The essence of calculus

Calculus seems like a mélange of complicated symbols when you first encounter it. However, behind the cryptic letters and symbols lies a simple idea: change.

Differential calculus

This is the branch of calculus which is often taught first. It can help find the gradient of a graph- even a curved one- with accuracy, but what does the gradient mean in the first place?

You may already know that gradient is calculated by the change in the y value divided by the change in the x value for two sets of coordinates. Now let's take a look at this but with a graph of distance against time.

The gradient of a graph of distance against time, where time is on the x axis and distance is on the y axis, is the change in distance divided by the change in time. This is also equal to the speed, which is the rate of change of distance! This implies that a graph's derivative shows its rate of change.

This idea can be applied to any graph: its derivative is its rate of change.

Integral calculus

If differentiation is shredding a picture into tiny pieces, integration is putting those little pieces together. Behind the elusive symbol and complex methods is simple summation- except it is continuous, rather than discrete. It adds an infinite number of infinitesimally small piece, while assuming the variable is continuous. The importance of abstract infinity here may be why it's so hard to wrap our heads around calculus, yet calculus is so crucial that we must learn to make peace with infinity. Or at least be able to tolerate it.

To conclude, calculus is much more than just fancy symbols and mysterious functions: it is change and accumulation, and it shows continuity.

In this post, we've only discussed calculus in two dimensions. Now what do you think it would mean to differentiate a 3-dimensional graph?