A coded animation portfolio. Thank you to Grant Sanderson for Manim (www.manim.community).
In mathematics, change is not merely an abstract idea but a dynamic, measurable force that governs how one variable interacts with another. The slope of a line, the rate at which a function shifts or evolves, embodies this concept of change, revealing a deeper, philosophical truth: all things are in flux, moving along paths that may seem linear or curved, but are always changing, always in motion. Understanding the nature of this mathematical change allows us to see how patterns emerge, repeat, and transform—both in numbers and in life.
Slope of a Line
The slope of a line is a measure of how steep the line is, or how much it rises or falls as you move along it. The slope is a crucial concept because it tells us the rate at which one variable changes in relation to another. Specifically, the slope is calculated as the “rise” over the “run,” which means how much the line goes up or down for a given horizontal movement.
The slope is often represented by the letter m in the equation of a line, which typically looks like this: y=mx+b. Here, m is the slope, and b is the y-intercept—the point where the line crosses the y-axis.
In this visualization, you can see a point moving along the line y=2x+1. The slope m=2 means that for every unit you move to the right (along the x-axis), the line goes up by 2 units (along the y-axis). This constant slope is illustrated by the straightness of the line: no matter where you are on the line, the rate of change remains the same.
Tangent Line Along a Curve
The tangent line represents a straight line that just touches a curve at a specific point without crossing it. This point of contact is crucial because the tangent line shows the “instantaneous rate of change” of the curve at that exact spot. Think of it like a snapshot of the curve’s direction at a single moment.
The concept of a derivative connects closely with this idea. The derivative of a function at a particular point tells us the slope of the tangent line at that point. In simpler terms, it tells us how fast the curve is rising or falling right there. If the derivative is a positive number, the function is increasing at that point; if it is negative, the function is decreasing. The greater the absolute value of the derivative, the steeper the slope of the tangent line.
This visualization helps illustrate this concept dynamically. As the dot moves along the curve, notice how the tangent line adjusts itself to always just touch the curve at the moving point.
Rate of Change in Exponential Growth and Decay
For exponential growth — like population growth or investment returns — the quantity increases at a rate proportional to its current size. This means the larger the quantity, the faster it grows. Mathematically, if we consider f(t) = et, the derivative at any point t is also et. This tells us that the rate of change (how fast the function is growing) is the same as the function’s value. As t increases, the rate of growth increases, making the curve steeper.
For exponential decay — like radioactive decay or cooling — the quantity decreases at a rate proportional to its current size. As the quantity gets smaller, the rate of decrease slows down. With f(t) = e-t, the derivative at any point t is -e-t. The negative sign indicates that the function is decreasing. Closer to zero, it decreases quickly, but as the value gets smaller, the rate of decay slows.
As the dot moves along the exponential growth curve f(t) = et, the tangent line becomes steeper, reflecting the increasing rate of change. Conversely, on the exponential decay curve f(t) = e-t, the tangent line gradually flattens, showing a decreasing rate of change over time.
Tangent Lines for Sine and Cosine
The sine and cosine functions are known as periodic functions, meaning their values repeat in a regular, predictable pattern over a specific interval. For both sine (sin(x)) and cosine (cos(x)), this interval is 2π (or 360 degrees). In other words, for every 2π units along the x-axis, the sine and cosine functions complete one full cycle and begin to repeat their values.
Now, let’s consider the tangent line. Since the sine and cosine functions repeat their values every 2π, the behavior of their slopes also repeats every 2π. This means that at corresponding points within each interval, the slope of the tangent line will be the same. For example, the slope of the tangent line for sin(x) at x = 0 will be the same as the slope at x = 2π, x = 4π, and so on.