Mathematics Applications and Interpretation SL (and HL) Lecture Notes

1.1 Numbers – Rounding – Scientific Form - % Error

• Notation for Sets of Numbers:

  • $N = {0, 1, 2, 3, 4, …}$ (natural numbers)

  • $Z = {0, \pm1, \pm2, \pm3, …}$ (integers)

  • $Q = {\frac{a}{b} : a,b \in Z, b \neq 0}$ (rational numbers, fractions of integers)

  • $R = \text{rational} + \text{irrational}$ (real numbers)

• Known Irrational Numbers:

  • $\sqrt{2}, \sqrt{3}, \sqrt{5}$ and all $\sqrt{a}$ where $a$ is not a perfect square.

  • $\pi = 3.14159…$

  • $e = 2.7182818…$

• Particular Subsets:

  • $Z^+ = {1, 2, 3, …}$ (positive integers)

  • $Z^- = {-1, -2, -3, …}$ (negative integers)

  • $Z^* = {\pm1, \pm2, \pm3, …}$ (non-zero integers, i.e., $Z^* = Z - {0}$)

  • Similar notations apply for other sets.

• Intervals of Real Numbers:

  • $x \in [a,b]$ for $a \leq x \leq b$

  • $x \in ]a,b[$ or $x \in (a,b)$ for a < x < b

  • $x \in [a,b[$ or $x \in [a,b)$ for a \leq x < b

  • $x \in [a,+\infty[$ or $x \in [a,+\infty)$ for $x \geq a$

  • $x \in ]-\infty,a]$ or $x \in (-\infty,a]$ for $x \leq a$

  • $x \in ]-\infty,a[ \cup [b,+\infty[$ for $x \leq a$ or $x \geq b$

Decimal Places vs. Significant Figures

• Consider the number $123.4567$. There are two ways to round it:

  • To a specific number of decimal places (d.p.)

  • To a specific number of significant figures (s.f.) (counting from the first non-zero digit).

• Rounding Rules:

  • If the following digit is 0, 1, 2, 3, or 4, the digit at the critical position remains as it is.

  • If the following digit is 5, 6, 7, 8, or 9, the digit at the critical position increases by 1.

• Examples:

  • $123.4567$ to 1 d.p. is $123.5$

  • $123.4567$ to 2 d.p. is $123.46$

  • $123.4567$ to 3 d.p. is $123.457$

  • $123.4567$ to 4 s.f. is $123.5$

  • $123.4567$ to 5 s.f. is $123.46$

  • $123.4567$ to 6 s.f. is $123.457$

  • $123.4567$ to the nearest integer is $123$

  • $123.4567$ to the nearest 10 is $120$

  • $123.4567$ to the nearest 100 is $100$

  • $123.4567$ to 2 s.f. is $120$

  • $123.4567$ to 1 s.f. is $100$

• Example 1: Consider the number $0.04362018$

  • To 2 d.p.: $0.04$

  • To 3 d.p.: $0.044$

  • To 4 d.p.: $0.0436$

  • To 6 d.p.: $0.043620$

  • To 2 s.f.: $0.044$

  • To 3 s.f.: $0.0436$

  • To 4 s.f.: $0.04362$

  • To 5 s.f.: $0.043620$

• Important Remark: In the final IB exams, answers should be given either in exact form or to 3 s.f.

  • Example:

    • $2$ (exact form) $\approx 1.41$ (to 3 s.f.)

    • $2\pi \approx 6.28$

    • $12348 \approx 12300$

The Scientific Form $a \times 10^k$

• Any number can be written in the form $a \times 10^k$, where 1 \leq a < 10.

• Move the decimal point after the first non-zero digit.

• Example: $123.4567 = 1.234567 \times 10^2$. The decimal point was moved 2 positions to the left, so $k = 2$.

• For a small number, e.g., $0.000012345 = 1.2345 \times 10^{-5}$. The decimal point was moved 5 positions to the right, so $k = -5$.

