Calculus for the rest of us – Part 3 July 3, 2007Posted by Jeff in Education.
The last two posts on Calculus for the rest of us (part 1 here, part 2 here) have focused on describing a real world use of calculus – finding the distance your car travels when you press on the gas pedal in different ways.
In part 1, we found the distance your car would go if you held the gas pedal down the same amount for a long time – in that case, you would go a certain speed (we used 70 miles per hour) for a certain amount of time (we used 12 hours). In part 2, we found the distance your car would go if you started from a stand-still and slowly pressed the gas pedal down over a long period of time (again, we used 12 hours) until you were going a certain speed (again, we used 70 miles per hour).
For both of these, we saw that figuring out the distance you traveled was as easy as figuring out the area under the “curve” of a chart of your speed. For part one, the “curve” was just a flat (horizontal) line, with an equation of y=70, because your speed was always 70 miles per hour, and the shape of the area under that curve was a simple rectangle. For part two, the “curve” was a line that ramped up from 0 mile per hour at hour 0 to 70 miles per hour at hour 12, and the shape of the area under that curve was a simple triangle.
Now in the real world, we don’t go 70 miles per hour constantly for 12 hours, and we don’t start from rest and ramp up to 70 miles per hour over the course of 12 hours. In the real world, we might start from rest, but we will accelerate very quickly to get up to our desired speed and then we will keep that speed relatively constant over time. The next example shows how to use calculus to figure out how far your car travels in a scenario that seems a little bit more like the real world. In this example, we’re going to assume that you accelerate quickly at first and then slower later on, more like you would in a car.
Let’s say your car speeds up like the medium blue curve in the following graph:
Now, this curve is a real curve. It’s not a rectangle or a triangle that make it easy to figure out the area under it, but just like the simple examples in the last two posts, the area under this curve – the light blue area – is the distance your car travels while you’re driving. For this example, we have a more complex equation for the curve:
y = (-70/144)x2 + (70/6)x
Don’t worry about what the equation is or how we got it. The idea is that that equation is roughly the shape of how your accelerated your car. In practice, you sometimes have to figure out an equation like this through experimentation or through approximation. The point is that you have an equation to start with.
The next thing is to find the area under that curve. (Finally I’m getting around to saying this after only 1,671 words! Hopefully by now you know why we want to find the area under the curve – it is the distance the car travels on your trip.) Since we cannot just multiply length by width – like we did with the rectangle – to get the area, we have to have another way to get the area. The way to do this is with an “integral”. This is one of the calculus things that scares people, but integrals are not scary.
I’m not going to explain the math behind how to figure out what an integral is, but suffice it to say the following: If you were going to figure out the area under that curve by hand, one way to do it might be to divide it up into a bunch of areas that you could calculate the area of. For example, you already know how to find the area of a rectangle (length × width), so you could divid the curve up into a bunch of rectangles, like in the following graph, calculate the area of all the rectangles and add them all up:
That technique would get you pretty close to the actual area under the curve, but it wouldn’t be exact. (The edges of the rectangles are not really lined up with the edge of the curve, so the area under the curve might be more or less than the areas of all of the rectangles.) Remember, one of the reasons you might be doing this exercise is to create a more fuel efficient car. Being exact is important for things like that, and integrals are the tools we use to get an exact answer for the area under the curve. Integrals, essentially, are mathematical tools that divide up a curve into a whole bunch of tiny rectangles -similar to the graph above – and then add up all of their areas.
Mathematicians like to derive things like integrals, but that can get a little tedious. So, I’m going to skip a few steps, and you’ll have to trust me. The total distance your car travels is the integral of (-70/144)x2 + (70/6)x between 0 hours and 12 hours. Putting that in math notation, we get this:
12 Area = ∫ ( (-70/144)x2 + (70/6)x ) dx 0
The integral of that equation is something that you can look up in a math book, but for the sake of simplicity, I’m going to give the answer below. If you want to know how I got this, just ask me.
3 2 | 12 Area = (-70/432)x + (70/12)x | | 0
The “|” characters at the end of that line, followed by the 12 and 0, just mean that we need to make x equal to 12 and solve the equation and then make x equal to 0 and solve the equation again. The latter subtracted from the former is the area we’re looking for. Therefore:
3 2 Area = ( (-70/432)(12) + (70/12)(12) ) 3 2 - ( (-70/432)(0) + (70/12)(0) )
Area = 560 - 0 = 560 miles
So, when you accelerate your car like this, you end up traveling 560 miles in 12 hours.