Comprehensive IB Maths AA SL & HL Syllabus

Comprehensive IB Maths AA SL & HL Syllabus

Chapter 1: Number and Algebra 

Chapter Number IB Points to Understand
1.1 A one variable linear equation in ? can be written in the form ?? + ? = 0, ? ≠ 0, and we will have exactly one solution, ? = − ?/a.

On a number line, the absolute value of difference between any two numbers is used to define the distance between those two points.

1.2 A function from ? (independent variable) to ? (dependent variable) is a relation ? between ? and ? in such a way that for a specific value of ?, a single value of ? is determined. 

The domain of a function is the set of all real numbers for which the given expression is defined as a real number. 

Range of a function is determined by analyzing the output of the function for all values of input (domain).

1.3 A function that is obtained from simpler functions by applying one after another in a specific way is called a composite function.

Decomposing and finding domain of a composite function

1.4 The inverse of a function will only exist if the function is one-to-one.

The inverse of a function ? is denoted ?−1.

1.5 Graphs of common functions

Reflection

Stretching and shrinking

1.6 Sequence- A sequence is a list of numbers that is written in a defined order, ascending or descending, following a specific rule. 

Series- A series is the sum of all the terms in a sequence. 

A finite sequence has a fixed number of terms. 

An infinite sequence has an infinite number of terms. 

1.7 A term in a sequence is named using the notation ??, where ? is the position of the term in the sequence.

A term in a sequence is named using the notation ??, where ? is the position of the term in the sequence.

1.8 A sequence formed when each term after the first is found by adding a fixed non-zero number is called an arithmetic sequence.

A sequence in which every term is obtained by multiplying or dividing a non-zero number with the preceding number is known as a geometric sequence.

This non-zero number is called the common ratio .

When ? > 1, the sequence is diverging.
And when ? < 1, the sequence is converging.

Adding up the terms of an arithmetic sequence gives us arithmetic series.

A geometric series with ? < 1 and with infinite number of terms is called an infinite geometric series.

1.9 Compound interest, Population growth
1.10 Permutation relates to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging its elements, a process called permuting

The combination is a way of selecting items from a collection, such that (unlike permutations) the order of selection does not matter.

A binomial is a polynomial with two terms.

Pascal’s triangle is a triangular array constructed by summing adjacent elements in preceding rows. 

1.11 Partial fractions: Maximum of two distinct linear terms in the denominator, with degree of numerator less than the degree of the denominator.
1.12 Complex numbers: the number i, where i2 = − 1. Cartesian form z = a + bi; the terms real part, imaginary part, conjugate, modulus and argument.
1.13 Modulus–argument (polar) form: z = r(cosθ + isinθ) = rcisθ.

Euler form: z = reiθ

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

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

Powers and roots of complex numbers.

1.15 Proof by mathematical induction 

Proof by contradiction

1.16 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.

 

Chapter 2: Functions

Chapter number IB Points to Understand
2.1 Axis of symmetry– Every parabola has symmetry about a vertical line, called axis of
symmetry
Vertex– The axis of symmetry passes through a point on the parabola called vertex.Concavity– If the leading coefficient ?, of the quadratic function is positive, then the parabola opens upwards (concave up) and the ?-coordinate of the vertex is the minimum value of the function. General form can be converted to vertex form by completing the square method.The solution or the root to the equation ??2 + ?? + ? is called the zero of the equation.?2 − 4??, is called the discriminant
2.2 Reciprocal functions: The reciprocal of a number is 1 divided by that number.

Rational functions: Functions of the form ?(?) = ?(?), where ? and ? are polynomials are called rational ?(?) functions.

2.3 To solve equations involving radicals, take the radical term to one side, and the left over terms to another and then raise it to a power which removes the radical. 

To solve equations involving fractions, we have to take LCM (Least common multiple) of all the fractions

2.4 Exponents– A quantity representing the power to which a given number or expression is to be raised.

Exponential Functions- It involves powers, where an independent variable is used as an exponent.