• Notices:

  • Give the number in scientific form to 3 s.f. Example: $1.2345 \times 10^2 \approx 1.23 \times 10^2$, $1.2345 \times 10^{-5} \approx 1.23 \times 10^{-5}$

  • Many calculators use the symbol $E\pm$ for scientific notation: $1.2345E+02 = 1.2345 \times 10^2$, $1.2345E-05 = 1.2345 \times 10^{-5}$

• Example 2:

  • (a) Give the scientific form of the numbers:

    • $x = 100000 = 1 \times 10^5$

    • $y = 0.00001 = 1 \times 10^{-5}$

    • $z = 4057.52 = 4.05752 \times 10^3$

    • $w = 0.00107 = 1.07 \times 10^{-3}$

  • (b) Give the standard form of the numbers:

    • $s = 4.501 \times 10^7 = 45010000$

    • $t = 4.501 \times 10^{-7} = 0.0000004501$

• Example 3: Given $x = 3 \times 10^7$ and $y = 4 \times 10^7$, give $x+y$ and $xy$ in scientific form.

  • $x+y = 7 \times 10^7$ (add 3+4, keep the same exponent)

  • $xy = 12 \times 10^{14} = 1.2 \times 10^{15}$ (multiply 3x4, add exponents, modify a so that 1 \leq a < 10)

• Example 4: Given $x = 3 \times 10^7$ and $y = 4 \times 10^9$, give $x+y$ and $xy$ in scientific form.

  • For addition, modify $y$ (or $x$) to achieve similar forms: $y = 4 \times 10^9 = 400 \times 10^7$

  • $x+y = 403 \times 10^7 = 4.03 \times 10^9$ (add 3+400, keep the same exponent, modify a so that 1 \leq a < 10)

  • For multiplication, no need to modify $y$: $xy = 12 \times 10^{16} = 1.2 \times 10^{17}$ (multiply 3x4, add exponents, modify a so that 1 \leq a < 10)

Percentage Error

• When approximating a value, we cannot avoid an error.

• Given an exact value $v<em>E$ and an approximate value $v</em>A$, the absolute error is $|v<em>E - v</em>A|$.

• The percentage error $(\varepsilon)$ is given by the formula:

  • $\varepsilon = \frac{|v<em>E - v</em>A|}{v_E} \times 100\%$,

• Example: The value of $\pi = 3.14159265…$ to 3 s.f. is $\pi \approx 3.14$.

  • The absolute error is $\pi - 3.14 = 0.00159265…$

  • The percentage error is $\frac{|\pi - 3.14|}{\pi} \times 100\% \approx 0.050695\% \approx 0.05\%$.

• Example 5: For two numbers A and B, the exact and approximated values are provided, demonstrating the significance of percentage error.

  • Exact values: A = 1003, B = 1,000,003

  • Values to 3sf: A = 1000, B = 1,000,000

  • Absolute error: 3 , 3

  • $\varepsilon<em>A = \frac{|1003 - 1000|}{1003} \times 100\% \approx 0.3\%$. $\varepsilon</em>B = \frac{|1000003 - 1000000|}{1000003} \times 100\% \approx 0.0003\%$.

  • The deviation is more severe for number A.

1.2 Exponents

• The exponential $2^x$ is defined as $x$ moves along the sets $N$, $Z$, $Q$, $R$.

  • If $x = n \in N$, then $2^n = 2 \cdot 2 \cdot 2 \cdot … \cdot 2$ ($n$ times). For example, $2^3 = 8$

  • If $x = -n$, where $n \in N$, then $2^{-n} = \frac{1}{2^n}$. Thus, we know $2^x$ for any $x \in Z$. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

  • If $x = \frac{m}{n}$, where $m, n \in Z, n \neq 0$, then $2^{\frac{m}{n}} = \sqrt[n]{2^m}$. Thus, we know $2^x$ for any $x \in Q$. For example, $2^{\frac{2}{3}} = \sqrt[3]{2^2} = \sqrt[3]{4}$. Also, $2^{\frac{1}{2}} = \sqrt{2}$.

