# CLEAR Calculus

## Lab 1: Rate of change - part 1

Chort and Frey ran a marathon (26.2 miles). Chort ran at a perfectly uniform pace of eight-minutes-per-mile. Frey took exactly eight minutes and one second to complete each one-mile interval. This refers to all one-mile intervals, including, for example, the interval from 5.63 miles to 6.63 miles. Nevertheless, Frey finished ahead of Chort. Explain how.

We clearly need to understand rate of change more deeply!

Below is a graph of various data for a rocket launched straight up into space. The red line is velocity $$V$$ (in feet per second) as indicated by the scale on the left hand side as a function of time elapsed since launch $$T$$ (in minutes:seconds) as indicated by the scale along the bottom of the graph. Only the highlighted points are accurate; the straight lines filling in between are not.

Lab Preparation: This lab is about the meaning of various types of speed. Write your answers to the following questions and be prepared to share them with the rest of the class at the beginning of lab.

1. Pick a point on the Velocity graph (red circles) and explain its meaning.
2. Pick two points on the Altitude graph (green triangles) and compute the average speed between the two. Explain the meaning of your answer.

Lab: Work through the following questions with your group.

1. At 2 minutes and 5 seconds, the Velocity graph (red circles) has a point at 4600 ft/s. Two people off the street were stopped and asked to explain what it means to say that “the instantaneous speed of the rocket is 4600 ft/s” and gave the following responses:

Hannibal: The rocket’s speedometer is pointing at the 4600.

Kylo: The rocket travels 4600 feet every second.

Choose one of these responses and write a sentence or two describing why it does not provide a good explanation for the meaning of the statement that “the rocket is traveling at 4600 ft/s.”

2. We need to back up and talk about constant speed. The top speed of the rocket is 4600 ft/s. Two people off the street were stopped and asked what it means to say that “the rocket is traveling at a constant speed of 4600 ft/s” and gave the following responses:

Verbal: The instantaneous speed is a constant 4600 ft/s (not changing). There is no acceleration.

Sméagol: The rocket travels 4600 feet every second.

1. Choose one of these responses and write a sentence or two describing why it does not provide a good explanation for the meaning of the statement that “the rocket is traveling at a constant speed of 4600 ft/s.”
2. If the rocket is traveling at a constant speed and you fix a duration of time, what can you say about the distance traveled during any time interval of that duration? What if you triple that duration of time? What if you cut the duration in half?
3. If the rocket is traveling at a constant speed of 4600 ft/s, how far will it travel in 1 second? 3 seconds? ½ second?

3. We will use “delta-notation” to indicate differences in quantities. Specifically:
A time interval starting at time $$t_1$$ and ending at time $$t_2$$, will have a duration of
$\Delta t=t_2-t_1.$
A distance starting at height $$h_1$$ and ending at height $$h_2$$, will have a length of
$\Delta h=h_2-h_1.$
1. Write a formula relating $$\Delta h$$ and $$\Delta t$$ for a rocket traveling at a constant speed of 4600 ft/s, with height represented by h (in ft.) and time since launch represented t (in seconds).
2. The graph of distance vs. time for constant speed is a straight line. Why does the slope of that line correspond to the speed?
3. What is wrong with connecting the dots in the height vs. time graph (green line) with a straight line? What should the shape of that graph be? Why?

4. The entire graph covers the first 130 seconds of the rocket launch. We all know how to compute average speed over this time interval:
Δht = 290,000/125 = 2320 ft/s.
Two people off the street were stopped and asked to explain what it means to say that “the average speed of the rocket is 2320 ft/s” and gave the following responses:

Joffrey: The average of all the rocket’s speeds is 2320 ft/s.

Iago: The rocket travels 2320 ft/s most of the time.

1. Choose one of these responses and write a sentence or two describing why it does not provide a good explanation for the meaning of the statement that  “the average speed of the rocket is 2320 ft/s.”
2. Imagine that another rocket WAS traveling at a constant speed of 2320 ft/s. What would happen if the two rockets were launched at the same time?
3. Now give a correct explanation of the meaning of an average speed of 2320 ft/s as “the constant speed required to…

5. Draw a graph of distance, d, vs. time, t, for a car that starts from rest at one traffic light and stops at another traffic light.
1. What do the steeper parts of the graph mean? Explain using Δd and Δt.
2. What is the “inflection point” in the graph? What does this mean in the context of the car? Explain using Δd and Δt.