Essential Knowledge 2.1A3

provided this limit exists.

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I guess the point of this Essential Knowledge is just to know that the limit of the difference quotient we did in EK2.1A2 is called a "derivative". Since we wanted to know the velocity (rate of change of position) at an exact moment in time (instant), derivatives are also known as "instantaneous rates of change".

We'll do the question again, but with a twist. Instead of solving for just one particular time, we'll create an equation that will give us the speed at any time.

Non-Constant Speed

If I drop a ball off of a 1000 foot tall building, how fast will the ball be traveling at any point in time? The equation for a free falling object released from 1000 feet is:

y = -16t² + 1000

This time, let's solve using the other form of the difference quotient. See EK2.1A1 for a demonstration of why both difference quotients are the same thing.

Let's make this formula apply to our example. The variable in our example is time, so we'll let x = t. Remember h is the change in x, so that means h = Δt.

This equation says, "For a given time t, what is the change in position over change in time if the interval of time is very small?" Since change in position over change in time is the definition of "velocity", and a very small interval of time is also known as an "instant", this equation will still give us "instantaneous velocity". So you can see we get the same result with either difference quotient.

The graph below shows the meaning of Δt, and why we would want it to become zero.

Let's substitute.

This means that to know how fast the ball is falling at any given time, simply use the equation v = -32t. Does this agree with what we found earlier? For t = 2, v = -32(2) = -64, which is exactly what we got in the previous Essential Knowledge.

Now that we have an equation, we could easily find all of the other velocities too, and add them to the table. This is the power of calculus.

Let's solve this with the other difference quotient formula

For this, a would be t₁, which we are calling t so that we will get an equation. x is t₂. It really doesn't matter which one is t₁ and t₂, in some ways it makes more sense to say x = t₁, that way x = t since x usually implies the main variable.

solving

As you can see, both formulas are capable of giving us an equation for the derivative of a function. An equation is extremely useful because you can quickly substitute values to find rates at different points.