I once had a student working through a midterm exam problem in calculus, about a bike ride:
A bicyclist sets off for school. She takes fifteen minutes to get up to cruising speed, 16 miles per hour. After fifteen minutes of cruising, she remembers she forgot her presentation for class at home: time to turn around! She turns around almost instantaneously and her velocity is suddenly 16 miles per hour in the other direction. She keeps going at 16 miles per hour until she gets home.
(a) Draw a graph of the bicyclist’s velocity on the following axes. Label your units.
(b) Use geometry or Riemann sums to estimate the distance the bicyclist traveled in her first 45 minutes. Show your work clearly.
(This is from a real exam I gave at St. Olaf College about four years ago.)
She had a quick question about something so I got to look at her work — and I was horrified to see that she’d flipped the miles and the minutes upside down and mixed up minutes and hours. Her answer was off by a factor of 1000 or something crazy like that. But the integral and the rest of the process was fine! I felt so bad for her — but it was a test, and I couldn’t say anything.
Units can save you from silly conversion mistakes. In any application problem like this, make a habit to write down the units next to every number you’re using. Speed: 16 — is that 16 minutes per hour or miles per hour? WRITE IT DOWN! 16 m/h is 16 miles / 60 minutes. Writing down the units will remind you to convert things: you’ll save yourself from writing 15*16 = 240 so the biker went 240 miles before she even turned around…. ?!
Units can help you figure out what to do. You need velocity. You need distance. You have a shaky grasp on the whole calculus thing but you know you need to do an integral and a derivative. Velocity is miles per hour with a sign (unlike speed) and distance is just miles. Miles are…. miles per hour times hours. Hm. In a Riemann sum you’re going to multiply miles per hour by time (hopefully hours!) — that will give you miles. Ok! Check! For part (a) you’ll draw the speed with a sign, and then for part (b) you’ll draw boxes and triangles to find the area between the curve and the horizontal axis.
This could help in the other direction, too. If you have distance f(t), then the derivative of f(t) with respect to t is d f(t) / dt — (instantaneous) change in distance over change in time. That’s miles per hour, so velocity is the derivative of the distance function.
You can even figure out crazy physics formulas just by making sure the units match up. You have some nonsense problem that requires you figure out something in Newtons? Well, Newtons are kilograms times meters per second squared. I bet they gave you a mass (kilograms) and an acceleration (meters per second squared). There’s an 80% chance you’ll get this problem right just by multiplying those numbers, because it’ll make the right units for the answer.
Start this habit right away in precalculus if it’s not too late. It will help so much!
