### Video Transcript

By finding the average of the right and left approximations for the derivative, use the function table to estimate π prime of one.

The question wants us to estimate the first derivative of π evaluated at one. And it wants us to do this by finding values for the right and left approximations of the derivative by using the function table given.

To start, we need to recall what the right and left approximations of the derivative of a function π are. Weβll start with the left approximation. If π is less than π, then we can approximate the first derivative of π evaluated at π by first finding π evaluated at π, subtracting π evaluated at π, and then dividing this by π minus π. And the reason this is an approximation for the first derivative of π evaluated at π is that this is the slope of the line connecting the point π, π of π to the point π, π of π.

So, as we take values of π closer and closer to the value of π, weβre getting closer and closer to the slope of our curve. We can do the same for the right approximation. This time, weβll take some value of π which is bigger than π. And again, we can approximate the first derivative of π evaluated at π by finding the slope of the line connecting the point π, π of π to the point π, π of π. And this slope is equal to π evaluated at π minus π evaluated at π divided by π minus π.

And as we take our values of π closer and closer to the value of π, the slope of this line will get closer and closer to the slope of the tangent line when π₯ is equal to π. The question wants us to estimate the first derivative of π when π₯ is equal to one. So, weβll set our value of π equal to one. And we can find the column in our table for π₯ is equal to one.

Letβs start by finding a suitable value for π in our left approximation. Remember, we want π to be less than π, but we also want to choose π as close as possible to π. In our table, we can see the closest value to one, which is smaller than one, is negative one. We can do the same to find a suitable value for π in our right approximation. We want π is bigger than π, but we also want to pick π as close as possible to π. And we can see in our table the closest value of π₯ to one, which is bigger than one, is when π₯ is equal to three.

We now have all the information we need. Remember, the question wants us to find the average of the right and left approximations of the derivative. So, π prime evaluated at one is approximately equal to our left approximation plus our right approximation all divided by two. Using π is equal to one and π is equal to negative one, we got our left approximation is π evaluated at one minus π evaluated at negative one divided by one minus negative one.

And by using π is equal to three and π is equal to one, we get the right approximation of our derivative is π evaluated at three minus π evaluated at one divided by three minus one. And we want to take the average of these two values, so we add them together and divide by two. From the table, we see that π evaluated at one is negative 76. Similarly, from the table, we see that π evaluated at negative one is negative 84. Finally, the table shows us that π evaluated at three is negative 64.

Weβre now ready to evaluate this expression. First, we get our left approximation, negative 76 minus negative 84 divided by one minus negative one is equal to four. Then, we add our right approximation, negative 64 minus negative 76 divided by three minus one, which we can calculate to give us six. And then, we take the average of these two values. This gives us 10 divided by two, which is equal to five. Therefore, by finding the average of the right and left approximations for the function π of π₯ given to us in the table, we were able to show that π prime of one is approximately equal to five.