Calculus for the rest of us – Part 2 July 2, 2007Posted by Jeff in Education.
The last “Calculus for the rest of us” post calculated how far you drive when driving at the same speed for a certain amount of time. If you make a chart of your speed over time, you’re looking at a horizontal line between the time you start (hour zero in the example) and the time you finish (hour twelve in the example).
Let’s say that instead of driving the same speed for all 12 hours, you actually start from zero miles per hour and accelerate – speed up – for the whole trip, until you are going 70 miles per hour by the time you get there. Again, this is not really how one would drive, but we’re getting there.
For a trip like this, if you made a chart of your speed over time – starting from 0 mph and working up to 70 mph after 12 hours – it would look something like this:
Just like in the last example, we created a graph that has a medium blue line with the area under it filled in with a lighter blue color.
Again, you may not intuitively think of it this way, but that medium blue line has an equation, and its equation is:
y = (70/12)x
Reminder: y is the speed of your car in miles per hour and x is the number of hours you’ve been driving. So, if you want to know how fast you were going after 6 hours, you can plug that in to the equation of the line:
y = (70 / 12) × (6) = 35 mph
In our last example, where your car was going 70 miles per hour the whole time, the total distance your car traveled was equal to the speed of your car times the time you were driving (70 mph × 12 hours = 840 miles). This was equal to the area under the medium blue line – or the “curve” as the mathematicians like to call it. This time, the car is not always going the same speed – it’s speeding up the whole time. However, we can still figure out how far your car went by calculating the area under the medium blue line.
To do this, take a look at the “curve” in the above graph. It is just a straight line that slants upward from zero mph at hour zero to 70 mph at hour twelve. The area under it – in light blue – is just a triangle, and figuring out the area of a triangle is a cinch! The area of a triangle is one-half times the length of the triangle times the height of the triangle, or:
Area = (1/2) × Length × Width
The triangle here has the same length and width as the rectangle in the last post. The length is 12 hours and the width is 70 miles per hour. Using the formula above for the area of the triangle, we see that:
Area = (1/2) × 12 × 70 = 420 miles
So, if your car was completely stopped and you started accelerating very slowly and very steadily for 12 hours, until your car was going 70 miles per hour, you would have gone 420 miles.
In the next post, we’ll figure out the same thing, but we’ll think of a more realistic situation, where you speed up pretty quickly and then travel at the same speed for the rest of the trip.