IB Mathematics: Applications and Interpretation Ultimate Guide

TOPIC 1: Number & Algebra

Natural Numbers (N)

• The set of positive counting numbers.

Examples: 1, 2, 3, 4, 5, ...

• Used for discrete quantities.

Integers (Z):

• The set of all whole numbers, including negatives and zero

Examples: -2, -1, 0, 1, 2, ...

• Includes natural numbers and their negatives.

Rational Numbers (Q):

• The set of all numbers that can be expressed as fractions a/b.

Examples: ½, 0.75, -3.1

Irrational Numbers:

• The set of all numbers that are not expressible as fractions. Their decimal expansions are non-terminating and non-repeating.

Examples: sqrt(2), π, e

Real Numbers (R):

• The set of all rational and irrational numbers.

• Represents every point on the number line.

Rounding Up

• Numbers that are greater than 4 should be rounded up.

• Numbers that are less than or equal to 4 should be rounded down.

• Examples: 4.5 will be rounded up to 5.

            4.1 will be rounded down to 4. 

Significant Figures (s.f.) 

• Non-zero digits are always significant.

  • Example: 123.45 has 5 sig figs.

• Zeros between nonzero digits are significant.

  • Example: 1002 has 4 sig figs.

• Leading zeros (zeros before the first non-zero digit) are not significant.

  • Example: 0.0045 has 2 sig figs.

• Trailing zeros in a number with a decimal point are significant.

  • Example: 45.600 has 5 sig figs.

• Trailing zeros in a whole number without a decimal point are not significant unless specified by a bar or underline.

  • Example: 1500 has 2 sig figs (unless written as 1500̲, which indicates 4 sig figs).

• Default: 3 s.f.

Operations with s.f.:

• Addition/Subtraction:

  • The result should have the same number of decimal places as the number with the fewest decimal places.

  • Example: 12.11 + 0.3 = 12.4 (1 decimal place).

• Multiplication/Division:

  • The result should have the same number of sig figs as the number with the fewest sig figs.

  • Example: 4.56 × 1.4 = 6.4 (2 sig figs).

Scientific Notation/Standard Index Form:

a*10^k, 1 ≤ a < 10, k is integer

• Calculator format: 5.678112E14 = 5.67 * 10¹⁴ (3 s.f.)

• Ex: 167400 = 1.67 * 10⁵

Percentage Error:

Accuracy of an approximate value compared to the exact value, expressed as a percentage.

Formula: | (V_A-V_E)/V_E | * 100%

V_A - Approximate value

V_E - Exact value

Arithmetic Sequences:

A sequence with a constant difference (d) between consecutive terms.

Formula: U_n=U₁+d(n-1)

U₁: First term of sequence

d: common difference (U₂-U₁, U₃-U₂, etc)

U_n: n^th term of the sequence

• Sum formula: S_n=n*(U₁+U_n)/2

  • n is the position of the term you’re summing up to

Geometric Sequences:

A sequence with a constant ratio (r) between consecutive terms.

Formula: U_n=U₁*r^(n-1)

U₁: First term of sequence

r: The common ratio between the terms (U₂/U₁, U₃/U₂, etc.)

U_n: The n^th term

• Finite sum formula: S_n=U₁*(1-rⁿ)/(1-r)

  • n is the position of the term you’re summing up to

• Infinite sum formula: S=U₁/(1-r)

  • Converges if |r| < 1

Sigma Notation:

A mathematical way to represent the sum of a sequence of terms in a compact form using the Greek letter Σ (sigma).
Σ f(k), from k = a to b
where:

• Σ represents summation

• k is the index of summation (dummy variable)

• a is the lower limit (starting value of k)

• b is the upper limit (ending value of k)

• f(k) is the function being summed

Examples:

1. Sum of first 4 squares:
   Σ k², from k = 1 to 4
   = 1² + 2² + 3² + 4²
   = 1 + 4 + 9 + 16
   = 30

2. Sum of an arithmetic sequence:
   Find Σ (2k + 3), from k = 1 to 5
   2(1)+3=5, 2(2)+3=7,...
   5,7,...
   U₁=5, d=2
   S₅= 5*(U₁+[U₁+d(n-1)])/2
       = 5*(5+[5+2(5-1)])/2
       = 5*18/2
       = 45

Exponents and Logarithms:

Exponent Rules:

• a^m * aⁿ = a^m+n

• a^m/aⁿ = a^m-n

• (a^m)ⁿ = a^mn

• a⁰ = 1

• a^-n = 1/aⁿ

Logarithms:

• If a^b=c, then log_ac=b

• Rules:

  • log_a(mn) = log_am + log_an

  • log_a(m/n) = log_am - log_an

  • log_a(mⁿ) = nlog_am

  • Change of base: log_ab = (log_cb)/(log_ca)

• Natural log is when the base is e 

  • If e^a = b, then ln(b) = a

• Common log is when the base is 10

  • If 10^a=b, then log(b)=a

Binomial Theorem and Pascal's Triangle

Binomial Theorem

The Binomial Theorem provides a formula to expand expressions of the form (a + b)ⁿ without direct multiplication.

Formula:

(a + b)ⁿ = Σ [ C(n, k) a⁽ⁿ⁻ᵏ⁾ bᵏ ], from k = 0 to n

where:

• n is the exponent

• k is the term index (starting from 0)

• a and b are the terms in the binomial

• C(n, k) (binomial coefficient) is given by:
  C(n, k) = n! / [k!(n - k)!], which represents the number of ways to choose k elements from n.

Example Expansion:

Expanding (x + y)³:

(x + y)³ = C(3,0) x³ + C(3,1) x²y + C(3,2) xy² + C(3,3) y³

= 1x³ + 3x²y + 3xy² + 1y³

= x³ + 3x²y + 3xy² + y³

Pascal’s Triangle

Pascal’s Triangle is a triangular array where each row contains the binomial coefficients for a given exponent n.

Construction Rules:

1. Start with 1 at the top.

2. Each row begins and ends with 1.

3. Each middle term is the sum of the two terms directly above it.

Pascal’s Triangle Structure:

       1

      1 1

     1 2 1

    1 3 3 1

   1 4 6 4 1

  1 5 10 10 5 1

• Row n gives coefficients for (a + b)ⁿ.

• The nᵗʰ row contains n + 1 elements.

Connection to Binomial Theorem

Each coefficient in the expansion (a + b)ⁿ corresponds to a number in Pascal's Triangle.

Example for (x + y)⁴:

• Pascal's Triangle row: 1 4 6 4 1

• Expansion:
  x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴

Compound Interest and Depreciation

Compound Interest

Compound interest is the interest calculated on both the initial principal and the accumulated interest from previous periods.

Formula:

A = P(1 + r/n)^(nt)

Where:

• A = Final amount after interest

• P = Initial principal (starting amount)

• r = Annual interest rate (decimal form)

• n = Number of times interest is compounded per year

• t = Number of years

Example:

If you invest $1,000 at an annual interest rate of 5%, compounded quarterly for 3 years, the final amount is:

A = 1000(1 + 0.05/4)^(4×3)
A = 1000(1.0125)^12
A ≈ 1157.63

After 3 years, the investment grows to $1,157.63.

Depreciation

Depreciation is the reduction in the value of an asset over time, often due to wear and tear or obsolescence.

Formula:

A = P(1 - r)ⁿ

Where:

• A = Final value after depreciation

• P = Initial value of the asset

• r = Depreciation rate (decimal form)

• n = Number of years

Example:

A car worth $20,000 depreciates by 15% per year. Its value after 3 years is:

A = 20000(1 - 0.15)³
A = 20000(0.85)³
A ≈ 12262.50

After 3 years, the car’s value drops to $12,262.50.

Loans and Annuities

Make/receive payments along the way

Loan - Borrow money to be paid back at regular intervals

Savings annuity - save money and make regular contributions

Gift annuity - receive a gift that is paid out at regular intervals

Use Finance app on calculator:

n - The number of payments made

I% - annual interest rate

PV - amt of money involved at the start

PMT - regular payment

FV - amt of money involved at the end

P/Y - frequency, per year, which payment is made

C/Y - frequency, per year, which interest is calculated

TOPIC 2: Functions

Formulas

Standard Level

• Equations of a straight line: y=mx c; ax+by+d=0; y-y₁=m(x -x₁)

• Gradient formula: m=y₂-y_1×2-x₁

• Axis of symmetry of the graph of a quadratic function: f(x)=ax²+bx+c

• Axis of symmetry is x= -b/(2a)

Higher Level

• Logistic Function: f(x)=L/(1+Ce^-kx), where L, k, and C>0

Section A: Forms of Linear Lines

Linear Lines

• Refers to straight lines and can be expressed in various forms:

  • slope-intercept form 

  • point-slope form

  • standard form.

Slope-Intercept Form

• y = mx + c, where m is the gradient (slope) of the line and c is the y-intercept.

• Useful for its simplicity; allows easy identification of slope and y-intercept.

