Very fast calculus: Consider a standard car with a speedometer (reports how fast are going) and an odometer (reports how far have gone).

Easily enough we can take the speedometer readings, say, 1 time each second, and calculate a good approximation to the odometer readings. That is a 1 second approximation to the calculus operation of integration.

Similarly we can take the odometer readings, say, 1 time each second, and calculate a good approximation to the speedometer readings. That is a 1 second approximation to the calculus operation of differentiation.

If we use smaller time intervals than just 1 second, then we will usually get a more accurate approximation. It is a theorem that, under mild assumptions, as we let the lengths of the time intervals shrink toward 0, the results of the operations will reach limits and quit changing.

Those limiting values are the actual definitions of differentiation and integration.

No big surprise, under mild assumptions, if we start with the odometer readings, differentiate to get the speedometer readings, and integrate to get back the odometer readings, then we really will get back the odometer readings. That is the fundamental theorem of calculus.

Some common mild assumptions are basically that the speedometer readings change only continuously (no jumps) over time and we are working over only finitely long time intervals.

Newton's second law of motion

force = mass x acceleration

essentially guarantees the continuity of the speedometer readings and, thus, justifies the integration back to the odometer readings.

Of course, calculus and Newton's second law of motion are close cousins in both theory and applications -- no big surprise since Newton essentially created both (might mention Leibniz and some others).

Can quickly show that if we integrate time t, we get (1/2)t^2. So if we differentiate (1/2)t^2 we will get back t.

A calculus course will show how to differentiate and integrate a wide variety of mathematical expressions, polynomials, sines and cosines, products, quotients, composite expressions, etc.. E.g., differentiate sine(t) and get cosine(t). Differentiate cosine(t) and get -sine(t). Can also find many cases of arc lengths, areas, volumes.

Suppose we are starting a business. At time t, let the revenue be y(t). Suppose we have argued that as we reach all our target customers, our monthly revenue will be b. Suppose we argue that due to word of mouth advertising the rate of growth is proportional to both the number of happy customers talking and the number of target customers not yet customers listening. Denote the rate of growth of y(t), that is the derivative, by y'(t). Then for some constant of proportionality we should have

y'(t) = k y(t) ( b - y(t) )

Of course we know current revenue, say, at time t = 0, that is, y(0).

Then by the first weeks of calculus, can show that, with TeX syntax,

y(t) = { y(0) b e^{bkt} \over y(0) \big
( e^{bkt} - 1 \big ) + b }

More generally

y'(t) = k y(t) ( b - y(t) )

is an example of an initial value problem
of a first order, linear, ordinary differential
equation and an introduction to a course in ordinary differential equations.

Calculus has wide applications to physical science, engineering, economics, finance,
spread of diseases, etc.