Given that 𝑦 equals 𝑓 of 𝑥 is a function for three known values where 𝑓 of one is 1.5, 𝑓 of two is 2.75, and 𝑓 of three is 3.25, estimate 𝑓 prime of two.
So we don’t know the function. But we’ve been given some 𝑥 values with the corresponding values of 𝑓 of 𝑥. Let’s sketch these points on a graph to help us visualize what we’ve got. So we’ve drawn some axes and marked on the points one, 1.5; two, 2.75; and three, 3.25. So our function may look something like this. It could of course have an entirely different shape that still go through these points. However, this would be the most obvious graph. And we want to estimate the value of 𝑓 prime of two, so the value of the derivative, the value of the slope at 𝑥 equals two.
So how do we do this? It’s gonna be hard to guess the slope of this tangent line. So instead, we connect the two points either side. This is called a secant line. And we’ll estimate the slope at 𝑥 equals two by working out the slope of this line instead. We remember that the slope is the changing 𝑦-coordinates over the change in 𝑥-coordinates. So for our points — one, 1.5 and three, 3.25 — the slope of the line joining those points together is a 3.25 minus 1.5 over three minus one which is 1.75 over 2 or 0.875. And we must remember that this is just an estimate of course.