Essential Knowledge 2.1A2

, provided that the limit exists. These are common forms of the definition of the derivative and are denoted f '(a).

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Non-Constant Speed

Let's take a look at this situation again, this time bringing in the calculus. I'll restate the question.

If I drop a ball off of a 1000 foot tall building, how fast will the ball be traveling at any point in time? For example, how fast will the ball be traveling after falling for one second? Two seconds? Five seconds? Any number of seconds?

So you do an experiment. You can't measure speed, but you can measure position. You buy camera and an extremely long tape measure. You extend the tape measure from the top of the building to the ground, then you film the ball traveling down along the tape measure. You make a table of the distance the ball has traveled after each second, and the results are as follows. Remember the building is 1000 feet tall.

In the previous Essential Knowledge, we estimated the velocity at t = 2 by using the difference quotient. Let's take a look at the first difference quotient.

Remember, you can think of x as x₂ and a as x₁. Then the equation would look like this:

You may have noticed something interesting about the difference quotient formula. Since "f(x) =" and "y =" are basically the same thing, we could make the first difference quotient look like this:

Look familiar? I sure hope so. This is the equation for the slope of a line.

This means we can look at the slope of a line connecting point 1 and point 2, and that slope will be the same value as the answer we get from the difference quotient.

Let's try it out. We used t = 2 and t = 3 for one estimate, and t = 2 and t = 1 for another estimate. If we graph those lines, we see the following

Sure enough, they have slopes -48 and -80, which are the same exact values we got when we used the difference quotient.

Now, we want the speed of the ball 2 seconds after being let go. We get different answers using these other lines because we bring in times other than 2 seconds. What we really want is a line that only touches the line at t = 2. This is called a tangent line. It would look like this:

This would require that t₁ = 2 and t₂ = 2. Unfortunately, when we try to use the difference quotient, here's what happens.

undefined.

(Zero over zero... that should be a pretty big hint as to what we're going to do next.)

So here's what mathematicians did. They said, fine, we can't use t = 2 twice. So let's use t = 2 and something extremely close to t = 2.

Try using the GeoGebra applet to see what this looks like. Point 1 is locked in at t = 2. We can move point around to pick our second time. If we move it really close to point 1, we would get a really good estimate for the velocity. But unfortunately, if you put it directly at t = 2, the line will disappear, because you really do need 2 unique points to make a line (to do this, you'll have to attempt to drag point 2 onto the point where it says "Drag Mouse Here").

As you do this, you'll notice that the closer you can bring the points without being in the same spot, the slope gets closer and closer to -64.

We can also accomplish this using a table.

The table also shows that as t₂ gets closer to t₁ = 2, the difference quotient approaches -64.

Time to make things official.

Something as close as possible without actually being t = 2. Tables of values near t = 2 without actually being 2. Do these types of sentences sound familiar? These are the exact types of sentences we used to describe a limit!

So, we'll use a = 2 and x will be the limit as x→2. This means we'll take the difference quotient and add a limit in front

If we do this for the question at hand, we would change x to t, since our equation uses time, and let a = 2, since that's the time we're solving for.

and, surprise surprise, we get 0/0 :(

But limits were made to overcome this. Let's do some factoring.

Wow we did it. The graph, the table, and the official limit method all lead us to a velocity of -64 when the ball has been falling for 2 seconds.

You might be thinking, "that took a really long time". Things will get faster. We spent a lot of time thinking about the equation, experimenting with graphs, and looking at tables. This limit method also has many shortcuts. Once you understand the questions and learn the shortcuts, I promise you'll be able to answer questions like these in under 10 seconds.