A rock is shot upward from a volcano on a different planet with a velocity of eighty
meters per second. The height after t seconds is given by the function h of t equals eighty plus
forty-eight t minus zero point five t squared. We want to find the average velocity over the given
time intervals here on the left and then predict the instantaneous velocity after thirty seconds.
The average velocity over the given intervals will be the slope of the secant lines passing through
the two endpoints of each interval, and the instantaneous velocity will be the slope of the
tangent line at t equals thirty. So if we consider the two endpoints for each of these intervals,
notice how the two points are getting closer and closer together, approaching t equals thirty.
So we can use these average velocities to make a prediction about the instantaneous velocity. To
find the average velocities, we can use the slope formula, or this formula here, where
we'll find the change in the height and divide by the change in time. But before we do this,
let's take a look at an animation to get a better idea of how this is going to work.
Looking at this black function here, the slope of this blue tangent line at this blue point would
be the instantaneous rate of change. And if we consider the slope of the red secant line passing
through the blue point and red point, notice as the two points get closer and closer together,
the slope of the secant line is going to approach the slope of the tangent line, as we see here.
So using the slopes of the secant lines, we're going to make a prediction about the
slope of the tangent line. As you can see, as the two points get closer and closer together,
the slope of the secant line will approach the slope of the tangent line.
To find the slope of the secant lines using the endpoints of each of these intervals will take
quite a bit of work, and because of this, I've already set most of it up. Notice how for this
first interval, t sub one is equal to thirty, t sub two is equal to thirty-one. In the second
interval, t sub one is equal to thirty, t sub two is equal to thirty point five.
In the third interval, t sub one is thirty,
t sub two is thirty point one. In the last interval, t sub one is thirty,
t sub two is thirty point zero one. So the average velocity for each of these intervals
will be the change in the height, or the change in the function value, divided by the change in time.
And I do want to show how we can save quite a bit of time by performing these
calculations on a graphing calculator. We first want to enter the height function,
so we'll press y equals, and we'll type in our function. Instead of using t,
we'll use x. So we'll have eighty plus forty-eight x minus zero point five x squared.
Now we'll go back to the home screen to determine these average velocities. To
find the change in height, because our function is stored in y one, we're going to press vars,
right arrow, enter, enter. This brings up y one. So to find h of thirty-one,
we can enter y one of thirty-one and now we'll subtract y one of thirty.
So again, vars, right arrow, enter, enter, in parentheses thirty,
press enter. Here is the change in the height over this first interval,
and we'll divide by the change in time. Thirty-one minus thirty is equal to one, giving
us an average velocity on this first interval of seventeen point five meters per second.
Let's go ahead and do one more of these intervals, then you can try
the last two on your own. So we want y one of thirty point five minus y one of thirty,
and we'll divide this by the change in time, or divide by zero point five.
So we're going to press vars, right arrow, enter, enter, y one of thirty point five this time,
and then we'll subtract y one of thirty. So here's the change in height. We'll
divide by zero point five, which is the change in time, which gives us an average
velocity on this interval of seventeen point seventy-five meters per second.
So notice as the two endpoints of each interval get closer and closer together, approaching t
equals thirty, the average velocity appears to be approaching eighteen meters per second at t
equals thirty, which would be our prediction for the instantaneous velocity at t equals thirty.
I hope you found this example helpful.