MAT 21A Lecture : MAT 21A – Lecture 1 – Rates of Change
MAT 21A – Lecture 1 – Rates of Change
• A topic like Rates of Change is seen in a daily life example such as: “I drive east
for 2 hours and end up 108 miles away.” We can conclude that the average
speed was 54 miles per hour (or 108 miles per 2 hours).
• A mathematical way of expressing the average speed is:
Average rate of change =
=
.
• However, there is also “instantaneous rate of change.” Using the above example:
At one moment you may be going 60 mph at a given moment, but 10 seconds
later, you may be going at 64 mph.
• Example: The population in the U.S. in 1980 was 226 million. Then population
grew to 248 million by 1990. Calculate the average rate with respect to time.
=
=
= 2.2 million people/year.
• Referring to the previous scenario about growth of U.S. population, an example
of an instantaneous rate of change is: “At noon on January 11, 1987, the U.S.
population was changing at a rate of 1.8 million people per year.” This is because
it is referring to a specific time or moment, which is why it is instantaneous.
• Example: What if you want to know your car’s speed at exactly 4:03:00 PM? The
basic idea is to make an estimate of the speed at an instant by taking the average
speed over a very short time interval.
Using
, you could pick time interval to be from 4:03 to 4:05.
Then pick a time closer to 4:03:00 such as 4:03:01. The shorter the time
interval, the better your estimate of the speed at 4:03:00 is.
• Problem: Suppose your location at time, t, minutes after 4 PM is equal to 1 + t2 -
t miles east of home. Find both the average and instantenous speeds between
4:03 PM and 4:05 PM.
Average speed =
=
=
=
=
7 miles/minute.
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