IB Maths: Analysis and Approaches HL 166711

Operations with numbers in the form $a×10^k$ where $1≤a<10$ and $k$ is an integer.

Calculator or computer notation is not acceptable. For example, 5.2E30 is not acceptable and should be written as $5.2×10^{30}$.

Arithmetic sequences and series.

Use of the formulae for the $n^{th}$ term and the sum of the first $n$ terms of the sequence.

Use of sigma notation for sums of arithmetic sequences.

Spreadsheets, GDCs and graphing software may be used to generate and display sequences in several ways.

If technology is used in examinations, students will be expected to identify the first term and the common difference.

Applications.

Examples include simple interest over a number of years.

Analysis, interpretation and prediction where a model is not perfectly arithmetic in real life.

Students will need to approximate common differences.

Geometric sequences and series.

Use of the formulae for the n th term and the sum of the first n terms of the sequence.

Spreadsheets, GDCs and graphing software may be used to generate and display sequences in several ways.

Use of sigma notation for the sums of geometric sequences.

If technology is used in examinations, students will be expected to identify the first term and the ratio.

Link to: models/functions in topic 2 and regression in topic 4.

Applications.

Examples include the spread of disease, salary increase and decrease and population growth.

Financial applications of geometric sequences and series:

• compound interest
• annual depreciation.

Examination questions may require the use of technology, including built-in financial packages.

The concept of simple interest may be used as an introduction to compound interest.

Calculate the real value of an investment with an interest rate and an inflation rate.

In examinations, questions that ask students to derive the formula will not be set.

Compound interest can be calculated yearly, half-yearly, quarterly or monthly.

Link to: exponential models/functions in topic 2.

Laws of exponents with integer exponents.

Examples:

• $5^3×5^{−6}=5^{−3}$,
• $6^4÷6^3=6$,
• $(2^3)^{-4}=2^{-12}$,
• $(2x)^4=16x^4$,
• $2x^{-3}=\frac{2}{x^3}$.

Introduction to logarithms with base 10 and e.

Numerical evaluation of logarithms using technology.

Awareness that $a^x=b$ is equivalent to $\log{a}{b}=x$, that $b>0$, and $\log_{e}{x}=\ln{x}$.

Simple deductive proof, numerical and algebraic; how to lay out a left-hand side to right-hand side (LHS to RHS) proof.

The symbols and notation for equality and identity.

Example: Show that $\frac{1}{4}+\frac{1}{12}=\frac{1}{3}$. Show that the algebraic generalisation of this is

$\dfrac{1}{m+1}+\dfrac{1}{m^2+m}≡\dfrac{1}{m}$

LHS to RHS proofs require students to begin with the left-hand side expression and transform this using known algebraic steps into the expression on the right-hand side (or vice versa).

Example: Show that $(x-3)^2+5≡x^2−6x+14$.

Students will be expected to show how they can check a result including a check of their own results.

Laws of exponents with rational exponents.

$a^{\frac{1}{m}} = \sqrt[m]{a}$, if $m$ is even this refers to the positive root. For example: $16^{\frac{3}{4}}=8$.

Laws of logarithms.

• $\log_a{xy} = \log_a{x} + \log_a{y}$
• $\log_a{\dfrac{x}{y}} = \log_a{x} - \log_a{y}$
• $\log_a{x^m} = m\log_a{x}$

for $a, x, y > 0$

$y=a^x⇔x=\log_a{y}$; $\log_a{a}=1$, $\log_a{1}=0$,

$a, y ∈ ℕ, x ∈ ℤ$

Link to: introduction to logarithms (SL 1.5)

Examples:

• $\dfrac{3}{4}=\log_{16}{8}$,
• $\log{32}=5\log{2}$
• $\log{24}=\log{8}+\log{3}$
• $\log_3{\dfrac{10}{4}}=\log_3{10}−\log_3{4}$
• $\log_4{3^5}=5\log_4{3}$

Link to: logarithmic and exponential graphs (SL 2.9)

Change of base of a logarithm.