2.5 The domain of an exponential function is all Real numbers; hence the graph will be continuous.

Exponential Growth Curve continually and when 0 < ? < 1 there is a continually decreasing Exponential Decay Curve.

2.6 The number ‘e’ is one of the most important numbers in mathematics. It is an irrational number, with an approximate value of 2.718281
2.7 The logarithmic function with base 10 is called Common Logarithmic Function (denoted by log). And when ‘e’ is used as a base, then we call it the Natural Log (denoted by ln).
2.8 A radian is the measure of the angle with its vertex at the center of the circle and two radii with their end points on the circumference.

Arc length: For a circle of radius ?, a central angle ? subtends an arc of the circle of length ? given by ? = ??, where ? is measured in radians.

2.9 The ? and ?-coordinates of the unit circle are used to define the trigonometric ratios sine, cosine and tangent. (sin, cos, tan).
2.10 Sine, Cosine, Tangent curve
2.11 Exact, Graphical, Analytic solutions

Trigonometric identities

2.12 Polynomial functions, their graphs and equations; zeros, roots and factors.

The factor and remainder theorems.

Sum and product of the roots of polynomial equations.

2.13 Rational functions of the form

ax+b

f(x) = cx2 + dx + e, and f(x) =

ax2 +bx+c dx + e

2.14 Odd and even functions.

Finding the inverse function, f −1(x), including domain restriction.

Self-inverse functions.

2.15 Solutions of g(x) ≥ f (x), both graphically and analytically.
2.16 The graphs of the functions, y = | f (x)| and

y= f(|x|), y= 1 , y= f(ax+b), y=[f(x)]2.

 

Chapter 3: Geometry and trigonometry

Subtopic Subtopic Number IB Points to Understand
3.1 Distance between two points in 3-D

Midpoint of line segment in 3-D

Volume and surface of 3-D solids

3.2 Right-angled triangles

Trigonometry with triangles

Angle of elevation is the angle up from the horizontal and the line from the object to the person’s eye.

Angle of depression is the angle down from the horizontal and the line from the object to the person’s eye.

A bearing is used to indicate the direction of an object from a given point.

3.3 Areas of triangle

Equations of lines and angles between two lines

3.4 Sine rule: ????/a = ????/b = ????/c

Cosine rule:

c?? ? = ??+??−?? / ???

??? ? = ??+??−?? / ???

??? ? = ??+??−? / ???

3.5 Definition of cosθ, sinθ in terms of the unit circle

Definition of tanθ as sinθ/cosθ.

3.6 The Pythagorean identity cos2θ + sin2θ = 1. Double angle identities for sine and cosine.

The relationship between trigonometric ratios.

3.7 The circular functions sinx, cosx, and tanx; amplitude, their periodic nature, and their graphs

Composite functions of the form f (x) = asin(b(x + c)) + d.

3.8 Solving trigonometric equations in a finite interval, both graphically and analytically.
3.9 Pythagorean identities: 1 + tan2θ = sec2θ

1 + cot2θ = cosec2θ

The inverse functions f (x) = arcsinx, f (x) = arccosx, f (x) = arctanx; their domains and ranges; their graphs.

3.10 Compound angle identities. Double angle identity for tan.
3.11 Relationships between trigonometric functions and the symmetry properties of their graphs.
3.12 Concept of vector

Representation of vectors using directed line segments.

Base vectors i, j, k. Components of a vector:

v1

v= v2 =v1i+v2j+v3k.

3.13 The definition of the scalar product of two vectors. The angle between two vectors.

Perpendicular vectors; parallel vectors.

3.14 Vector equation of a line in two and three dimensions:

r = a + λb.

3.15 Coincident, parallel, intersecting and skew lines, distinguishing between these cases.

Points of intersection.

3.16 The definition of the vector product of two vectors.

Properties of the vector product.

