More on the Slope of a Function

In this post I describe two computation techniques that use the slope of a function in powerful ways. These techniques are just the beginning of deeper methods used by mathematicians (and the calculators they program) to compute numerical values.

Let's approximate the cube root of 9. The high school algebra approach would be to find a cube close to 9 and use that to provide a first estimate.  Since 8 = 2^3, we know that the cube root of 8 is 2. Since the cube root function rises as x rises, we know that the cube root of 9 must be slightly more than 2. How much more?

Method #1 for estimating the cube root of 9

If we know the slope of the curve y = x^(1/3) at the point (8,2) then we can guess the change in the y value as x moves from 8 to 9.  It so happens -- take my word for it! -- that the slope of cube root curve at (8,2) is 1/12.  So as x takes one step to the right, going from 8 to 9, y must rise by 1/12, going from 2 to 2+1/12 = 25/12.

So 25/12 = 2.0833333... is a good approximation for the cube root of 9.

Method #2 for estimating the cube root of 9

We can approximate the cube root of 9 in another manner.  Consider the function g(x) = x^3-9.  The graph of this function crosses the x-axis at the cube root of 9, that is, the cube root of 9 is a zero of g(x).  We can approximate this function by finding a good first guess at the zero of g(x) and then using function slope to improve our guess.

Again, the fact that 2^3=8 is a nice fact to use.  This means that g(2) = 2^3-9 = -1 and so (2,-1) is a point on the curve y = x^3-9.  This is a nice point because not only are the coordinates integers but the point is already close to the x-axis and it is the x-intercept that is our goal.

The curve y = x^3-9 has slope 12 at the point (2,-1).  (Again, you must, at this time, take my word for this!)  The tangent line (linearization) of x^3-9 at (2,-1) is, by the point-slope formula,

y+1 = 12(x–2).

Instead of finding the x-intercept of y = x^3-9 we replace that curve by the line y+1=12(x-2) and find the x-intercept of this line.

This is easy to do.  Set y=0 and solve for x.  We get x = 2+1/12=25/12!

Once again, 25/12 is a good approximation for the cube root of 9.

Getting greater precision

We can improve on these approximations using generalizations of these two techniques. (This is what our calculators do.) The first method explicitly used the slope of the original cube root function.  The slope of a function at a point is the derivative of the function at that x-value.  If the function value at x=8 and the derivative at x=8 give us important information, then so will the "second derivative" and "third derivative" and so on ... and the representation of a function by a sequence of derivatives leads to the concept of Taylor series.  This is a topic for a second semester calculus class. (For those curiousabout Taylor series, here, from WolframAlpha, are the first few terms of the Taylor series about x=8 for the cube root function:

The second method described above involved a "first guess" for a zero, guessing x=2 and getting the point (2,-1).  That guess and the associated linearization led us to x=25/12 and if we were to look at 

g(25/12) = (25/12)^3-9 = 73/1728 

we would discover that the y-value is not zero but very close to zero.  Another point on the curve is

(25/12, g(25/12)) = (25/12, 73/1728)

We could then linearize the curve at that new point and continue!  This leads us to Newton's method, to be developed towards the end of this class.

But before we explore any of the paths in the wilderness of calculus, we need to learn how to compute the derivative, that is, the function slope.  Tomorrow we will begin this by looking at the ARC and IRC.

For those in my MATH 1420 class this semester, we are now into the second day of the course and already there have been more than a dozen posts on our class discussion board (on Blackboard)! Excellent!

If you have not yet looked at our course discussion board, I offer the following picture that came across my Facebook feed just this morning!
 :-)