Analyzing Functions: Intersections and Critical Points

Understanding mathematical behavior often requires comparing multiple functions simultaneously or pinpointing specific coordinates. Our updated EquationGraph component now supports multi-series plotting and coordinate points.

1. Finding Intersections

Consider the problem of finding where a parabola $y = x^2$ intersects with a line $y = x + 2$.

Solving $x^2 = x + 2$: $x^2 - x - 2 = 0$ $(x-2)(x+1) = 0$ So, the intersections are at $x = -1$ and $x = 2$.

Let’s visualize this:

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2. Visualizing Derivatives

In Calculus, it’s helpful to see a function alongside its derivative. Here is $\sin(x)$ and its derivative $\cos(x)$.

Notice how the peaks of the sine wave line up with the zero-crossings of the cosine wave.

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3. Optimization Problems

Visualizing the maximum area of a rectangle inscribed in a semicircle. If the circle is $x^2 + y^2 = 4$ (radius 2), the area of a rectangle with corner $(x, y)$ is $A(x) = 2x \cdot \sqrt{4-x^2}$.

This graph clearly shows the peak area occurs at $x = \sqrt{2} \approx 1.414$, where the Area is exactly 4.

Implementation Code

Here is how you can use the component in your MDX files:

<EquationGraph 
    series={[
        { function: "sin", color: "blue" },
        { function: "cos", color: "green" }
    ]}
    points={[
        { x: 0, y: 0, label: "Origin" }
    ]}
/>