  • Example:y=(2/3)x+2

Point-Slope Form

• y - y₁=m(x -x₁), where (x₁, y₁) is a point on the line and m is the slope.

• useful when the y-intercept is not given, requiring a specific point on the line.

  • example: y-4=23(x-3)

Standard-Form

• Ax + By = C, where A, B, and C are constants, and A and B are not both zero.

• all terms must be on one side of the equation and integer coefficients.

  • example: -2x+3y-6=0

Things to Note

• Often, questions in IB exams may ask for the final answer in standard form, necessitating rearrangement from other forms.

• The process for converting from point-slope or slope-intercept to standard form involves expansion, grouping like terms, and sometimes multiplication to eliminate fractions.

• Both slope-intercept and standard forms represent the same equation, just arranged differently.

• Depending on the question's requirements, either form may be suitable for the final answer.

Section B: Gradients and Intercepts

Gradient (Slope)

• Represents the steepness of the line.

• Analogous to the slope of a hill; steepness determines the rate of change.

• Gradient (m) = rise/run    m=y2-y1x2-x1

• Δy represents the change in y and Δx represents the change in x between two points on the line.

Intercept Basics

• X-intercept: Where the line intersects the horizontal axis (x-axis).

• Y-intercept: Where the line intersects the vertical axis (y-axis).

• A straight line will have only one x-intercept and one y-intercept.

• Positive gradient: Line slopes upwards from left to right. Negative gradient: The line slopes downwards from left to right.

Section C: Parallel and Perpendicular Gradients

Parallel Lines

• Lines that are always equidistant and never intersect, resembling train tracks.

• The gradients of parallel lines are equal to each other.

• Example: If Line 1 has a gradient of 2, then Line 2, parallel to Line 1, also has a gradient of 2.

Perpendicular Lines

• Lines that intersect at a right angle (90 degrees), forming a "T" shape.

• The gradients of perpendicular lines are negative reciprocals of each other.

• Negative reciprocal means taking the negative of the reciprocal of the gradient of one line.

• Example: If Line 1 has a gradient of 2, then Line 2, perpendicular to Line 1, has a gradient of -1/2.

Section D: Perpendicular Bisectors

Perpendicular Bisector

• Application primarily appears in Voronoi diagrams.

• Definition: A line segment that cuts another line segment into two equal parts at a right angle.

• The midpoint of the line segment defines the point through which the perpendicular bisector passes.

Example:

• Given a line segment AB, find the equation of the perpendicular bisector.

  • Step 1: Find the midpoint of AB using the midpoint formula.

  • Step 2: Determine the gradient of AB using the gradient formula.

  • Step 3: Apply the concept of perpendicular lines to find the gradient of the perpendicular bisector.

  • Step 4: Use the point-slope form of a linear equation to find the equation of the perpendicular bisector.

• Example Solution:

  • Given line segment AB with A(-1,1) and B(3,3).

  • Step 1: Midpoint = (1,2)

  • Step 2: Gradient of AB = ½

  • Step 3: Gradient of perpendicular bisector = -1 / (1/2) = -2

  • Step 4: Equation of perpendicular bisector = y = -2x + 4

Section E: Types of Functions

• A function is described as a mathematical rule denoted by f(x)=something

• It represents the relationship between two variables, where:

  • x is the input (independent variable) and

  • y is the output (dependent variable).

  • Example:

    • Throwing a rock down a ditch 

    • Time (input) influences height (output).

    • Modelled with a linear function:

      • H(t)=10a+50

Types of Functions

• Linear Functions: Have a constant rate of change, forming straight lines.

• Quadratic Functions: Involve squares of the variable, often representing parabolic curves.

• Exponential Functions: Involve a constant raised to a variable power, used for growth or decay processes.

• Logarithmic Functions: The inverse of exponential functions, used to model decreasing growth rates.

• Sinusoidal Functions: Represent periodic oscillations, such as waves or tides.

• Logistic Functions: Combine exponential and logarithmic functions, useful for modelling growth that levels off.

• Piecewise Functions: Combinations of different functions defined over specific intervals, useful for modeling complex scenarios with changing behaviors.

HL Only: Domain & Range, Composite, Inverse

Function Representation

• Functions can be represented in various ways, such as:

• (x)=x₂−2, f:x↦x₂−2 f:x↦x₂−2, or y=x₂−2y=x₂−2.

• All these representations convey the same meaning, depending on the context or textbook used.

Domain and Range

• Domain refers to the set of possible input values (X), while range refers to the set of possible output values (Y).