3.17 Vector equations of a plane:

r = a + λb + μc, where b and c are non-parallel vectors within the plane.

r · n = a · n, where n is a normal to the plane and a is the position vector of a point on the plane.

Cartesian equation of a plane ax + by + cz = d.

3.18 Intersections of: a line with a plane; two planes; three planes.

Angle between: a line and a plane; two planes.

 

Chapter 4: Probability and Statistics

Subtopic Subtopic Number IB Points to Understand
4.1 Statistics deals with data collection, organization, analysis, interpretation and presentation.

Sampling is the selection of a subset of individuals from within a statistical population to estimate characteristics of the whole population.

A population may refer to an entire group of people, objects, events, hospital visits, or measurements. 

A variable is any characteristics, number, or quantity that can be measured or counted. 

4.2 Numerical or quantitative data measures a numerical quantity. Types: Discrete and Continuous

Qualitative data measures a quality or characteristic of the experimental unit. 

A frequency distribution table is one way you can organize data so that it makes more sense.

Bar charts are suitable for discrete data, while a histogram is used for continuous data.

Cumulative frequency graph is a line graph. 

4.3 Random sampling is a part of the sampling technique in which each sample has an equal probability of being chosen. 

Non-random sampling is a way of selecting units based on factors other than random chance.

4.4 A measure of central tendency is a summary statistic that represents the center point or typical value of a dataset. 

The mean is the arithmetic average.

The mode is the value that occurs most frequently in a set of data.

The median is the middle data value when the data values are arranged in order of size.

4.5 Variability refers to how “spread out” a group of scores is.

The range is simply the highest observation minus the lowest observation

The variance is defined as the average squared difference of the scores from the mean.

Therefore, statisticians often find the standard deviation, which is the square root of the variance. 

A percentile is simply a measure that tells us what percent of the total frequency of a data set was at or below that measure.

A practical way of seeing the significance of the standard deviation can be demonstrated by empirical rule. The empirical rule applies to a normal distribution.

4.6 Bivariate statistics is a type of inferential statistics that deals with the relationship between two variables. 

Correlation- A correlation exists between two variables, ? and ?, when a change in ? corresponds to a change in ?.

Causation – Causation indicates a relationship between two events where one event is affected by the other. 

Measuring correlation- Sometimes it is difficult to make judgements about the strength of a correlation by means a scatter plot.

A line of best fit can be determined by drawing a straight line on a scatter plot so that the number of points above the line and below the line is about equal 

Interpolation– We could use our function to predict the value of the dependent variable for an independent variable that is in the midst of our data.

Extrapolation– We could use our function to predict the value of the dependent variable for an independent variable that is outside the range of our data. 

The least square regression line has the smallest possible value for the sum of the squares of the residuals.

4.7 Probability– Probability concerns numerical descriptions of how likely an event is to occur or how likely it is that a proposition is true. 

Theoretical approach-To find the probability of an event using theoretical probability, it is not required to conduct an experiment. 

Experimental approach– This is an approach that relies upon actual experiments and adequate recordings of occurrence of certain events.

Sample space– The set of all the possible outcomes is called the sample space of the experiment and is usually denoted by ?.

A random experiment is an experiment where there is uncertainty concerning which of two or more possible outcomes will result.

Probability is the study of randomness and uncertainty.

A Venn diagram is an illustration that uses circles to show the relationships among sets.

4.8 Mutually exclusive events are events whose outcomes cannot occur at the same time.

When the likelihood of happening of two events are same they are known as equally likely events. 

4.9 Dependent Events are where what happens depends on what happened before, such as taking cards from a deck makes less cards each time

Two events A and B are independent if the occurrence of one does not affect the chance that the other occurs. 

?(?|?) is known as conditional probability

4.10 Equation of the regression line of x on y.
4.11 Formal definition and use of the formulae:

P(A|B) = P(A B) for conditional probabilities, and

P(B)

P(A|B) = P(A) = P(A|B′) for independent events.