  • If $x$ is irrational, the definition is beyond the scope and we use technology. We know $2^x$ for any $x \in R$. For example, $2^{\pi} = 8.8249779$ (given by a calculator).

  • $2^0 = 1$

• In general, if a > 0, we define:

  • $a^0 = 1$

  • $a^n = a \cdot a \cdot a \cdot … \cdot a$ ($n$ times)

  • $a^{-n} = \frac{1}{a^n}$

  • $a^{\frac{m}{n}} = \sqrt[n]{a^m}$

  • $a^x$ is given by a calculator for any $x \in R$

• Example 1:

  • $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$

  • $(\frac{-1}{5})^{-2} = (-5)^2 = 25$

  • $(\frac{-2}{5})^3 = (\frac{5}{-2})^3 = \frac{25}{9}$

  • $\sqrt[3]{8^2} = \sqrt[3]{64} = 4$

• Notices:

  • If a < 0, $a^x$ is only defined for $x = n \in Z$

  • $0^x = 0$ only if $x \neq 0$

  • $0^0$ is not defined

Properties of Exponents

• All known properties of powers are valid for exponents $x \in R$. Here a, b > 0 and $x, y \in R$.

  • $(1) a^x a^y = a^{x+y}$

  • $(2) \frac{a^x}{a^y} = a^{x-y}$

  • $(3) (ab)^x = a^x b^x$

  • $(4) (\frac{a}{b})^x = \frac{a^x}{b^x}$

  • $(5) (a^x)^y = a^{xy}$

• Example 2: Express the following as single powers:

  • $a^3 a^2 = a^5$

  • $\frac{a^6}{a^4} = a^2$

  • $\frac{x^5}{x^3} = x^2$

The Number e

• There is a specific irrational number which plays an important role in mathematics, especially in exponential modelling. The number $e$ is almost as popular as $\pi = 3.14…$

• An approximation of $e$ is given by the expression $(1 + \frac{1}{n})^n$

  • For $n = 1$, the result is $2$

  • For $n = 2$, the result is $2.25$

  • For $n = 10$, the result is $2.5937424…$

  • For $n = 100$, the result is $2.7048138…$

  • For $n = 1000$, the result is $2.7169239…$

  • For $n = 10^6$, the result is $2.7182804…$

  • As $n$ tends to $+\infty$, this expression tends to $e = 2.7182818…$

• Example 3: Express the following as single powers of $e$:

  • $(e^2)^3 e^3 = e^6 e^3 = e^9$

  • $e^{x+1} e^{3x} = e^{4x+1}$

  • $\frac{e^{4x}}{e^x} = e^{3x}$

Simple Exponential Equations

• If $a \neq 1$, then $a^x = a^y$ implies $x = y$.

• Example 4: Solve the following equations:

  • (a) $2^{3x-1} = 2^{x+2} \Leftrightarrow 3x - 1 = x + 2 \Leftrightarrow 2x = 3 \Leftrightarrow x = \frac{3}{2}$ (already a common base)

  • (b) $2^{3x-1} = 4^{x+2} \Leftrightarrow 2^{3x-1} = 2^{2(x+2)} \Leftrightarrow 3x - 1 = 2x + 4 \Leftrightarrow x = 5$ (write $4 = 2^2$)

  • (c) $4^{3x-1} = 8^{x+2} \Leftrightarrow 2^{2(3x-1)} = 2^{3(x+2)} \Leftrightarrow 6x - 2 = 3x + 6 \Leftrightarrow 3x = 8 \Leftrightarrow x = \frac{8}{3}$ (write $4 = 2^2$ and $8 = 2^3$)

  • (d) $\sqrt{2}^{3x-1} = 4^{x+2} \Leftrightarrow 2^{-\frac{1}{2}(3x-1)} = 2^{2(x+2)} \Leftrightarrow -3x+1 = 4x+8 \Leftrightarrow x = -\frac{3}{5}$

  • (e) $\sqrt{2}^{3x-1} = 4^{x+2} \Leftrightarrow 2^{\frac{3x-1}{2}} = 2^{2x+4} \Leftrightarrow 3x-1 = 4x+8 \Leftrightarrow x = -9$

1.3 Systems of Linear Equations

• A system of 2 linear equations in 2 unknowns has the form:

  • $a<em>1x + b</em>1y = c_1$

  • $a<em>2x + b</em>2y = c_2$

• We can solve using GDC (Graphical Display Calculator).

