2013-10-17 Finding Instantaneous Rate of Change Exactly

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NAME DATE BAND

FINDING IROCEXACTLY

CALCULUS |PACKERCOLLEGIATE INSTITUTE

Section 1: The problem with exactitude, dude

Until this point, we could approximate the instantaneous rate of change. We could either pick two points

pretty close to each other and calculate the average rate of change between these two points or we could

draw a tangent line as best as our human hands allowed us and we could try to approximate the slope of

that line.

But calculus you will see is aboutprecision. And honestly, its a little pathetic if all we could do is

approximate the instantaneous rate of changebut never find it exactly. So today were going to be putting

the pieces together.

Lets start with2( )f x x . Our goal is to find the IRoCexactly at 1x .

We have two points we want to find the slope between.

One point is: _________________

The second point is a bit trickier to find. The problem is that if we pick an x-value of 1.000001, we will still only

get an approximate answer for the IRoC. Because although we are bringing the second point close to the first

point, we still have a slight distance between them. And if we pick an x-value of 1.000000000000000001, that

will still be approximate, for the same reason. AAAAAAAAAAAAARGH.

We want the second point to be infinitely close to the first point.

Heres where the really extraordinary insight comes into play And even though its such a simple thing, it

truly revolutionized everything. It may not seem like much, but watch this beautifulness unfold.

Section 2: The Problematic Second Point

Were going to define the second point to be: ___________________

What does h represent?


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Check if you understand whats going on

In the following pictures, the second point is plotted. For each, what is the value of h ?

The second point is:

(_______, _______) and

h


(_______, _______) and

h


(_______, _______) and

h


(_______, _______) and

h

Right now in the four graphs above the second point is pretty far away from the point we care about! Sad!

So what would we have to do to h to bring the second point closer and closer to the first point?

Okay were going to file that away in our brains for now.

Can h be zero? Why or why not?

Section 3: Average Rate of Change Again?

Lets start with what we know. We know that if you calculated the average rate of change of a function

between two points close together, wed get a good approximation for the instantaneous rate of change. So

lets start by calculating the average rate of change between our two points:

The ARoC between (_______, _______) and (_______,_______) is:


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Simplify the ARoC:

Now we are going to remember our insight. Right now we have the ARoC between two points. But we need

the two points to be infinitely close to each other. Go back to Section 2 and look at our conclusion about h .

Also recognize that we said that h is not zero.

Notes Marginalia


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Now we have the IRoC exactly at x=1!

Section 4: Practice!

That was a lot of work because we took everything slowly. Do me a favor and try to find the IRoC of2( )f x x

at 5x .

Two Points: _________________ and ____________________

The ARoC is:

To get to the IRoC you have to:

Thus, the IRoC is (do work and write answer below):

2013-10-17 Finding Instantaneous Rate of Change Exactly

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