4.12 Standardization of normal variables (z- values).

Inverse normal calculations where mean and standard deviation are unknown.

4.13 Bayes’ theorem can be used for a maximum of three events.
4.14 Variance of a discrete random variable.

Continuous random variables and their probability density functions.

 

Chapter 5: Calculus 

Subtopic Subtopic Number IB Points to Understand
5.1 Differential calculus deals with the study of the rates at which quantities change.

Limit– A value we get closer and closer to, but never quite reach.

One-sided limits are those limits that only converge to a single value on one side of a function.

5.2 Tangent line and slope

Basic differentiation rules

Derivatives of sine and cosine functions

5.3 The derivative of a function may be used to determine whether the function is increasing or decreasing on any intervals in its domain.

Critical points are the points where ?′ is zero or undefined.

This method of checking if the function is increasing or decreasing and about which point is called the first derivative test.

A maximum or minimum is said to be local if it is the largest or smallest value of the function, respectively, within a given range.

Point of inflection: These are critical points of a function. These are the points where ?′(?) or the slope of the curve is zero.

Given a function ?(?), the derivative ?′(?) is known as the first derivative of ?(?). The gradient of the first derivative, ?′′(?) is called the second derivative of ?(?).

5.4 A tangent to a cruve is a line that touches the curve at one point and has the same slope as the curve at that point.

A normal to a curve is a line perpendicular to a tangent to the curve.

Finding equation of tangent and normal

5.5
  • In this unit we carry out the process of differentiation in reverse. 
  • Given any antiderivative of ?(?), all others can be obtained by simply adding a different constant. 
5.6
  • Integration by Substitution (also called “u-Substitution” or “The Reverse Chain Rule”) is a method to find an integral, but only when it can be set up in a special way.
5.7
  • The area under the curve can be found using integration and this is what is known as a definite integral. 
5.8
  • The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function.
5.9
  • If ? is the velocity function for a particle moving along a straight line, then: ∫?2 ?(?) ?? = ?(?2) − ?(?1) is the displacement from ?1 to ?2.
5.10
  • A random variable, usually written ?, is a variable whose possible values are numerical outcomes of a random phenomenon.
  • A probability distribution is a list of all of the possible outcomes of a random variable along with their corresponding probability values.
  • The expected value (or mean) of ?, where ? is a discrete random variable, is a weighted average of the possible values that ? can take, each value being weighted according to the probability of that event occurring.
5.11
  • A binomial distribution can be thought of as simply the probability of a SUCCESS or FAILURE outcome in an experiment or survey that is repeated multiple times.
5.12
    • Normal distribution is a continuous probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.
    • Probability density function (PDF), is a function whose integral is calculated to find probabilities associated with a continuous random variable.
  • The Standard normal distribution is a normal distribution where ? = 0 and ? = 1.
  • An inverse normal distribution is a way to work backwards from a known probability to find an ?-value.
5.13 The evaluation of limits of the form lim f (x) and x a g(x)

lim f (x) using l’Hôpital’s rule or the Maclaurin series

5.14 Implicit differentiation. Related rates of change. Optimisation problems.
5.15 Indefinite integrals of the derivatives of any of the above functions.

The composites of any of these with a linear function.

5.16 Integration by  substitution, by parts and repeated integration by parts
5.17 Area of the region enclosed by a curve and the y– axis in a given interval.

Volumes of revolution about the x-axis or y-axis.

5.18 First order differential equations. Numerical solution of dy = f (x, y) using Euler’s method. 

Homogeneous differential equation dy = f ( y )

using the substitution y = vx.

5.19 Maclaurin series to obtain expansions for

xp e , sinx, cosx, ln(1+x), (1+x) ,

p ∈ Q.

Use of simple substitution, products, integration and differentiation to obtain other series.

 

Leave a Reply