# Interactive Math Demos

In this website, you'll find some interactive math demos for students under 19. Click on one of the other tabs to begin.

# Calculus

Let's start from graphs. Choose a function to plot its graph below.

It is often of interest to know how much the value of a function changes $$(\Delta y)$$ for a given change in the input $$(\Delta x)$$. Using computers these variations can be computed quickly for all the points.

# Tracking $$\frac{\Delta y}{\Delta x}$$ in $$y=x$$

$$y(x)$$
$$\frac{\Delta y}{\Delta x }$$
$$\Delta x$$

Hovering over the function's plot shown in red, computes $$\Delta y / \Delta x$$ and shows it in green below. As the interval $$\Delta x$$ is reduced to very small values using the slider on the right, the green curve approaches the blue curve. The blue line corresponds to the true derivative and often referred as $$dy/dx$$. This is written as: $$\frac{dy}{dx} = \lim_{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}$$ The process of computing the derivative for a given function is termed differentiation. In simple words, differentiation or differential calculus deals with the study of increase or decrease of a function. It turns out that this is quite useful in many applications. For example, the derivative of distance travelled $$x$$ as a function of time $$t$$ gives us $$v$$, the instantaneous velocity (velocity at a specific time instant). $$v = \frac{dx}{dt} = \lim_{\Delta t \rightarrow 0} \frac{\Delta x}{\Delta t}$$

# Eigen vectors and Eigen values (Work in progress)

Consider the linear transformation: $$y= Ax$$ where, a square matrix $$A$$ acts upon a vector $$x$$ to form a new vector $$y$$. Supposing, $$A = \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}$$

For a given matrix A, eigen vectors are those special vectors which when transformed by $$A$$ result in no change in direction. In other words, the resulting vector has the same direction as $$x$$. We can write this as:  In this demo, we will consider a $$2 \times 2$$ matrix to demonstrate the concept of Eigen vectors. A square matrix $$A$$ acting upon a vector $$x$$ results in a new vector say $$y$$. This is often called a transformation as $$A$$ transforms the vector $$x$$ into $$y$$. For every square

# Newton Raphson method demo (Work in Progress)

Newton Raphson method provides a quick way to compute the roots of a continuous and differentiable function provided its derivative is known. The steps involved are:

1. Pick an initial point
2. Compute the slope at the point
3. Draw a line from the point with this slope until it meets the independent axis
4. Select this as the new point
5. Repeat

Here is a demo of this process: