IA Topics

Sample Research Questions

How can regression techniques be used to explain the relationship between xxx and yyy?

•

Predicting the yyy value based on multiple input factors x1x_1x1​, x2x_2x2​, x3x_3x3​?

•

How can interpolation methods can be used to generate the simplest possible curve passing through nnn data points?

How can [Regression Technique] be used to explain the relationship between [explanatory variable #1, #2, #3...] and [dependent variable]

Linear Regression

Suppose we have

n

non-collinear data points,

(x_1, y_1)

,

(x_2, y_2)

,…,

(x_n, y_n)

.

A linear regression model,

\hat y=ax+b

, may be defined to predict a value of

y

for each corresponding input value of

x

.

Depending on how the line is drawn, some predicted values

\hat {y_i}

are less than the actual value

y_i

.

\hat {y_1}=ax_1+b

and

\hat y_3=ax_3+b

underestimate the value of

y_1

and

y_3

respectively.

In other cases, the predicted value of

\hat {y_i}

is greater than the actual value of

y_i

.

\hat {y_2}=ax_2+b

, for example, overestimates the value of

y_2

.

Given that an infinite combination of coefficients

(a, b)

are possible, which linear equation

\hat y=ax+b

would be the curve of best fit offering the most accurate representation of all individual data points?

Curve Fitting & Regression Techniques

Sample Research Questions

•

Designing a geometrical 3D shape (cylinder, cone, etc.) to maximize volume for fixed surface area

•

Minimizing the surface area required to construct a 3D shape that can hold a given volume of contents.

Problem Definition

An optimization problem can be formulated by incorporating the following elements.

Decision Variables

The controllable inputs of an objective function that represent decisions to be made within the problem.

Objective Function

Mathematical expression that calculates the objective value.

Objective Value

Constrained Optimization

Dynamic Programming

Dynamic programming algorithms, colloquially referred to as divide-and-conquer algorithms, are problem-solving paradigms that utilize memoization to efficiently solve problems by breaking them down into overlapping sub-problems and solving each sub-problem only once, storing their solutions to avoid redundant computations.

Dynamic programming algorithms offer improved speed and efficiency when it comes to solving complex problems that involve recursive functions.

Recursive Functions

In programming, a recursive function calls upon itself to solve a problem by breaking it down into smaller and simpler sub-problems until the base case is reached. Repeated function calls create entries for all variables and constraints in a function stack, leading to suboptimal use of memory space as redundant values have to be stored until the function returns the final base case output.

Memoization Tables

Memoization is a data structure that stores the results of expensive function calls and returns the cached result when the same inputs occur again, enabling efficient computing resource usage when optimizing recursive functions.

Fibonacci Sequence

F(n) = \begin{cases} F(n-1)+F(n-2),~ n \geq3 \\ 1,~n=1,2 \end{cases}

As can be seen in the above example, repetitive function calls and calculations must be made until the base case F(1) and F(2) are reached.

Knapsack Problem

Knapsack Problem (AI-friendly)

Envelope Condition

Pencils refer to a family of curves that can be generated by varying a single parameter

t

within an equation in the form

F(x,y,t)=0

.

Varying parameter

t

can lead to different shapes and configurations within the family.

The collection of points

(x,y)

that satisfy:

\begin{cases} F(x,y,t)=0 \\ \frac{ ∂F(x,y,t)}{∂t}=0 \end{cases}

form the envelope for the family of curves,

F(x,y,t)=0

.

Envelope of Lines

F(x,y,t)=yt+(11-t)(x-t)

Describes a family of lines.

\frac{∂F(x,y,t)}{∂t}=\frac{∂}{∂t}(yt+(11-t)(x-t))

Solving for

\frac{∂}{∂t}(yt+(11-t)(x-t))=0

yields:

-(x-y)^2+22(x+y)-121=0

Family of Curves & Envelop Theorem

Bisection Method

Numerical technique that approximates the root (zero) of a given function by finding values of

x

for which

f(x)=0

.

The bisection method repeatedly bisects an interval and determines which subinterval the root lies in, eventually narrowing down the interval that the zero resides in to a desired level of precision.