$\log_a{x} = \dfrac{\log_b{x}}{\log_b{a}}$, for $a, b, x > 0$

Examples:

• $\log_4{7}=\dfrac{\ln{7}}{\ln{4}}$
• $\log_{25}{125}=\dfrac{\log_5{125}}{\log_5{25}} (=\dfrac{3}{2})$

Solving exponential equations, including using logarithms.

Examples:

• $(\frac{1}{3})^x=9^{x+1}$,
• $2^{x−1}=10$.

Link to: using logarithmic and exponential graphs (SL 2.9).

Sum of infinite convergent geometric sequences.

Use of $|r|<1$ and modulus notation.

Link to: geometric sequences and series (SL 1.3).

The binomial theorem:

expansion of $(a+b)^n, n ∈ ℕ$.

Counting principles may be used in the development of the theorem.

Use of Pascal’s triangle and $^nC_r$.

$^nC_r$ should be found using both the formula and technology.

Example: Find $r$ when $^6C_r=20$, using a table of values generated with technology.

Counting principles, including permutations and combinations.

Not required: Permutations where some objects are identical. Circular arrangements.

Extension of the binomial theorem to fractional and negative indices, ie $(a+b)^n, n∈ℚ$.

$(a+b)^n=\big(a(1+\frac{b}{a})\big)^n=a^n(1+\frac{b}{a})^n, n∈ℚ$

Link to: power series expansions (AHL 5.19)

Not required: Proof of binomial theorem.

Partial fractions.

Maximum of two distinct linear terms in the denominator, with degree of numerator less than the degree of the denominator.

Example:

$\dfrac{2x+1}{x^2+x−2}≡\dfrac{1}{(x−1)}+\dfrac{1}{(x+2)}$.

Link to: use of partial fractions to rearrange the integrand (AHL 5.15).

Complex numbers: the number $\text{i}$, where $\text{i}^2 = -1$.

Cartesian form $z=a+b\text{i}$; the terms real part, imaginary part, conjugate, modulus and argument.

The complex plane.

The complex plane is also known as the Argand diagram.

Link to: vectors (AHL 3.12).

Modulus-argument (polar) form:

$z=r(\cos{θ}+\text{i}\sin{θ}) = r\text{cis}{θ}$

Euler form:

$z=re^{\text{i}θ}$

Sums, products and quotients in Cartesian, polar or Euler forms and their geometric interpretation.

The ability to convert between Cartesian, modulus-argument (polar) and Euler form is expected.

Complex conjugate roots of quadratic and polynomial equations with real coefficients.

Complex roots occur in conjugate pairs.

De Moivre’s theorem and its extension to rational exponents.

Includes proof by induction for the case where $n∈ℤ^+$; awareness that it is true for $n∈ℝ$.

Powers and roots of complex numbers.

Link to: sum and product of roots of polynomial equations (AHL 2.12), compound angle identities (AHL 3.10).

Proof by mathematical induction.

Proof should be incorporated throughout the course where appropriate.

Mathematical induction links specifically to a wide variety of topics, for example complex numbers, differentiation, sums of sequences and divisibility.

Proof by contradiction.

Examples:

• Irrationality of $\sqrt{3}$;
• irrationality of $\sqrt[3]{5}$;
• Euclid’s proof of an infinite number of prime numbers;
• if $a$ is a rational number and $b$ is an irrational number, then $a+b$ is an irrational number.

Use of a counterexample to show that a statement is not always true.

Example: Consider the set $P$ of numbers of the form $n^2+41n+41,n∈ℕ$, show that not all elements of $P$ are prime.

Example: Show that the following statement is not always true: there are no positive integer solutions to the equation $x^2+y^2=10$.

It is not sufficient to state the counterexample alone. Students must explain why their example is a counterexample.

Solutions of systems of linear equations (a maximum of three equations in three unknowns), including cases where there is a unique solution, an infinite number of solutions or no solution.

These systems should be solved using both algebraic and technological methods, for example row reduction or matrices.

Systems which have no solution(s) are inconsistent.

Finding a general solution for a system with an infinite number of solutions.

Link to: intersection of lines and planes (AHL 3.18).