Calculus without Plotting Graphs

Learn how to calculate gradients mathematically.

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 Δx\Delta x to the mathematical expressions that define these things.

To recap, speed is a function of the time we know to be s=t2s = t^2. We want to know how the speed changes as a function of time. We’ve seen the slope of ss when it is plotted against tt.

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

Its height is (t+Δx)2(tΔx)2(t + \Delta x)^2 - (t - \Delta x)^2, as we saw before. This is just s=t2s = t^2, where tt is a bit below and above the point of interest. That amount of bit is Δx\Delta x.

What is the extent? As we saw before, it’s simply the distance between (t+Δx)(t + \Delta x) and (tΔx)(t - \Delta x), which is 2Δx2 \Delta x.

So, we have:

st=heightextent\frac{\partial s}{\partial t} = \frac {\text{height}}{\text{extent}}

st=(t+Δx)2(tΔx)22Δx\frac{\partial s}{\partial t} = \frac {(t+\Delta x)^2 - (t - \Delta x)^2}{2 \Delta x}

Get hands-on with 1200+ tech skills courses.