From the usual source:

Can you post a calculus problem and solution so we can see what it looks like? Also, how and where would you use it in the world?

There are three major ideas of Calculus. I’ll tell you about the first two, since they’re the ones that are easy to wrap your mind around. Then I’ll finish with the third. The first two are:

- Derivative and
- Integral

### derivative

A derivative is a slope. Pure and simple. Just as slope in seventh and eighth grade is the rate of change of one variable with respect to the other, derivative is the rate of change in calculus. I don’t mean this metaphorically; they are the same thing.

In algebra, slope has to do with straight lines. If Emile walks 75 meters in 30 seconds, you have no trouble figuring out how many meters per second he walks. That’s his walking rate; it’s the slope of his line. In calculus, we would say that this is the *derivative* of his walking function.

But what if his rate isn’t constant? What if he starts off slowly and speeds up as time goes by? What if his walking function is more complicated than *y*=2.5*x*? You get a taste of how to figure rates (slopes) of non-linear relationships when you work on first and second differences in studying quadratic or exponential relationships.

But first and second differences give us *average* rates of change. Calculus is interested in *instantaneous* rate of change-not just *what is the average speed between 2 and 3 seconds?* but *how fast is this thing moving right now?*

Consider the video below:

How does the speedometer work? Calculus answers that question using derivatives.

If we model the car’s distance traveled as a function of time:

We read this as *The derivative of f with respect to t*. When we specify *f* as a function, the derivative is a function. When we also specify a number *t*, then the derivative has a numerical value. That value is the speed at the time we selected.

### INtegral

Going back to Emile who walks 2.5 meters per second. If you know his walking rate and how long he walks, you can figure out how far he walked. And you do that frequently in algebra.

But what if his walking rate is changing? Now it’s a much harder problem, and it requires more advanced mathematical techniques. Go back to the video:

How far did the car travel in going from 0 to 60? Now, strictly speaking you don’t need Calculus to answer this. You could mark the beginning and end of the run and measure it. But what if it’s an airplane and you need to know how long to make the runway? What if it’s the Space Shuttle? These would be harder to measure directly. The Calculus idea of *integral* solves these kinds of problems.

If we model the car’s speed as a function of time from time 0 to time *a* (meaning whenever the car gets to 60 mph):

And we read this as *The integral from 0 to a of g of t*. After specifying *g* as a function and *a* as a number, this integral has a numerical value-the distance the car traveled.

### limit

Now think about what you know in algebra. To find Emile’s speed, you divided his distance by his time. No matter what two points you choose on Emile’s graph, the difference in distance divided by the difference in time will be the same. But when his walking rate is changing, that’s not true. The *average* speed will depend on which points you choose. And as a general rule, the average speed between two times will not be the exact speed he is traveling at either time.

So mathematicians get a better approximation of the speed at a given time by taking averages over small intervals of time. And they imagine an answer to the question, *What if we averaged his speed over 0 seconds?* This would mean dividing by zero, which is impossible. The third major idea of Calculus-*limit*-gets around this technicality.

Derivatives are defined in terms of limits.

And so are integrals for a similar reason. Our attempts to precisely answer questions about how far the car travels in getting to 60 mph inevitably lead to adding up infinitely many numbers. We literally do not have time for that. *Limits* get us around this problem, too.

This is outstanding, and highlights one of my favorite parts of the CMP curriculum- the first and second differences discussion.

Invokes Silvanus P. Thompson’s classic ‘Calculus Made Easy‘:

‘Considering how many fools can calculate, it is surprising that it should be thought either a difficult or a tedious task for any other fool to learn how to master the same tricks. Some calculus tricks are quite easy. Some are enormously difficult. The fools who write the textbooks of advanced mathematics –and they are mostly clever fools– seldom take the trouble to show you how easy the easy calculations are. On the contrary, they seem to desire to impress you with their tremendous cleverness by going about it in the most difficult way.’