Download Exercises for Thursday and Friday

The University of Sydney
MATH1111 Introduction to Calculus
Semester 1
Week 2 Exercises (Thurs/Fri)
2017
Important Ideas and Useful Facts:
(i) Sets and elements: A set is a collection of objects, referred to as elements. A set may
be represented, for example, by a list of elements surrounded by curly brackets and
separated by commas, or using set builder notation {. . . | . . .}, where the vertical line is
an abbreviation for “such that”. For example, { x | x is a natural number less than 5 }
and {0, 1, 2, 3, 4} represent the same set, whose elements are precisely 0, 1, 2, 3 and 4.
(ii) Element symbol: The symbol ∈ is an abbreviation for “is an element of”, and 6∈ is an abbreviation for “is not an element of”. For example, if A = { x | x is a natural number less than 5 },
then 2 ∈ A, but 5 6∈ A.
(iii) Subset symbols: If A and B are sets and we write A ⊆ B or B ⊇ A , then we mean that
every element of A is also an element of B, and say that A is a subset of B. For example
{1, 2, 3} ⊆ {1, 2, 3, 4} and {1, 2, 3, 4} ⊇ {1, 2, 3}, but {1, 2, 3, 4} 6⊆ {1, 2, 3}.
(iv) Equality of sets: If A and B are sets then A = B if and only if A ⊆ B and B ⊆ A, that
is, A and B have precisely the same elements. Order and repetition are not important.
For example, {1, 2, 3, 4} = {4, 1, 3, 2} = {4, 1, 3, 1, 2, 3}.
(v) Intersection, union and slash: If A and B are sets then put
(a) A ∩ B = { x | x ∈ A and x ∈ B }, called the intersection of A and B.
(b) A ∪ B = { x | x ∈ A or x ∈ B }, called the union of A and B.
(c) A\B = { x | x ∈ A and x 6∈ B }, called A slash B, the result of removing from A all
elements from B.
(vi) Empty set: The empty set is the set without any elements and denoted by ∅. If A is any
set then A ∪ ∅ = A, A ∩ ∅ = ∅ and A\A = ∅.
(vii) Natural numbers: The set N = {0, 1, 2, 3, . . .} of natural numbers forms a number system,
closed under addition and multiplication. A prime number is a natural number greater
than 1 that has no factors other than itself and 1. For example, 2, 3, 5, 7, 11 and 13 are
prime numbers.
(viii) Integers: The set Z = {0, ±1, ±2, ±3, . . .} of integers forms a number system, closed under
addition, subtraction and multiplication.
(ix) Rationals: The set Q = {a/b | a, b ∈ Z, b 6= 0} of rational numbers forms a number system,
closed under addition, subtraction, multiplication and division by nonzero elements. To
add and multiply rational numbers, use the rules
ad + bc
a c
+ =
b d
bd
and
1
a c
ac
· =
.
b d
bd
(x) Decimal expansion of a real number: A real number α has a decimal expansion
α = bn bn−1 . . . b2 b1 · a1 a2 a3 . . .
where the bi and aj are digits from the set {0, 1, 2, . . . , 9}. A real number is rational if and
only if it has a recurring decimal expansion, which means the pattern of digits repeats
forever from some point onwards.
(xi) The real number line: The real numbers form a number system R that is closed under
addition, subtraction, multiplication and division by nonzero elements. We visualise R as
a continuous number line, called the real line, with zero in the middle, negative numbers
to the left and positive numbers to the right.
√ √
(xii) Irrationals: A real number that is not rational is called irrational. For example, 2, 3,
π and e (Euler’s number) are irrational, though the proofs for π and e are difficult. In a
sense that can be made precise, most real numbers are irrational.
(xiii) Significant figures: Real numbers may be approximated by rational numbers with finite
decimal expansions. The number of digits counted to the right from the leftmost positive
digit is called the number of significant figures. For example, real numbers represented
by 26.103, 0.00304 and 0.003040 are quoted to 5, 3 and 4 significant figures respectively.