• For example, for f(x)=x₂−2

  • Domain: x∈R, where x∈R means all real numbers

  • Range: y ≥-2 (all real numbers greater than or equal to -2).

Composite Functions

• Composite functions, denoted as

  • f∘g, or g∘f, involve applying one function within another function.

• For example, if f(x)=x₂−2 and g(x)=x+3

  • f∘g: Replace x in f(x) with g(x), i.e., (x+3)2−2

  • g∘f: Replace x in g(x) with f(x), i.e., (x₂−2)+3

Inverse Functions

• The inverse function, denoted as f-1, reverses the action of the original function.

• To find the inverse of a function:

  • Rewrite the function in the form y=f(x)

  • Swap x and y and solve for y.

  • The resulting equation is the inverse function.

• For example, if f(x)=x2−2

  • Rewrite as y=x₂−2

  • Swap x and y to get x=y₂−2

  • Solve for y to get f-1(x)=x+2

Understanding Functions

• Composite functions and inverse functions have deeper meanings and are essential in understanding the behavior and relationships between functions.

• Inverse functions represent reflections across the line y=x, while composite functions involve applying one function's output as another function's input.

HL Only: Transformation of Functions

Functions Representation

• Functions are often represented as f(x), where x is the input variable and f(x) is the output.

• Knowing a coordinate on the function, such as (2,2), helps understand its behavior.

Translation (Shift)

• Translations involve shifting the function horizontally or vertically.

• Horizontal shift: f(x±a)

  • f(x±a) shifts a units left or right.

• Vertical shift: f(x)±a

  • f(x)±a shifts a units up or down.

• Remember that signs inside and outside the function bracket indicate horizontal and vertical shifts, respectively.

Stretch

• Stretches involve making the function skinnier or wider.

• Vertical stretch: a⋅f(x)

  • a⋅f(x) multiplies all y values by a.

• Horizontal stretch: f(ax)

  • f(ax) stretches the function horizontally by a factor of 1a

  • Horizontal stretch by 1a means compressing if a>1 or stretching if 0<a<1.

Reflection

• Reflections involve flipping the function over the x-axis or y-axis.

• Reflection in the x-axis: −f(x)

  • −f(x) reflects all y values vertically.

• Reflection in the y-axis: f(−x)

  • f(−x) reflects all

  • x values horizontally.

Summary

• Horizontal shifts affect x values inside the function.

• Vertical shifts affect y values outside the function.

• Stretches scale the function vertically or horizontally.

• Reflections flip the function over the x-axis or y-axis.

Plotting Functions and Analysis Tools

Plotting a Quadratic Function

• Example: Height of a cannonball over time.

  • Enter function:

    • −2x²+20x+8

  • Adjust zoom settings for accurate visualization. (I suggest [ZOOM] -> 0)

Analysis Tools:

• Maximum Height:

  • Use "Maximum" tool.

  • After 5 seconds, cannonball reaches 58 meters.

• Time to Reach Ground:

  • Utilize "Zero" tool by pressing [2ND] + [TRACE] + 2

  • “Left bound?” -> Move “blinky” to left side of line from where it touches at zero

  • “Right bound?” -> move “blinky” to right side from where it touches at zero

  • “Guess?” -> place “blinky” on top of where it touches zero

  • Result: 10.38 (round up by three sig-figs) -> 10.4 

  • Interpretation: Cannonball touches ground at 10.4 seconds 

• Initial Height:

  • Plug in 0 for function; in most cases, getting an initial height can be found by isolating the constant alone.

  • In context: initial height is 8 meters.

• Time to Reach Specific Height:

  • Plot horizontal line (e.g., at 40 meters).

  • Use "Intersection" tool.

  • Cannonball reaches 40 meters in 2 seconds.

Tips: Numeric Solver

• Numeric Solver (N Solve) is a valuable function on TI-84 for solving equations quickly and accurately.

• It's particularly essential for topics like exponential, logarithmic, quadratic, and cubic functions.

Accessing N Solve

• Access the function by navigating to [MATH] > [C:Numeric Solver] 

Entering Equation

• Input the equation directly into the calculator, ensuring correct syntax and using 'X' as the variable.

Specifying Bounds

• Specify appropriate bounds to narrow down the search range for the solution.

• Bounds should be close to the expected solution for faster convergence.

Initiating Calculation

• Press [GRAPH] button when ready to start “guessing”

• Input any value for X (as long as it fits in context to problem)

• Result will be displayed in the same area you input a value in

Viewing Result

• The calculator displays the approximate solution once the calculation is complete.

