Calculus without Plotting Graphs

Calculate slopes mathematically

We said earlier that calculus is about understanding how things change in a mathematically precise way. Let’s see if we can do that by applying this idea of ever smaller $\Delta x$ to the mathematical expressions that define these things.

To recap, speed is a function of the time we know to be $s = t^2$. We want to know how the speed changes as a function of time. We’ve seen the slope of $s$ when it is plotted against $t$.

This rate of change $\partial s / \partial t$ is the height divided by the extent of our constructed lines, but is where the $\Delta x$ gets infinitely small.

Its height is $(t + \Delta x)^2 - (t - \Delta x)^2$, as we saw before. This is just $s = t^2$, where $t$ is a bit below and above the point of interest. That amount of bit is $\Delta x$.

What is the extent? As we saw before, it’s simply the distance between $(t + \Delta x)$ and $(t - \Delta x)$, which is $2 \Delta x$.

So, we have:

$$\frac{\partial s}{\partial t} = \frac {\text{height}}{\text{extent}}$$

$$\frac{\partial s}{\partial t} = \frac {(t+\Delta x)^2 - (t - \Delta x)^2}{2 \Delta x}$$

Let’s expand and simplify that expression:

$$\frac{\partial s}{\partial t} = \frac {t^2 + \Delta x^2 + 2t \Delta x - t^2 - \Delta x^2 + 2t\Delta x}{2 \Delta x}$$ ...