This exercise will help familiarize your instructor with your way of approaching and thinking through problems. Most students at this point will have some difficulty with at least some of the problems. Give this your best shot, but don't worry if you occasionally fall short. However do memorize the given definition.
Memorize the following definition and be able to write it down or recite it, word for word, from now on. This definition consists of 18 words (over half of them monosyllables, none over two syllables) and five symbols. Know them all. At any time. Period.
This is not an insurmountable task. It's not even difficult. It just takes a little time, practice and attention.
Applying the definition in the various circumstances you will encounter in this course is challenging. Learning to write it down or recite it is not. You just need to do it. Before the next class.
Now we start to understand why this definition is important. Here's a simple example:
A dog weighs 20 pounds at age 4 months and 35 pounds at age 7 months. How many pounds per month was the dog gaining, on the average, during this time?
You should easily be able to figure out that the dog gained 15 pounds in 3 months so gained an average of 5 pounds per month. This was an average rate of gain. We don't know anything about how the gain was spread over the various months.
Now a very similar question could have been posed for this information.
We could have asked for the average rate of change of the puppy's weight with respect to time, calculated in terms of the above definition.
The key phrase is 'with respect to', which we compare with the language of the definition you will soon know. The quantity that precedes this phrase in the definition is A and the quantity that follows it is B. In the question about the puppy the quantity that precedes the phrase 'with respect to' is weight, and the quantity that follows it is time.
We conclude that in the question about the puppy, the A quantity is weight and the B quantity is time.
So 'change in A' means 'change in weight', while 'change in B' means 'change in time'. So the rate of change, being (change in A) / (change in B), is (change in weight) / (change in time).
More specifically, by interpreting the definition we conclude that the average rate of change of weight with respect to time is just what we would think, (change in weight) / (change in time).
We actually need to be a little more specific with the definition. For now, this would involve more words than you will be required to memorize. It will be enough for now that you understand the idea of an interval, as discussed next:
You should know this, but you don't need to memorize it word-for-word.
Defining the word 'event' gets way too technical for a course at this level, but the general idea is not difficult:
In the case of the puppy, the interval in question begins when the puppy is 4 months old and weighs 20 pounds, and ends when the puppy is 7 months old and weighs 35 pounds. So the initial event of this interval is the puppy weighing 20 pounds at 4 months of age, and the final event is the puppy weighing 35 pounds at 7 months of age.
Our interval therefore contains a time interval of 3 months, and a weight interval of 15 pounds.
When we calculate the average rate of change of weight with respect to time, we naturally calculate
ave rate = (change in weight) / (change in time),
and we have verified that this natural calculation is consistent with our definition of average rate of change. In terms of interval terminology, we are calculating weight interval / time interval.
Let's apply the definition of average rate to another situation. Answer the following, inserting your answer to each question between the **** line and the #$&* line. You should provide your detailed reasoning on each question, along the lines of the reasoning presented in the preceding examples:
If we turn on a tap at 11:24 a.m. and direct it into an empty 5-gallon bucket, and find that the bucket first reaches the full level at 11:28 a.m., at what average rate is water flowing into the bucket with respect to clock time?
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#$&*
Specifically how does the definition of average rate apply to this situation? What is the A quantity? What is the B quantity?
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#$&*
What event begins the interval, and what event ends it?
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#$&*
Answer the following as well:
Water flows through household plumbing at a typical average rate of about 3 gallons per minute. Had the tap been turned on at 11:28, and had water flowed at this rate, when would the bucket have been full?
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And answer the following:
A ball rolls from rest down a 30 cm ramp in 6 'beats'. Starting with the definition of rate, reason out your answers, showing the details of your reasoning:
What is the average rate at which the position of the ball changes during the interval starting with release and ending when the ball passes the end of the ramp?
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#$&*
How fast do you conjecture that the ball is probably rolling when it reaches the end of the ramp?
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#$&*
How quickly would you therefore conclude the velocity of the ball is changing?
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#$&*
What is the average rate at which the velocity of the ball changes during the interval starting with release and ending when the ball passes the end of the ramp?
****
#$&*
At this point of the course there are no formulas, only these concepts and definitions. These definitions lead to formulas, but for right now we aren't going to pollute your sensibilities with formulas that will invite you to bypass understanding.
... calculating change in A given ave rate and change in B ... when would the bucket be full at 3 gal / min
... leading to velocity and acceleration, the former being pretty easy to understand, the second not much so
At this point there are no formulas, only these concepts and definitions. These definitions lead to formulas, but for right now we aren't going to pollute your sensibilities with formulas that will invite you to bypass understanding.