(xiv) Scientific notation: A positive real number α is expressed in scientific notation if it has
the form
α = b · a1 a2 . . . am × 10k
where m is nonnegative and k is an integer. For example, 193.034 and 0.003040 become
1.93034 × 102 and 3.040 × 10−3 respectively in scientific notation. The number of digits
used in scientific notation is the number of significant figures being quoted, and this avoids
ambiguity in the case of large whole numbers (with zeros as place-holders).
(xv) Inequalities: If α, β ∈ R then we write α < β if β − α is positive, so that α appears to the
left of β on the real number line. In this case we say that α is less than β and β is greater
than α. We write α ≤ β if α = β or α < β.
(xvi) Properties of inequalities: Let a, b, c ∈ R.
(a) If a < b and c ≤ d then a + c < b + d, so that, in particular, a + c < b + c.
(b) If a < b and c > 0 then ac < bc. If a < b and c < 0 then ac > bc.
(c) If 0 < a < b then a1 > 1b > 0 .
(xvii) Solution sets: The solution set (within the real number system R) of an equation or an
inequality involving a variable x is the set of all real numbers x that satisfy the given
equation or inequality. If no real numbers satisfy the given equation or inequality then
the solution set is empty.
(xviii) Interval notation: If a, b ∈ R and a < b then
(a)
(b)
(c)
(d)
(e)
(f)
[a, b] = { x ∈ R | a ≤ x ≤ b }, called a closed interval.
(a, b) = { x ∈ R | a < x < b }, called an open interval.
[a, b) = { x ∈ R | a ≤ x < b }, called half-open or half-closed.
(a, b] = { x ∈ R | a < x ≤ b }, also called half-open or half-closed.
[a, ∞) = { x ∈ R | a ≤ x } and (a, ∞) = { x ∈ R | a < x }.
(−∞, b] = { x ∈ R | x ≤ b } and (−∞, b) = { x ∈ R | x < b }.
2
Exercises labelled with an asterisk are suitable for students aiming for a credit or higher.
Tutorial Exercises:
1.
Use your calculator to evaluate the following expressions, giving your answers to an appropriate number of decimal places or significant figures:
(i) 32.67 + 3.5556
(ii) 32.67 − 3.5556
110.435
0.0062
(iii) 110.435 × 0.0062 (iv)
In parts (iii) and (iv), note that the zero digits in the measurent 0.0062 are place holders
only and not counted as significant.
2.
3.
Solve for x in each of the following cases:
(i) 3x = 63
(ii) 3x − 17 = 4
(iii) 2x = 5x + 6
(iv) −(x − 1) = 2x + 4
x x−1
x
3x 9 − x
1
22
(v)
−
=7
(vi)
=
−
(vii)
−4=
2
3
4
5
10
x−3
7
Find solutions sets, within the real number system R, for the following equations:
(i) 2x = 3
4.
(ii) (x − 1)(x − 2) = 0
(iv) x2 = −7
(iii) 4x(x + 1)(x − 3) = 0
Locate each of the following sets on the real number line and express each as an interval
or as a union of (disjoint) intervals:
(i) { x ∈ R | 2 ≤ x ≤ 4 } (ii) { x ∈ R | − 1 < x ≤ 4 or x ≥ 5 }
(iv) [2, 5)∩(3, 6] (v) [−4, 0)∪[−5, 1] (vi) [−4, 0)∩[−5, 1]
5.
(v) 3 ≤ 4 − x < 7
6.
(vii) (−∞, −1)∩(−3, 6]
Solve the following inequalities, expressing the solution sets as intervals or unions of
intervals, and locate them on the real number line:
(i) 3x − 1 < 8
∗
(iii) [2, 5] ∩ [3, 6]
(ii)
x
2
+ 6 ≥ 14
(iii) 4 + 5x ≤ 3x − 7
(vi) −2 ≥ 6 + 8x ≥ −10
(iv) 7x − 1 > 7 − x
(vii) x2 ≥ 9
(viii) x2 < 5
How many significant figures are being quoted in the following numerical expression:
3, 644, 700, 000 ?
Give several interpretations in scientific notation. Avogadro’s number is estimated to be
(6.0221415 ± 0.0000010) × 1023 .