• REMEMBER TO USE THREE SIG-FIGS

Considerations

• Numeric Solver provides the left-hand solution for equations with multiple solutions.

• For equations with multiple solutions or complex functions, consider using other analysis tools like plotting functions and intersection analysis.

TOPIC 3: Geometry & Trigonometry

1. Introduction to Geometry and Trigonometry

Geometry and Trigonometry are essential branches of mathematics that deal with the properties, relationships, and measurements of shapes, angles, and distances. In the IB Mathematics curriculum, these topics form the foundation for various real-world applications, including physics, engineering, architecture, and navigation.

2. Geometry

2.1 Basic Concepts

• Point: A location in space with no dimensions.

• Line: A one-dimensional figure extending infinitely in both directions.

• Line Segment: A part of a line with two endpoints.

• Ray: A part of a line that starts at a point and extends infinitely in one direction.

• Plane: A flat, two-dimensional surface extending infinitely.

• Collinear Points: Points that lie on the same line.

• Coplanar Points: Points that lie in the same plane.

2.2 Types of Angles

• Acute Angle: Less than 90°.

• Right Angle: Exactly 90°.

• Obtuse Angle: Greater than 90° and less than 180°.

• Straight Angle: Exactly 180°.

• Reflex Angle: Greater than 180° and less than 360°.

• Complementary Angles: Two angles whose sum is 90°.

• Supplementary Angles: Two angles whose sum is 180°.

• Vertically Opposite Angles: Angles opposite each other when two lines intersect, always equal.

2.3 Properties of Triangles

• Sum of Interior Angles: 180°.

• Pythagorean Theorem: In a right-angled triangle, where and are the legs, and is the hypotenuse.

• Congruent Triangles: Triangles with equal corresponding sides and angles (SSS, SAS, ASA, RHS).

• Similar Triangles: Triangles with equal corresponding angles and proportional sides (AA, SAS, SSS).

• Isosceles Triangle: A triangle with at least two equal sides.

• Equilateral Triangle: A triangle with all sides and angles equal.

• Scalene Triangle: A triangle with all sides of different lengths.

• Right-Angled Triangle: A triangle with one 90° angle.

2.4 Quadrilaterals and Polygons

• Sum of Interior Angles of a Polygon: , where is the number of sides.

• Regular Polygons: All sides and angles are equal.

• Properties of Quadrilaterals:

  • Parallelogram: Opposite sides are equal and parallel.

  • Rectangle: Opposite sides are equal, and all angles are 90°.

  • Rhombus: All sides are equal, opposite angles are equal.

  • Square: A rectangle with all sides equal.

  • Trapezium: One pair of parallel sides.

  • Kite: A quadrilateral with two pairs of adjacent sides equal.

2.5 Circles

• Circumference: the length of the circle's boundary

• Area: the region occupied by the circle in a two-dimensional plane

• Chord, Arc, Sector, Segment: Various parts of a circle.

• Circle Theorems: Key properties related to angles, tangents, and chords.

• Tangent Properties: A tangent to a circle is perpendicular to the radius at the point of contact.

3. Trigonometry

3.1 Trigonometric Ratios

For a right-angled triangle:

3.2 Special Angles

3.3 Trigonometric Identities

3.4 The Sine and Cosine Rules

For any triangle:

• Sine Rule: a/sin A = b/sin B = c/sin C

• Cosine Rule: a² = b² + c² - 2bc cos A

3.5 Applications of Trigonometry

• Angles of Elevation and Depression

• Navigation and Bearings

• Physics Applications (e.g., wave motion, oscillations)

• Construction and Architecture

• Astronomy: Calculating distances of celestial bodies

• Computer Graphics: Used in transformations and animations

TOPIC 4: Statistics & Probability

Descriptive Statistics

Population and Sample

• In statistics, when we talk about a population, we mean every single person or thing we're interested in studying. Think of it as the whole pie. A sample, on the other hand, is just a slice of that pie – a smaller group taken from the population to represent the whole. For instance, if we want to know about the heights of all students in a school (the population), we might measure just 100 students picked randomly (the sample) to get an idea.

Random Sampling

• To put it simply, a random sample is like picking names out of a hat where everyone in the group has the same chance of getting their name drawn. This fairness is really important because it helps make sure that the small group we pick actually reflects the bigger group we're interested in.

Data Types

• Data comes in two main types: discrete and continuous. 

  • Discrete data is like counting whole things – it can only be certain separate values, usually whole numbers. Think of it as things you can count, like the number of students in a class. 

  • Continuous data, however, is like measuring things – it can be any value within a range. Imagine the height of students; it could be any value within a certain span, not just whole numbers.