• Example 1: George buys 3 burgers and 5 sandwiches and pays 21.4 euros. Catherine buys 2 burgers and 3 sandwiches and pays 13.6 euros. Find the prices of each burger and of each sandwich.

  • Let B be the price of each burger and S be the price of each sandwich.

    • $3B + 5S = 21.4$

    • $2B + 3S = 13.6$

  • The GDC gives the solution:

    • B = 3.8 (euros)

    • S = 2 (euros)

System of 3 Linear Equations in 3 Unknowns

• A system of 3 linear equations in 3 unknowns has the form:

  • $a<em>1x + b</em>1y + c<em>1z = d</em>1$

  • $a<em>2x + b</em>2y + c<em>2z = d</em>2$

  • $a<em>3x + b</em>3y + c<em>3z = d</em>3$

  • We can solve using GDC (Graphical Display Calculator).

• Example 2: The expression $A(t) = Pt^2 + Qt + R$ takes the value 9 when $t = 1$, 18 when $t = 2$, and 3 when $t = -1$. Find the values of $P, Q, R$.

  • For $t = 1: P + Q + R = 9$

  • For $t = 2: 4P + 2Q + R = 18$

  • For $t = -1: P - Q + R = 3$

  • The GDC gives the solution:

    • P = 2

    • Q = 3

    • R = 4

  • Therefore, the expression is $A(t) = 2t^2 + 3t + 4$.

1.4 Sequences in General – Series

• Sequence: An ordered list of numbers (terms in a definite order).

  • Example: 2, 5, 13, 5, -4, …

  • Notation: $u<em>n$ describes the n-th term. The terms of the sequence are denoted by $u</em>1, u<em>2, u</em>3, u<em>4, u</em>5, …$

• Series: A sum of terms:

  • $S<em>n = u</em>1 + u<em>2 + u</em>3 + … + u_n$ (the sum of the first n terms, partial sum)

  • $S<em>\infty = u</em>1 + u<em>2 + u</em>3 + …$ (the sum of all terms, infinite series).

• Example 1: Consider the sequence 1, 3, 5, 7, 9, 11, … (odd numbers).

  • Some of the terms are: $u<em>1 = 1, u</em>2 = 3, u<em>3 = 5, u</em>6 = 11, u_{10} = 19$

  • Also, $S<em>1 = 1, S</em>2 = 1 + 3 = 4, S<em>3 = 1 + 3 + 5 = 9, S</em>4 = 1 + 3 + 5 + 7 = 16$

  • Finally, $S_\infty = 1 + 3 + 5 + 7 + …$ (in this case the result is $+\infty$)

• Sigma Notation ($\sum<em>{n=1}^{k} u</em>n$):

  • Instead of writing $u<em>1 + u</em>2 + u<em>3 + u</em>4 + u<em>5 + u</em>6 + u<em>7 + u</em>8 + u<em>9$ we may write $\sum</em>{n=1}^{9} u<em>n$. It stands for the sum of all terms $u</em>n$, where n ranges from 1 to 9.

Examples Using Sigma Notation

• $\sum_{n=1}^{3} n^2 = 1^2 + 2^2 + 3^2 = 2 + 4 + 8 = 14$

• $\sum_{n=1}^{4} \frac{1}{n} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{12 + 6 + 4 + 3}{12} = \frac{25}{12}$

• $\sum_{k=1}^{3} \frac{1}{2^k} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} = \frac{4 + 2 + 1}{8} = \frac{7}{8}$

• $\sum_{n=3}^{6} (2n + 1) = 7 + 9 + 11 + 13 = 40$

• There are two basic ways to describe a sequence:

General Formula

• We describe the general term $u_n$ in terms of $n$.

