7 Math IA Topics for a 7

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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

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.

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.

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)

\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

Family of Curves & Envelop Theorem

Bisection Method

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

x

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.

For any continuous function

f(x)

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

[x_1,x_2]

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.)

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.

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

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

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

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*}

Vectors and Matrices