MS120 Definition of Derivative
We are going to consider the derivative in one of its fundamental roles: the slope of a tangent line.
Since I can’t do graphs easily here, I will refer to the figures on pages 228 and 229 of the book.
For the function fin the figure in the shaded box on p. 229, I want to figure out how to write the equation of the
tangent line where x=a. In order to write the equation of a line, I need 2 facts: its slope and a point that is on the
line. The point is known... It is (a,f (a)), so I need to find the slope of the imaginary tangent line that is shown as a
blue line in the figure. So the problem becomes: How to find the slope of a line. Usually to find the slope of a line, I
need 2 points; but how can we draw the tangent accurately enough to be sure that it is the tangent? (If I just guess at
it, and pick 2 points, I will not be very accurate.... For example, if someone graphed a line such as y=3x−1, plot-
ting the points (−5,−16) and (5,14), could you look at the x-intercept and tell for sure that it was 1
3? Probably not!)
So, here is the procedure: We move a “small” distance haway from a. This gives us an x-value of a+h. The point
on fthat corresponds to this is (a+h, f (a+h)). (Remember to find the y-value of an ordered pair, you substitute
the given x-value. The y-value is then the same as “fof that x-value.”) (See the figure). Now a line is drawn
through our original point and the new point. This is the red line in the figure and is called a secant line because it
intersects the graph in 2 points. Now I can easily find the slope of this secant line:
Slope of secant line = f(a+h)−f(a)
(a+h)−a(Difference of y-values divided by difference of x-values.)
Notice that this reduces to: f(a+h)−f(a)
h
Now we will use an intuitive idea of limits to see what happens:
Picture making hsmaller.... This will move the point (a+h, f (a+h)) closer to our original point, but we will still
have 2 points so that we can calculate slope. The idea of a few steps of this procedure are shown in the top figure
on page 228. As hgets smaller and smaller, the secant line is a better and better approximation of the tangent line.
The points are not labeled in this figure, because doing so would clutter up the picture so much that we wouldn’t be
able to let our intuition follow the progression of secant lines “approaching” the tangent line.
We find the slope of the tangent line at x=aby:
lim
h→0
f(a+h)−f(a)
h
As we said in class, this value of hcan be positive or negative, so it would make sense that we should get the
same limit whether we approached 0 from the left or from the right in order to call this “the limit”. You saw this
demonstrated in problems 1 thru 4 of 3.5 where the sequences of numbers were approaching the same limit whether
the values of hwere positive or negative, as long as we let hget “smaller and smaller.” the figure mentioned on
page 229 only shows a picture of happroaching 0 from the right, that is his positive and “small.” To picture the
limit as happroaches 0 from the left, that is his negative and “small”, we would have our point (a+h, f(a+h)) on
the left side of (a, f (a)), and follow the same procedure. It is beyond the scope of this course to “prove” that these
2 limits exist and are equal for continuous functions. What is presented here is an intutive idea of the process involved.