While it is robust and guaranteed to converge to a solution as long as the function is continuous and changes sign over the interval, the bisection method converges slowly compared to other methods.

Algorithm

For any continuous function

f(x)

, there must exist at least one zero within a closed interval

[x_1,x_2]

, where

f(x_1)\cdot f(x_2)<0

(function values are opposite signs at the ends of the interval, so the continuous graph must have crossed the x-axis at at least one point.)

1.

xi+2=xi+xi+12x_{i+2}=\frac{x_i+x_{i+1}}2xi+2​=2xi​+xi+1​​ to find the midpoint of the current interval

2.

If f(xi+2)=0,(xi+2,0)f(x_{i+2})=0, (x_{i+2},0)f(xi+2​)=0,(xi+2​,0) is the function’s root (zero).

3.

If f(xi+2)≠0f(x_{i+2})\neq 0f(xi+2​)=0,

4.

Repeat process until f(xi)=0f(x_i)=0f(xi​)=0 or f(xi)<εf(x_i)<εf(xi​)<ε, where εεε is a predefined threshold condition for terminating the bisection algorithm.

Investigation

\begin{align*} &\text{Zero of function lies in closed interval } [x_3, x_2] \text{ as } f(x_3) \text{ and } f(x_2) \text{ are opposite signs.} \\ &\text{Bisect the } [x_3, x_2] \text{ into two subintervals, } [x_3, x_4] \text{ and } [x_4, x_2]. \\ &\text{Zero of function lies in closed interval } [x_3, x_4] \text{ as } f(x_3) \text{ and } f(x_4) \text{ are opposite signs.} \\ &\text{Repeat process until sufficiently precise interval containing zero is obtained.} \end{align*}

Numerical Root Finding Techniques (AI-friendly)

Applications of differential equations in real life

Second-order partial differential equations are used to model standing waves on string, the distribution of heat along a solid, or even the quantum behavior of subatomic particles. While they provide useful information about a physical system, partial differential equations are often tricky to solve, requiring the use of advanced techniques beyond those introduced in the IB math HL syllabus.

Wave Equation:

\frac{∂^2u(x,t)}{∂t^2}=c^2\frac{∂^2u(x,t)}{∂x^2}

Heat Equation:

\frac{∂^2T(x,t)}{∂t^2}=c^2\frac{∂^2T(x,t)}{∂x^2}

Partial Differential Equations (PDEs)

Solution by Separation of Variables

A common technique to solving PDEs is to assume that the multivariate function can be expressed as the product of two single variate functions:

u(x,t)=F(x)G(t)

For example, when solving the Schrodinger equation to find a particle’s wave function

Ψ(x,t)

, which has bivariate input (position and time), the wave function is assumed to be the product of two single variate functions,

ψ(x)

and

φ(t)

, of position and time respectively.

Statistical Interpretation of the Wave Function

Advanced Calculus

System of Equations

Many math IA topics frequently involve setting up and solving large systems of equations (see: extension from topic 1: “Systematic approaches to reliably obtain solutions to large systems of equations with

m

equations and

n

unknown variables (in the HL syllabus, we typically worked with

m ≤ 3

,

n ≤ 3

)” Managing complex (linear) systems of equations that have a large number of equations and unknown variables requires the use of simplifying notation and standardizing procedures

Augmented Matrices

\begin{align*}a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n &= b_1 \\a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n &= b_2 \\\vdots \quad\quad \vdots \quad\quad &\vdots \\a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n &= b_m\end{align*}

\begin{bmatrix}a_{11} & a_{12} & \cdots & a_{1n} & b_1 \\a_{21} & a_{22} & \cdots & a_{2n} & b_2 \\\vdots & \vdots & \ddots & \vdots & \vdots \\a_{m1} & a_{m2} & \cdots & a_{mn} & b_m\end{bmatrix}

Elementary Row Operations

Three types of algebraic operations can be performed on equations to produce a succession of increasingly simpler systems that have the same solution set as the original system

1.

Multiply an equation through by a nonzero constant

2.

Interchange two equations

3.

Add a multiple of one equation to another

Matrices and Vectors