A ball is thrown into the air
by a baby alien on a planet in the system of
Alpha Centauri with a velocity of 31 feet per second.
Its height in feet after t seconds
is given by y equals 31 t minus 15 t squared.
We're asked to find the average velocity
for the time period beginning when t equals one
and lasting a variety of time intervals.
But notice how the time intervals are getting
shorter and shorter, or smaller and smaller.
And then we're asked to use these average velocities
to estimate the instantaneous velocity at t equals one.
As the time intervals get smaller and
smaller, the average velocities will
approach the instantaneous velocity.
Graphically, the average velocity is
a slope of a secant line, the instantaneous
velocity is a slope of a tangent line.
Before we set this up, let's look at this graphically.
The function y of t is graphed in red.
The purple line is tangent to the curve at t equals one,
which means a slope of the tangent line
is the instantaneous velocity at t equals one.
The green line is a secant line, and the slope
of the secant line is equal to the average velocity
over this particular time interval.
As a time interval gets smaller and smaller,
or as this point approaches the point of tangency,
the slope of the secant line, or average velocity,
approaches the slope of the tangent line,
or the instantaneous velocity, which we see here.
So going back to our work, the average velocity
over a particular time interval is equal to
the change of y divided by the change of t, which we
can express using function notation, as we see here.
This formula should look familiar.
In algebra, it's often expressed as the quantity
y sub two minus y sub one divided by
the quantity x sub two minus x sub one.
And for all these intervals, we're always
using one point where t equals one.
Let's begin by determining what this point would be.
So when t equals one, the point is one comma y of one.
To determine y of one, we substitute one into
the given function, which gives us 31 times one
minus 15 times one squared, which equals 16.
So for all the average velocities, we will always be using
the point one comma 16 as the first point,
but it's also the point of tangency.
The first time interval has a length of .01 seconds,
which means the second point on the secant line
is going to be 1.01 comma y of 1.01.
We obtain 1.01 by adding .01 to one,
which means the average velocity over this time interval
is y of 1.01 minus y of one divided
by the quantity 1.01 minus one.
To determine the function value y of 1.01,
we substitute 1.01 into our function y of t,
which gives us 31 times 1.01 minus 15 times 1.01 squared.
This expression is y of one, which we know is equal to 16.
Let's determine the slope, which is the average
velocity over this time interval, using the calculator.
And we'll show two ways to evaluate this.
One way is to enter the entire
expression on the home screen.
But let's enter this in two parts.
Let's first enter the numerator,
and then we'll divide by the denominator.
In the numerator, we have open parenthesis,
31 times 1.01, close parenthesis, minus
15 times 1.01 squared, close parenthesis,
minus, we know this is y of one, which is just 16,
let's just enter minus 16, Enter.
This is the numerator.
Now we need to divide by the denominator.
So we press the division key, and notice
how it shows answer divided by, this answer
is the previous answer here of .0085.
Then in parentheses, we have 1.01 minus one,
close parenthesis, and Enter.
So the average velocity over the first
time interval is .85 feet per second.
But I also want to show how we can find this average velocity
using function notation, as shown here.
To use function notation, though, we need to
enter our function y of t into the calculator
under y one, but we use x instead of t.
We press y=, and notice how I've already
entered 31 x minus 15 x squared in Y1.
Now we can go back to the home screen and determine
the average velocity using this notation here.
So we'll press second mode to go to the home screen.
To access y sub one, we press vars, right arrow,
Enter, Enter, which brings Y1 back to the home screen.
Then in parentheses, we have 1.01, close parenthesis,
minus y of one, which we know is 16, but I'll go ahead
and enter y of one, so we press vars, right arrow,
Enter, Enter, in parentheses, one, Enter.
And now we divide by again the quantity
1.01 minus one, which of course is just .01,
but I'll enter it in as shown here.
So in parentheses, 1.01 minus one.
And of course we get the same answer, 0.85 feet per second.
So going back to our home screen,
let's record the first average velocity.
We do not enter the units here, though.
And now we need to do the same thing
for the three remaining time intervals.
Notice the next time interval is .005 seconds,
followed by .002 seconds, and then .001 seconds.
Let's go ahead and show the setup of all of these.
For the time interval of .005 seconds, again we know
the first point is always going to be one comma 16,
the second point is going to be 1.005 comma y of 1.005.
Again, the 1.005 comes from taking the starting t value
of one and adding the particular time interval,
which means the average velocity over this time interval
is y of 1.005 minus y of one divided
by the quantity 1.005 minus one.
Again, this is the change of y, or the change in height,
divided by the change in t, or the change in time.
Because there are so many of these to evaluate, we'll be
using the function notation on the calculator to save time.
This time on the calculator,
let's enter all of this in at one time.
To do this, we would need a set of parentheses
around the numerator and the denominator.
So let's go ahead and clear this.
We have open parenthesis, now we need to enter y of 1.005.
So vars, right arrow, Enter, Enter,
and in parentheses, 1.005, minus y of one.
So again, vars, right arrow, Enter, Enter, in parentheses,
one, another close parenthesis for the numerator, and then
we'll divide by the quantity 1.005 minus one, or we could
just divide by .005, but I'll enter the difference.
So in parentheses 1.005 minus one, close parenthesis, Enter.
Notice now the average velocity is .925 feet per second.
So we have 0.925 feet per second.
Of course we know, for the first time interval,
it was 0.85 feet per second.
Now let's find the average velocity
where the time interval is .002 seconds.
Notice how again the first point is one comma 16,
now the second point is 1.002 comma y of 1.002,
which means this is the slope of the secant line,
or the average velocity for this time interval.
So going back to the calculator, another shortcut
to save some time is to press 2nd, Enter,
which brings up the previous entry,
but we can now edit the entry,
which means we can just change 1.005 to 1.002.
So if we scroll to the left, change this to 1.002,
keep scrolling left...
and change this 1.005 to 1.002, press Enter,
and now we have the average velocity over
this time interval, which is .97 feet per second.
The last time interval has a length of .001 seconds,
and therefore the second point is 1.001 comma y of 1.001,
giving us an average velocity of this quotient.
Going back to the calculator, let's press
2nd, Enter, and change the 1.002 to 1.001.
So pressing the left arrow, make the change here,
and make the change here as well, and press Enter.
The last average velocity is .985 feet per second.
From here, we're asked to estimate the
instantaneous velocity at t equals one.
So looking at the average velocities as the time interval
gets smaller and smaller and approaches zero,
it does appear as if the instantaneous velocity
is approaching the value of one, we'll estimate
the instantaneous velocity as one foot per second.
Now I do want to mention in our homework,
if the average velocities don't come out to nice values,
you can round to four decimal places.
I hope you found this helpful.