Reliability and Bias

• In statistics, it's really important to think about how trustworthy your information is (reliability) and if your sample is fair (bias). If your sample doesn't truly reflect the whole group you're studying, that's called bias, and it can mess up your results.

Outliers

• In statistical analysis, outliers are defined as observations that deviate markedly from the predominant pattern of the dataset. These anomalous data points, being significantly disparate from the central cluster of values, possess the potential to exert a disproportionate influence on statistical outcomes and therefore warrant meticulous evaluation.

Data Presentation

Frequency Distributions

• A frequency distribution is a tabulation or graphical representation that organizes observed data to illustrate the incidence of each distinct value within a dataset. This method effectively summarizes the dataset by displaying the count of occurrences for every unique value, thereby revealing the underlying pattern of data dispersion.

Histogram

• In statistical representation, histograms function as bar graphs specifically designed to depict the frequency distribution of continuous data. In a histogram, the horizontal axis (x-axis) is scaled to represent the range of data values, typically grouped into contiguous intervals or bins, while the vertical axis (y-axis) quantifies the frequency, or count, of observations falling within each respective interval.

Cumulative Frequency Graphs

• also known as ogives, are graphical tools that illustrate the accumulated frequency of data values up to and including each interval. These representations are particularly valuable in statistical analysis as they facilitate the determination of key positional measures, such as the median and percentiles, directly from the visual depiction of the data's distribution.

Box and Whisker Diagrams

• commonly referred to as box plots, are graphical instruments employed to furnish a concise visual synopsis of a dataset's distribution. These diagrams effectively represent key statistical measures including the median, quartiles, and the identification of potential outliers within the data.

Measures of Central Tendency and Dispersion

Central Tendency

• Mean: This is the arithmetic average, calculated as the sum of all values in a dataset divided by the total number of values.

• Median: Representing the central value in a dataset, the median is identified as the midpoint of the ordered data, dividing the distribution into two equal halves.

• Mode: The mode denotes the value that appears with the highest frequency within the dataset, indicating the most typical or common observation.

Dispersion

• Range: Defined as the extent of data spread, the range is calculated by subtracting the minimum value from the maximum value in the dataset.  

• Variance: This measure quantifies the average of the squared differences of each data point from the mean of the dataset. It provides an indication of the overall dispersion around the mean.  

• Standard Deviation: Representing the square root of the variance, the standard deviation is a widely used measure of dispersion. It expresses the degree of data spread in the original units of measurement, offering a more interpretable measure of variability.

Linear Correlation and Regression

Correlation

• Correlation, in statistical terms, is a measure that quantifies both the intensity and the direction of the linear association between two paired variables. This relationship is encapsulated by the correlation coefficient, denoted as r, which is dimensionless and scaled to range from -1 to +1.  

• The interpretation of the correlation coefficient r is as follows:

  • A value of r = +1 signifies a perfect positive correlation, indicating that as one variable increases, the other variable increases proportionally in a perfectly linear manner.  

  • Conversely, a value of r = -1 represents a perfect negative correlation, denoting a perfectly linear inverse relationship where an increase in one variable is accompanied by a proportional decrease in the other.  

  • A correlation coefficient of r = 0 suggests no linear correlation between the two variables, implying the absence of a linear trend in their relationship, although it does not preclude the possibility of a non-linear association.  

Regression

• Linear regression is a statistical method used to model the linear relationship between two variables by determining the optimal straight line that best fits a given set of data points. This line is mathematically represented by the equation y = mx + c, where:

  • y represents the dependent variable.

  • x represents the independent variable.

  • m denotes the slope of the line, indicating the rate of change in y for a unit change in x.

  • c signifies the y-intercept, representing the value of y when x is zero.

Probability

Basic concepts

• Trial: A trial constitutes a singular performance or instance of a defined experiment.

• Outcome: An outcome is defined as a specific, observable result that may arise from the execution of a trial.

• Probability: Probability is a numerical quantification of the likelihood of a particular event's occurrence. It is expressed as a value on a scale from 0 to 1, inclusive, where 0 indicates impossibility and 1 indicates certainty

Probability Calculations

• Venn diagrams: These diagrams are primarily employed to visualize and calculate probabilities associated with set operations. They are particularly useful for problems involving the union, intersection, and complement of events, allowing for a graphical representation of relationships between events and their probabilities.

• Tree diagrams: Tree diagrams are specifically designed for analyzing sequential events, where the outcome of one event influences subsequent events. By branching out to represent each possible outcome at each stage, these diagrams facilitate the calculation of probabilities for compound events occurring in sequence.

