Intro To Limits: Average Speed vs Instantaneous Rate of Change
Seeing as Newton pioneered Calculus, or as he called it The Method of Fluxions, it’s no wonder that one of the first topics you’ll learn about in Calculus involves an application of physics: rates of change.
If these sorts of applications make your eyes cross, then this quick tutorial will hopefully ease the pain. We’re going to start with some very basic concepts and work our way, gently, up to the idea of the limit, which lies at the heart of Calculus.
Galileo’s Law
We’ll begin our exploration of Calculus by investigating a simple and classic concept: the rate of a falling object over a period of time.
In the 16th century, Galileo discovered an equation that relates the distance and time it takes a solid object dropped from rest to hit the ground. The equation accounts for the effect of Earth’s gravity on the object and assumes negligible air resistance. The equation is pretty famous, and you’ve probably seen it before. It says that the height or distance from the ground in feet,(y) is equal to the fall time (t) in seconds squared and multiplied by 16.
Falling Rocks
Let’s put our formula to use. Say we have a rock that breaks loose and falls from a cliff, and it takes 20 seconds for the rock to hit the ground. Using Galileo’s law we can calculate just how high that cliff was by simply substituting t=20 into the equation.
Therefore, we know that assuming negligible air resistance, the cliff has a height of 6400 feet. Not too difficult, right?
Calculating Average Speed
Next, we want to find the rock’s average speed over an interval.
One thing we know about gravity’s pull on falling objects is that the longer the object falls the faster it’s speed. In other words, free-falling is an acceleration. This means that the speed differs over different time intervals of the fall.
For this first example, we are going to find the average speed during the first 3 seconds of the fall.
Since we want to find the speed, we know we’ll be looking for an answer in distance over time. In this case, our units will be feet per second.
And we want the average, so we’ll average both the change in height from the beginning of the interval to the end, Δy, (pronounced “delta y” where the delta symbol represents “change”) as well as the change in time, Δt.
Our interval is the first three seconds, so we can think of this as the interval [0,3] in mathematical notation. I’ll label the first time in the interval as T1 and the second time as T2. That means we’ll be calculating the height (y) for T1 and subtracting it from the height of T2 to get the change in height, Δy, over the time period.
Similarly, I’ll also be subtracting the start time from the end time in the time period to get Δt. This will provide us with the change in height divided by the change in time, which, if you think about it, is the average speed over the interval.
Filling in the formula with T1 = 0 and T2 = 3, for the interval [0,3], we get:
Finishing out the arithmetic we get 48 ft/sec is the average speed during the first 3 seconds of the fall.
How to Find Instantaneous Rate of Change
So far this has all been pretty straightforward. Now it’s time to start working our way towards the concept of the limit.
Suppose we want to know the speed of the falling rock at a specific time during the fall, say we want to know the speed at 2 seconds. (A speed at a specific point in time is called the Instantaneous Rate of Change by the way.) Using what you know from above, how could we go about calculating this? Or better yet if we don’t know how to calculate it precisely, how could we make a really good estimate?
Any ideas??
Well, we already know how to calculate the average speed over an interval. So one way to tackle this problem would be to calculate the average rate of speed over a small interval beginning at 2 seconds to see if we can make an educated guess at what the instantaneous speed at 2 seconds is.
Let’s figure out what the average speed is in the 1-second interval starting at 2 seconds. AKA let’s calculate the average speed on the time interval [2,3]. Plugging in the values to our average speed formula we get:
Which equals 80 feet/second. This may surprise you since the average speed on the interval [0,3] was 48 feet/second, yet on the interval [2,3] it’s 80 feet/second. Remember that our rock is accelerating as it falls, so the last second of the [0,3] interval will have a greater speed than the entire interval.
80 feet/second is an alright estimate, but we could do better by analyzing a smaller interval. How about the interval [2, 2.1]?
Finishing out this calculation we get 65.6 feet/second. This is an even better estimate. But why stop there? We could do an even smaller interval to find an even better estimate. How about [2, 2.01]?
Now we get 64.16 feet/second. I bet you know what I’m going to say next…why don’t we do an even smaller interval, like [2, 2.001]?
This time we get an answer of 64.016 feet/second, which is actually pretty close to our last interval which resulted in 64.16 feet/second. Interesting… let’s try one more teeny tiny interval and see what happens. This time let’s try [2, 2.0001]?
This one has an answer of 64.0016 feet/second, barely any different than our last estimate of 64.016 feet/second. Since it seems that we’ve finally stumbled upon small enough intervals where the average speed has only negligible differences, we now have enough info to truly make an educated guess.
What would you guess the instantaneous speed at 2 seconds is?
The smaller we make the interval beginning at 2 seconds, the closer our average speed gets to 64 feet/second. Therefore we can guess that the instantaneous speed at 2 seconds is 64 feet/second.
Congratulations, You Found the Limit!
This idea that we can use smaller and smaller intervals to find the speed of a falling object at a specific point in time leads us directly to the concept of limits.
In this case, we are saying that 64 feet/second is the limiting value for the speed of the rock at 2 seconds. Which is why we call it the limit. It is the value that our function is nearing in on.
Limits are truly the backbone of Calculus. It’s the concept that both separated Calculus from the other branches of mathematics and laid the foundation for derivatives and integrals. When you first start your study of limits, you’ll begin like we did today. You’ll be asked to make tables, where you’re intervals, get smaller and smaller until you can determine what value your function is going to, but as you progress you’ll leave behind the tedious task of table-making for more efficient methods.
The first shortcut you’ll learn is the algebraic approach.
Finding the Limit Algebraically
Instead of making smaller and smaller intervals manually, we could assign a variable to represent a really, really, really tiny interval. We will denote this teensy, tiny interval as h.
We can add h onto the time we wanted to calculate the instantaneous change above to create a really small interval. In other words, we’re going to evaluate the interval [2, 2 + h] where h represents a really, really, really, really small value.
This is where the Algebra comes in. Now we have to FOIL out the binomial and collect like terms. Once you do all that you’ll get:
Since every term left has an h, we can reduce everything by h leaving:
Since we were able to remove the h from the denominator, we can do something really cool now. We can make h=0, which makes it the smallest possible interval and gives us the instantaneous speed at 2 seconds.
(Note: we couldn’t do this until after we canceled out h in the denominator because we would have had a divide by zero error, and wouldn’t have been able to complete the calculation.)
Plugging in zero for h we get:
Our instantaneous speed at 2 seconds is exactly 64 feet/second!
That’ s a lot to take in for one lesson! The good news is that through this one application we’ve learned all about the concept of limits AND we have come dangerously close to learning about the derivative. Take some time understanding this lesson, and when you get to learn about derivatives you’ll see a lot of similarities!
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Thanks for joining me!
❤ Brett
