Use the table to estimate 𝑓 prime of six.
The question wants us to use this table to approximate the value of 𝑓 prime of six. And we know that 𝑓 prime of six is the slope of our function when 𝑥 is equal to six. From our table, we can see that when 𝑥 is equal to six, 𝑓 of 𝑥 is equal to three. We can also see from our table we’re given values of 𝑓 of 𝑥 for 𝑥 less than six and values of 𝑓 of 𝑥 for 𝑥 is greater than six. This means we could choose to approximate our slope by using a left derivative or a right derivative.
To approximate the slope of our curve when 𝑥 is equal to six by using a left derivative, we’ll want to find the slope of the line connecting the point on our curve when 𝑥 is equal to six to a point to the left on our curve. And the closer this point is, the more accurate our estimate will be. And the closest value of 𝑥 to six in our table where 𝑥 is less than six is 𝑥 is equal to five. We can also do the same on the right-hand side. We see the closest value of 𝑥 to six where 𝑥 is greater than six in our table is when 𝑥 is equal to eight.
So this means we could approximate 𝑓 prime of six by taking the slope of the line between the point five, two and the point six, three. The slope of this line will be the change in 𝑦 divided by the change in 𝑥. That’s three minus two divided by six minus five. And we can calculate this to give us one. However, we can also approximate 𝑓 prime of six by the slope of the line between the point six, three and the point eight, six. And the slope of this line will be six minus three divided by eight minus six, which we can evaluate to give us three over two.
And these are both fairly reasonable estimates of 𝑓 prime of six. However, we can get a more accurate estimate by taking the average of these two values. We call this the average of the left and right approximations for the derivative. This gives us that 𝑓 prime of six is approximately equal to one plus three over two all divided by two. And if we calculate this, we get five divided by four, which we’ll write as 1.25.
Therefore, by taking an average of the left and right approximations for the derivative of the function 𝑓 of 𝑥 given to us in the table, we were able to show that 𝑓 prime of six was approximately equal to 1.25.