• Sample space diagrams: These diagrams serve as a systematic method for enumerating and visualizing all potential outcomes of an experiment. By mapping out the entire sample space, they provide a comprehensive framework for determining probabilities, particularly in scenarios with a limited number of possible outcomes.

Conditional Probability

• Conditional probability refers to the measure of the likelihood of an event, denoted as event A, occurring, given that another event, denoted as event B, has already taken place. This probability is symbolized as P(A|B), which is verbally expressed as "the probability of A given B".

• Mathematically, conditional probability is defined by the formula:

P(A|B) = P(A ∩ B) / P(B)

where:

• P(A|B) is the conditional probability of event A occurring given that event B has occurred.

• P(A ∩ B) is the joint probability of both event A and event B occurring.

• P(B) is the marginal probability of event B occurring.

Discrete Random Variables and Probability Distributions

• A discrete random variable is defined as a variable whose possible values are confined to a countable set. This implies that the variable can only assume specific, distinct values, often integers, with no intermediate values permissible within its range.  

• Associated with each discrete random variable is a probability distribution. This distribution mathematically delineates the likelihood of the random variable assuming each of its possible values. In essence, it provides a comprehensive mapping of all potential outcomes of the variable and their corresponding probabilities.

Expected Value

• The expected value, denoted as E(X), of a discrete random variable X represents the theoretical average value of X over a large number of repeated trials. It is calculated as the summation of the product of each possible value (xᵢ) that the random variable can assume and its corresponding probability (P(X=xᵢ)). The formula, E(X) = ∑xᵢP(X=xᵢ), essentially weights each potential outcome by its likelihood of occurrence, providing a measure of the variable's central tendency in a probabilistic sense.

• 𝐸(𝑋)=∑𝑥𝑖𝑃(𝑋=𝑥𝑖)

Binomial Distribution

• The binomial distribution is a discrete probability distribution that mathematically describes the number of successful outcomes in a predetermined number of independent Bernoulli trials. Each trial is characterized by an identical and constant probability of success.

• If X ~ B(n, p), where n is the number of trials and p is the probability of success on each trial:

𝑃(𝑋=𝑘)=(𝑛𝑘)𝑝𝑘(1−𝑝)𝑛−𝑘

Normal Distribution

• The normal distribution is a fundamental continuous probability distribution characterized by a symmetric, bell-shaped curve. This distribution is uniquely defined by two parameters: its mean (μ), which dictates the central location of the curve, and its standard deviation (σ), which governs the spread or dispersion of the distribution around the mean.  

Standard Normal Distribution

• The standard normal distribution is a specific instance of the normal distribution distinguished by a mean (μ) of zero and a standard deviation (σ) of unity. This particular form serves as a foundational reference within statistical analysis due to its simplified parameters.

• To facilitate comparisons and calculations across diverse normal distributions, any normally distributed variable can be transformed into a standard normal variable through a process of standardization. This transformation is achieved using the z-score, which is calculated as follows:

z = (x - μ) / σ

where:

• z represents the standardized value, or z-score.

• x is the raw score or observed value from the original normal distribution.

• μ is the mean of the original normal distribution.

• σ is the standard deviation of the original normal distribution.

• The z-score essentially quantifies the number of standard deviations that a particular raw score x is away from the mean μ of its distribution. This standardization process enables the use of standard normal distribution tables or computational tools to determine probabilities associated with any normal distribution, regardless of its original mean and standard deviation.

Normal Probability Calculations

• To determine probabilities associated with normal distributions, the standard procedure involves a two-step process:  

  • Standardization: Initially, the raw value of interest is transformed into a z-score. This is accomplished through the application of the z-score formula, which effectively converts the value from its original normal distribution to its corresponding position within the standard normal distribution.  

  • Probability Determination: Subsequently, the calculated z-score is utilized to ascertain the desired probability. This step typically entails consulting standard normal distribution tables or employing computational technology capable of evaluating cumulative probabilities for the standard normal distribution. These resources provide the area under the standard normal curve, which directly corresponds to the probability of observing a value less than or greater than the standardized value, or within a specified range of standardized values.  

Bayes’ Theorem

• Bayes' Theorem establishes a mathematical relationship between conditional probabilities, specifically articulating how to revise the probability of an event based on new evidence. The theorem is formally expressed as:  

P(A|B) = [P(B|A) * P(A)] / P(B)

Continuous Random Variables and Probability Density Functions

• In dealing with continuous random variables, we shift from using probability mass functions to probability density functions (PDFs). Unlike discrete variables where we can talk about the probability of a variable exactly equaling a specific value, for continuous variables, the chance of hitting any single, precise number is essentially zero. Instead, PDFs describe the relative likelihood of the variable taking on a value near a given point. To find the probability that a continuous variable falls within a range of values, we need to calculate the area under the PDF curve over that range, which is done using integration.