Recursive Relation

• Given $u<em>1$, the first term, and $u</em>{n+1}$ in terms of $u<em>n$. *Example 3:$u</em>n = n^2$ is the sequence 1, 4, 9, 16, 25, …
  *Example 4:$u<em>1 = 3$, $u</em>{n+1} = 2u_n + 5$. It is the sequence 3, 11, 27, 59, …

Recursive Formula

• $u<em>1 = 1, u</em>2 = 1$

• $u<em>{n+1} = u</em>n + u<em>{n-1}$ *in other words, we add $u</em>1, u<em>2$ in order to obtain $u</em>3$, we add $u<em>2, u</em>3$ in order to obtain $u_4$, and so on.
  *It is the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …

1.5 Arithmetic Sequence (A.S.)

• In an arithmetic sequence, the difference between any two consecutive terms is constant (common difference, d).

Question A: What is the general formula for $u_n$?

• If we know $u<em>1$ and $d$, then $u</em>n = u_1 + (n - 1)d$

Example

*In an arithmetic sequence let $u_1 = 3$ and $d = 5$. Find
*(a) the first four terms
*(b) the 100th term

*Solution
*(a) 3, 8, 13, 18
*(b) Now we need the general formula

• $u<em>{100} = u</em>1 + 99d$ $= 3 + 99 \cdot 5$
  $= 498$

• We know $u<em>1$, we need $d$. We exploit the information for $u</em>{16}$ first.
  $u<em>{16} = u</em>1 + 15d$ $145 = 100 + 15d$ $45 = 15d$ $d = 3$
  Therefore, $u<em>7 = u</em>1 + 6d$ $= 100 + 6 \cdot 3$ $= 118$

Question B: What is the sum $S_n$ of the first $n$ terms?

*It is directly given by

$(u<em>1 + u</em>n) \frac{n}{2}$
$S_n =$,

• otherwise by $S<em>n = [2u</em>1 + (n - 1)d] \frac{n}{2}$

Properties of Arithmetic Sequences

*Consecutive Terms: Let a, x, b be consecutive terms of an arithmetic sequence. The common difference is: x – a = b – x. Hence, 2x = a + b, that is x = $\frac{a + b}{2}$ (x is the mean of a and b).
*Examples
$((3x) - (x + 1) = (6x - 5) - (3x)\Rightarrow 2x - 1 = 3x - 5\Rightarrow x = 4$

1. Show That : 1 + 2 + 3 + … + n = $\frac{n(n + 1)}{2}$

*Solution
This is the simplest arithmetic series with $u_1 = 1$ and $d = 1$.
We ask for $S_n$

$S<em>n = (u</em>1 + u_n) / 2 = (1 + n) / 2 = n(n + 1) / 2$

1.6 Geometric Sequence (G.S.)

• Geometric Sequence : Multiply by a fixed number, say r =2, in order to find the next term. The following sequence is generated.
  5, 10, 20, 40, 80, …
  Such a sequence is called geometric. That is, in a geometric sequence the ratio between any two consecutive terms is constant.

QUESTION A: What is the general formula for $u_n$?

Examples

1. In a geometric sequence let $u_1 =3$ and $r =2$. Find
   (a) the first four terms
   (b) the 100th term

Solution

(a) 3, 6, 12, 24
(b) Now we need the general formula
$u<em>{100} = u</em>1 \cdot r^{99} = 3 \cdot 2^{99}$

QUESTION B: What is the sum $S_n$ of the first n terms?

Examples

Show that: $1/2 + 1/4 + 1/8 + 1/16 + … = 1$
*Solution This is an infinite G.S. with $u<em>1 = 1/2$ and $r =/2$. Since |r|<1 we obtain. $S = u</em>1 / 1-r = (1/2) / (1 - (1/2) = 1$