Tangent Lines and Slope Predictors in Edwards & Penney's Calculus

presentation of Edwards & Penney's work, by Corry Shores
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Edwards & Penney's Calculus is an incredibly-impressive, comprehensive, and understandable book. I highly recommend it.

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[I your author am not a mathematician; I am merely an admirer of Edwards & Penney's wonderful calculus book. Please consult the text or other references to be certain about anything in the summary below. I mean this emphatically.]

Summary of

Edwards & Penney
Calculus

Chapter 2. Prelude to Calculus
Section 2.1. Tangent Lines and Slope Predictors

Recall the tangent problem. Consider circle O with point P on its circumference, making radius OP.

[Thanks Edwards & Penney]

The line tangent to point P is "the straight line through P that is perpendicular to the radius (OP)."

When we have any general graph expressing y = f(x), we do not necessarily have some radius that will allow us to easily find a tangent. However, "the line tangent to the graph at the point P should be the straight line through P that has -- in some sense -- the same direction at P as the curve itself." (p.54a)

A line's slope tells us its direction. So to determine the line tangent to a curve, we will find a "slope-prediction formula" that will tell us the tangent's slope.

Example
"Determine the slope of the line L tangent to the parabola y = x² at the point P(a, a²)." (p.54b)

Below we see the graph for y = x²

[Thanks Edwards & Penney]

We can judge that the curve seems to be wanting to go in the direction of L at point P. We need now determine the slope of L.

Because we only know one point of line L [point P(a, a²)], we cannot calculate the slope. Instead we will begin with another line whose slope we are able to determine. The graph below displays secant line K. It passes through point P and as well through a nearby point Q(b, b²) along parabola y = x².

Notice here we have point P. It is at x-coordinate a. Point Q however is further down the x axis. It is at x-coordinate b. So between points a and b on the x axis, there is a change or increment of x, or delta-x, marked, Δx. We will call it h:

h = Δx = b - a

So recall the coordinates of Q.

Q(b, b²)

The first b is on the x-axis. Because it is found by adding to a the change in x, it is defined as

b = a + h

The b² is found on the y axis. It is found through the function y = x². So we define it as

b² = (a + h)²

because (a + h) is how we get the x value for the function y = x² at point Q.

And because the y coordinates are found through the function y = x², the change in y between P and Q is

Δy = b² - a² = (a + h)² - a²

The slope of a line we call m. And the slope is the change in y over the change in x.

We know both the change in y and the change in x for secant line K, so we can write the formulation.

In the numerator, we multiply (a + h) by itself to get (a² + 2ah + h²). We then subtract out the a's in the denominator to leave just h.

We then subtract out the two a²'s in the numerator to get 2ah + h².


Then finally we factor out the h from the numerator to get h(2a + h).


We then cancel the h's from the numerator and the denominator, and thus the slope of secant K is

m_PQ = 2a + h

Now consider if we move point Q toward point P along the curve, which by the way is the same as h (change in x) approaching zero.

Line K will continue to pas through points P and Q, pivoting around point P. As h approaches zero, secant line K moves closer to overlapping tangent line L. See this motion in the animation below.

(Thanks Dr. Siddique / faculty.uncfsu.edu)

We want to define tangent line L as "the limiting position of the secant line K." (p.55d) Then:

"As h approaches zero,
Q approaches P, and so
K approaches L; meanwhile,
the slope of K approaches the slope of L"

What we want to know is, as h approaches zero, what value is slope m_PQ = 2a + h approaching? So we are looking for the "limiting value" of 2a + h.
Here lim stands for "limit", and "h → 0" stands for "h approaches zero". What the above formulation asks is "What is the limit of 2a + h as h approaches zero?" (p.55b)

We first consider what if a were either 1 or -2.


We would see that if a were 1, then 2 + h tends toward 2 as h tends toward zero. And if a were -2, then 2 + h tends toward 2-4 as h tends toward zero.


So as we see for the values for slope 2a + h,

we may say, more generally, that

Thus the "slope m = m(a) of the line tangent to the parabola y = x² at the point (a, a²) is given by

m = 2a

." (p.56d)

The above formula is the "slope predictor" for tangents to parabola y = x². "Once we know the slope ofthe line tangent to the curve at a given point of the curve, we can then use the point-slope formula to write an equation of this tangent line." (p.56d)




Text summary and images from:
Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, pp. 54-56.

Animation credits:

Geometrical Derivative animation:
http://faculty.uncfsu.edu/msiddiqu/Maple_Animations.htm
http://faculty.uncfsu.edu/msiddiqu/images/images/Gif_Folder/Definition%20of%20Derivative18.gif
Thanks Dr. Siddique of Fayetteville State University

Lagrange in the History of the Calculus

presentation of Edwards & Penney's work, by Corry Shores
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Joseph Louis Lagrange was skeptical of the infinitely small; for him, the accuracy of calculus results from a “compensation of errors.” And yet, he rejected the limit concept on account of its poor metaphysical grounding. The tangent was not a limit, for Lagrange, because after becoming the tangent, the secant could very well continue to the other side of the point and become a secant again. Nor did he accept the method of fluxions, because it made use of “the irrelevant notion of motion.” He rejected Euler’s presentation of dx and dy as 0, because Lagrange felt that we do not have a clear and precise notion of the ratio of two terms which become zero. As a result, Lagrange sought out a simple algebraic method that was free from these objections.

Lagrange found his means in the Taylor Series.

The series f (x + h) = f (x) + f ' (x)h + f '' (x)h^2/2! had been known at least from the time of Taylor, whose name it bears. In this series, the coefficients of the powers of h involve the ratios of differentials, or of fluxions. However, the series can be derived without reference to these notions. What would be more natural than to define differentials and fluxions in terms of the coefficients of such a series? This procedure would (only on the surface, as we know now) obviate the necessity of introducing either limits or infinitesimals into the work, and the calculus would thus be reduced to simple algebraic operations. (252c)

Taylor thought that such an approach would free calculus from "all metaphysics and of any theory of infinitely small or vanishing quantities" (Boyer quoting Lagrange 252d). Although his method was not completely satisfactory, it had the advantage that it did not make use of ideas from geometry, mechanics, or philosophy (253b).

And yet, Lagrange was later criticized for "giving up, in favor of mathematical formalism, the 'generative' concept which has frequently been felt to be the basis of the methods of fluxions and differentials" (253bc).

But Lagrange continued seeking a formalization of the notion of limit based on Euler's function concept.

Incidentally, in so doing he focused attention for almost the first time upon the quantity which is now the central conception in the calculus -- that of the derived function, or the derivative, or the differential coefficient. Lagrange, in this connection gave not only the name from which the word derivative was adopted, but also the notation f 'x, modifications of which are still conveniently used. (253d)

Newton did not interpret the ratio of infinitesimals as such a single number or quantity (derivative), for he considered it more as a ratio of increments or fluxions (254a).

Similarly, Leibniz did not consider the ratio of infinitesimals as a single number, but instead as a quotient of "inassignables" (254b).

Lagrange's method was first to properly make use of the notion of derivative as "merely a single coefficient of a term in an infinite series" that is also "completely divested of any idea of ratio, or limiting equality" (254c).

Boyer, Carl B. The History of the Calculus and its Conceptual Development. New York: Dover Publications, 1949.