Properties of Continuous Distributions

• For a continuous random variable X with PDF f(x):

• Mode: The value of x where f(x) is maximum

• Median: the value m such that the area under the PDF to the left of m equals 0.5 (or 50% of the total area)

• Mean: The value of the expected value of X

  

• Variance: The expected value of the squared deviation from the mean

  

• Standard Deviation: The square root of the variance
  

  

Linear Transformations

• In the realm of linear transformations of random variables, if we define a new random variable Y as a linear function of another random variable X, such that Y = aX + b, where a and b are constants, then the statistical properties of Y, specifically its expected value and variance, are directly related to those of X through the following established relationships:

• Expected Value of Y: The expected value of Y, denoted as E(Y), is given by the linear transformation of the expected value of X. This is expressed as:

E(Y) = aE(X) + b

• This formula indicates that the expected value of the transformed variable Y is obtained by multiplying the expected value of X by the constant a and subsequently adding the constant b. In essence, linear transformations are preserved under the expectation operator.

• Variance of Y: The variance of Y, denoted as Var(Y), is related to the variance of X by the square of the constant multiplier a. This relationship is formalized as:

Var(Y) = a²Var(X)

TOPIC 4: Calculus

1. Introduction to Calculus

Calculus is the branch of mathematics that studies continuous change. It is divided into two major parts:

• Differential Calculus: Concerns the study of rates of change and slopes of curves.

• Integral Calculus: Involves accumulation of quantities and areas under and between curves.

Calculus is fundamental in various fields such as physics, engineering, economics, and biology.

2. Limits and Continuity

2.1 Definition of a Limit

The limit of a function as approaches is written as: This means that as gets arbitrarily close to , approaches .

2.2 Techniques for Evaluating Limits

• Direct Substitution: Plugging in directly.

• Factoring: Factor and simplify before substitution.

• Rationalization: Multiply by the conjugate if necessary.

• L'Hôpital's Rule: Used for indeterminate forms (e.g., ).

2.3 Continuity

A function is continuous at if:

1. exists.

2. is defined.

Discontinuities can be classified as:

• Removable Discontinuities: Can be fixed by redefining .

• Jump Discontinuities: Sudden jumps in function values.

• Infinite Discontinuities: Asymptotic behavior.

3. Differentiation

3.1 Definition of the Derivative

The derivative of is defined as: It represents the instantaneous rate of change of .

3.2 Differentiation Rules

• Power Rule:

• Product Rule:

• Quotient Rule:

• Chain Rule:

3.3 Higher-Order Derivatives

• Second derivative: represents acceleration in physics.

• Higher derivatives: , , etc.

3.4 Applications of Derivatives

• Tangent and Normal Lines: Equation of the tangent at is:

• Increasing and Decreasing Functions:

  • is increasing.

  • is decreasing.

• Concavity and Inflection Points:

  • is concave up.

  • is concave down.

• Optimization: Finding local maxima and minima.

4. Integration

4.1 Definition of the Integral

The integral is the reverse process of differentiation. The indefinite integral is: where is the constant of integration.

The definite integral is defined as: and represents the signed area under the curve from to.

4.2 Integration Rules

• Power Rule: (for ).

• Substitution Rule: .

• Integration by Parts: .

• Partial Fractions: Used for rational functions.

4.3 Applications of Integration

• Area Under a Curve: gives the area.

• Volume of Revolution:

  • Disk Method 

  • Shell Method

• Average Value of a Function

• Kinematics: Finding displacement and total distance travele

5. Differential Equations

5.1 Definition

A differential equation relates a function with its derivatives. It can be:

• Ordinary Differential Equations (ODEs): Involving one independent variable.

• Partial Differential Equations (PDEs): Involving multiple independent variables.

5.2 Solving Differential Equations

• Separable Equations: can be rewritten as:

• First-Order Linear Equations: In the form , solved using integrating factors.

5.3 Applications of Differential Equations

• Exponential Growth and Decay: leads to

• Newton’s Law of Cooling: .

• Logistic Growth: models population growth.

6. Conclusion

Calculus is an essential mathematical tool used to model and analyze change in various disciplines. Mastery of differentiation and integration, along with their applications, is crucial for solving real-world problems.