = y

Check: −14 − 4 = −18 and 2 × (−14) + 10 = −18

hence X

Example: This one presents a new problem: 3x + 2 = 2( x + 7) Normally, we perform the operations in the brackets first but there is nothing we can do with x + 7. We need to expand the brackets. We do this with the aid of The Distributive Law: a (b + c) = ab + ac where a, b and c all represent terms....