For the given graph, estimate 𝑓
prime of three.
Here, we have been asked to
estimate the derivative of 𝑓 at three. And we’ve been given a graph of 𝑓
of 𝑥. Now, our graphical method for
estimating a derivative is to draw a tangent at the point where we’re trying to find
the derivative and to find the slope of that tangent. Now, if you find the value of three
on our 𝑥-axis and look at its point on our graph of 𝑓 of 𝑥, we can see that a
tangent has already been drawn for this point. And so, what we need to do in order
to estimate the derivative of 𝑓 at three is to find the slope of this tangent. The most accurate way to find the
slope of this tangent is to pick the two points furthest apart on our tangent, which
we can accurately read of our axes.
We can see that we have a point at
five, seven. And we also have another point on
our tangent at one, negative one. Therefore, we can use these two
points in order to find the slope of this tangent. We use the fact that the slope of a
line is equal to the change in 𝑦 over the change in 𝑥. A change in 𝑦 for our tangent is
the difference in the 𝑦-values of the points which we’ve found. So that’s seven minus negative
one. And the change in 𝑥 is the
difference in the 𝑥-values for these same points. So that’s five minus one. Now, let’s remember that it’s
important to put the corresponding points on the same side. So the five and the seven came from
the point five, seven and they both go on the left. Whereas the one and the negative
one came from the point one, negative one, and they both go on the right.
Now, it doesn’t matter which way
around we put these points as long as they’re consistent with one another. For example, this fraction is also
equal to negative one minus seven over one minus five since the negative one and the
one are both on the left and the seven and the five are both on the right. We can simplify this fraction in
order to find the slope of our tangent is equal to eight over four, which is, of
course, equal to two.
Now that we found the slope of our
tangent, we can use this to estimate 𝑓 prime of three since the derivative is the
slope function. And we’ve found the slope of the
tangent at three. This gives us a solution that our
estimate for the derivative of 𝑓 at three is two. Now, the reason why this is an
estimate and not an accurate answer is because we do not know how accurate the
tangent is at the point three.