Interpret this in scientific notation to 7 significant figures.
Further Exercises:
7.
Factorise the expression x2 − 5x and hence solve the equation x2 − 5x = 0.
8.
Write down an algebraic expression, and simplify it, to explain the following mind-reading
exercise:
Think of a number between 1 and 10. Triple it. Add 30. Double what you
have. Now divide by three. Subtract 10. Halve what you have. Subtract the
number you first started with. You are now thinking of the number 5.
3
9.
Rewrite the set X = { x ∈ Z | x2 ≤ 5 } as a list of elements and then decide which of the
following are true:
(i) 0 ∈ X
(ii) −2 ∈ X
(vii) X ⊆ { x ∈ Z | x2 ≤ 9 }
∗
10.
11.
12.
(iv) 5 ∈ X
(v) X ⊆ Z
(vi) Z ⊆ X
(viii) { x ∈ Z | x2 ≤ 3 } ⊆ X (ix) X = { x ∈ Z | x2 ≤ 6 }
Use interval notation to express solution sets for the following inequalities:
(i) (x − 4)(x + 2) > 0
(ii) (2x − 3)(x + 4) < 0
4
4
4
(iv)
>0
(v)
≤0
(vi)
≤1
2−x
2−x
2−x
(iii) x(x + 1)(x − 1) > 0
x
(vii)
<4
x−3
Find the repeating decimal expansions of the following rational numbers:
10
3
(i)
∗
(iii) 3 ∈ X
(ii)
1
7
(iii)
22
7
11
9
(iv)
(v)
9
11
(vi)
8
13
(vii)
49
48
(viii)
120
41
Find exact rationals to represent each of the following repeated decimals:
(i) 0.6̇
(ii) 0.4̇
(iii) 0.1̇2̇
(iv) 0.1̇23̇
(v) 2.4̇323̇
(vi) 0.9̇
(vii) 99.99̇
Short Answers to Selected Exercises:
1.
(i) 36.23 (ii) 29.11 (iii) 0.68 (iv) 18, 000
2.
(i) 21 (ii) 7 (iii) −2 (iv) −1 (v) 40 (vi) 2 (vii)
3.
(i) { 32 } (ii) {1, 2} (iii) {0, −1, 3} (iv) ∅
4.
(i) [2, 4] (ii) (−1, 4] ∪ [5, ∞) (iii) [3, 5] (iv) (3, 5) (v) [−5, 1] (vi) [−4, 0) (vii) (−3, −1)
5.
] (iv) (1, ∞) (v) (−3, 1] (vi) [−2, −1]
(i) (−∞, 3) (ii) [16, ∞) (iii) (−∞, − 11
√ √2
(vii) (−∞, −3] ∪ [3, ∞) (viii) (− 5, 5)
6.
5 s.f. gives 3.6447 × 109 , 6 s.f. gives 3.64470 × 109 , 7 s.f. gives 3.644700 × 109 , etc.
Avogadro’s number is 6.022141 × 1023 , 6.022142 × 1023 or 6.022143 × 1023 to 7 s.f.
7.
x = 0, 5
8.
9.
(3x+30)×2
3
2
− 10
157
50
−x= 5
X = {−2, −1, 0, 1, 2} (i) T (ii) T (iii) F (iv) F (v) T (vi) F (vii) T (viii) T (ix) T
10.
(i) (−∞, −2) ∪ (4, ∞) (ii) (−4, 32 ) (iii) (−1, 0) ∪ (1, ∞) (iv) (−∞, 2) (v) (2, ∞)
(vi) (−∞, −2] ∪ (2, ∞) (vii) (−∞, 3) ∪ (4, ∞)
11.
(i) 3.3̇ (ii) 0.1̇42857̇ (iii) 3.1̇42587̇ (iv) 1.2̇ (v) 0.8̇1̇ (vi) 0.6̇15384̇ (vii) 1.02083̇
(viii) 2.9̇2682̇
12.
(i)
2
3
(ii)
4
9
(iii)
4
33
(iv)
41
333
(v)
737
303
(vi) 1 (vii) 100
4