Section 2.1 How do we measure speed?

Imagine a ball being thrown straight up in the air. –When is that ball going the fastest? –When is it going the slowest? –Is there any time where the velocity is 0?

What can we say about the speed of the ball? How does that differ from the velocity? The following table gives the height of the ball at given times So what is the average velocity of the ball over the first 3 seconds? What about between 1 and 2 seconds? t (sec) h (ft)

We have a line between 0 and 3 seconds Its slope is the average rate of change between 0 and 3 seconds Notice that the slope of the line is similar to the slope of the function initially (both are increasing…kind of)

We have a line between 1 and 2 seconds Its slope is the average rate of change between 1 and 2 seconds Notice that the slope of the line is much closer to the slope of the line over that interval

It turns out that the average velocity of an object is –Notice that this is the same as slope for a linear function –It is actually the slope of the secant line between the two points on the graph Now what if we wanted to know how fast the ball was traveling at 1 second? –Take a smaller interval t (sec) h (ft)

In the previous slide we could estimate the velocity at 1 second by finding the average rate of change between 1 and 1.25 seconds –This is what’s known as a forward difference and is represented in function notation by where h = 0.25 –Subbing that in we get

Another option for estimating the velocity at 1 second is averaging the average rates of change between 0.75 and 1 seconds and 1 and 1.25 seconds –This is what’s known as a symmetric difference and is represented in function notation by where h = 0.25 –Subbing that in we get

Both the forward and symmetric difference are ways of estimating the velocity at t = 1 –The symmetric difference requires we have values on both sides Let’s take a look at the forward difference We know this will give us an estimate of the velocity at 1 second How can we get a better estimate? –Let’s try this with the function

In general if we have f(t) be the position at time t then the instantaneous velocity at t = a is defined as –This also gives us the slope of the curve at that point (point a) –In many cases we can solve this limit algebraically Let’s take a look with our above example

Notice that the difference quotient is really just the slope formula as we’ve seen before The limit is necessary because the slope is being measured at a point Let’s find the instantaneous velocity of the following functions at t = 3