Introduction

Welcome to the first FiveHive lesson for AP Calculus AB. Calculus is a subject which I hope you may find beauty in as well. In this adventure we will gain new tools to analyze mathematical problems. Right now we are gonna start talking about change: average and instantaneous.

What is a rate of change?

A rate of change can describe how fast some quantity is changing. You deal with these all the time in your life! Your car going 40 miles per hour means its distance is increasing by 40 miles after each hour. Also if you get 5 hours of sleep every day (try to get more!) then your rate is 5 hrs/day, every day that passes you sleep for 5 hours. The quantity that is changing might not be completely relevant to you, here it is total hours of sleep. However, if we talk about someone getting $20/hr to work their job, suddenly the rate and the changing quantity are both important. With every hour that passes, 20 more dollars are being earned by the employee. 

What is an average rate of change? 

Obviously, things in life are not constant. The population of a town might increase due to economic opportunity and then decrease due to a plague. A person might get a raise at their job, and a car especially is not going the same speed the entire way through a trip. 

So we calculate average rates of change. You might have done this in an algebra class, so let this be a refresher.

Example 1:

In the graph above find the average rate of change of dollar value per month on the interval $[0,5]$.

$\text{Rate of Change}=\frac{\Delta y}{\Delta x}=\frac{7-4}{5-0}=\frac{3}{5}=0.60\text{ dollars per month}$

We found the change in the dependent variable (the value of the FHV stock) and divided it by our change in the independent variable (the time) and we got the average rate in which the FHV stock changes over time.

How is this useful?

Right now we know a general picture of the change of the stock. So we can say that in month 6 the value of the stock will be about $7.60. However, what if we only had access to the first 4 months of data?

Example 2: 

Using the same graph find the average rate of change of dollar value per month on the interval $[0,4]$.

$\text{AROC}=\frac{\Delta y}{\Delta x}=\frac{4-4}{4-0}=\frac{0}{4}=0\text{ dollars per month!!}$ 

If we only had this data, we would say at month 5 the stock would also be $4. So what gives?

Average rates of change are average. They include all the little increases and decreases that might not happen in the future. However, if we lower our range we might find that average rates of change can be more predictive of behavior. The average rate for FHV stock on $[3.99,4]$ is $1.99/month which is a lot more accurate! A smaller interval will give us a lot more insight to the behavior at the current time and is the best to base our decisions off of. We can find a good moment to buy or sell compared to an average hiding these spikes or dips.

What is an instantaneous rate of change?

As we get smaller it gets more accurate as seen when we lower our interval. So the best interval we could have would have 0 length, but that would not be possible. Average rates of change involve dividing by change in the independent variable, and if the change is 0, we are dividing by 0. However, an instantaneous rate of change tells us the rate of change at an instant; it is the slope of a tangent line to the function at that point (You will be more familiar with this in Chapter 2). For example if the velocity of an object is $2m/s$ that is the object’s rate of displacement (in meters) at that instant. This can tell us important information at the moment for the compared objects. Instead of averaging everything out we have access to how much a stock price, town population, or car is moving at the moment. For example, you would not give a car a speeding ticket in a neighborhood because its average speed was 35 mph, you judge its speed at that very moment.

How can we use limits?

Later in this chapter we will introduce the idea of a limit so we can learn how to calculate these instantaneous changes and know how fast something is moving without having to estimate and generalize like we would have to do with the stock prices. We will use limits to turn our average rates of change into instantaneous rates. This is the essence of calculus: using limits to understand change and accumulation of dynamic quantities. So to answer the question, yes change can occur at an instant and that is what we will discover throughout this course.

Practice

Now that you have made it through the first ever topic of calculus, it is time to put your learning to the test with some practice questions