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

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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

Preview: What Is Calculus? in Ewards and Penney's Calculus

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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.]

[This section introduces problems that we later discuss much further in detail.]

Summary of

Edwards & Penney
Calculus

Chapter 1. Functions, Graphs, and Models
Section 1.5. Preview: What Is Calculus?

Calculus ("the calculus") is a body of computational techniques. It revolves around two basic geometrical problems, and mathematicians have been dealing with these problems for over 2000 years. Both problems involve "the graph y = f(x) of a given function." (p.45)


Problem 1: The Tangent Problem

One problem calculus tries to solve is finding the 'line tangent' at a given point on a curve y = f(x).


Edwards and Penney then formulate the tangent problem with reference to this graph:


So let's first recall what a function is. Functions describe relationships between variables. The function describes the way that one variable varies with respect to another variable, often formulated y = f(x). So consider the function y = x².

We have two variables, x and y. Both are varying, both increase in value. But the way that the one increases with respect to the other is further describable. As the x value increases, the y value increases to the power of two. So when x has the value of 2, y has the value of 4. And when x is 4, y is 16, and so on. Edwards and Penney offer this definition for functions.

Function:

A real-valued function f defined on a set D of real numbers is a rule that assigns to each number x in D exactly one real number, denoted by f (x).


The set D of all numbers for which f (x) is defined is called the domain (or domain of definition) of the function f. The number f (x), read "f of x," is called the value of the function f at the number (or point) x. The set of all values y = f (x) is called the range of f . That is, the range of f is the set {y: y = f (x) for some x in D). [p.2d]

So recall again the graph Edwards and Penney are using to describe the tangent problem.


Here we see that x varies with respect to y in such a way that the series of their correlated variations makes a waving curve. Now we consider a point P along the continuous variation of x's and y's correlations. a point is described with the x and y coordinate values (x, y). But the way we find y is by finding the value of the function applied to the value of x. So the y value can also be noted as f(x), and thus the coordinates for point P along the function y = f(x) would be P(x, f(x)). In the case of y = x², we might consider the point (2, 4) for example. Thus the tangent problem:

The Tangent Problem
Given a point P(x, f(x)) on the curve y = f(x), how do we calculate the slope of the tangent line at P?

Finding the tangent gives us what is called the derivative of the function f, and later Edwards and Penney explain how we obtain derivatives. This is largely a matter of differential calculus.

Example


We are driving down a straight road. It takes us a certain amount of time (t) to go a certain distance, and the way that the distance increases as time increases can be described with the function y = f(t). If we find the slope at point (t, f(t)), then we find the velocity at time t.


Problem 2: The Area Problem

Consider this graph.



We might want to know what the area is below the curve between a and b.

The Area Problem
If f(x) ≧ 0 for x in the interval [a, b], how do we calculate the area A of the plane region that lies between the curve y = f(x) and the x-axis over the interval [a, b]?


Example

Consider this graph.

Unlike the prior example which correlated the way that distance varied with respect to time, this new graph shows how velocity varies with respect to time, given with the function y = f(t). So f(t) gives us the velocity of the car at time t. This means that the area under the curve within the time interval [a, b] gives us the distance the car travels between time a and time b.

Text summary and images from:
Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, pp.2; 45-47.

Transcendental Equations in Edwards & Penney's Calculus

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Transcendental Equations in Edwards & Penney's Calculus

What do transcendental equations got to do with you?

Because they are all transcendental equations, see the got-to-do entries for trigonometric, exponential, and logarithmic functions.

Brief Summary

The solution to transcendental equations with the form f(x) = g(x) is the intersections of the graphs of the functions.

Points Relative to Deleuze

Again, because they are all transcendental equations, see the got-to-do entries for trigonometric, exponential, and logarithmic functions.

Summary of

Edwards & Penney
Calculus

Chapter 1: Functions, Graphs, and Models
Section 1.4: Transcendental Functions

Subsection 5: Transcendental Equations

We previously examined trigonometric, exponential, and logarithmic functions. These are all types of transcendental functions. Equations that include transcendental functions within them may have infinitely many solutions. Yet they might also have just a finite number of solutions. Edwards and Penney note one approach to dealing with transcendental equations. We might render them as

f(x) = g(x)

"where both the functions f and g are readily graphed." (p.40d) Wherever graphs y = f(x) and y = g(x) intersect are the solutions to the equation.


Example

Consider these graphs


There is a single point where graphs y = x and y = cos x. This means that the equation x = cos x has only one solution. The graphs also tell us that the solution lies within the interval (0, 1).

Text summary and images from:
Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, pp.40-41.

The Nature of Logarithmic Functions. Logarithmic Functions in Edwards & Penney's Calculus

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The Nature of Logarithmic Functions
Logarithmic Functions in Edwards & Penney's Calculus

What do logarithmic functions got to do with you?

In the natural world around us, things often expand or tapper off in a steady way. Consider how sound reverberations taper off or how logarithmic spirals expand:

(Thanks wikipedia )

Logarithms can help us study many phenomena in the natural world around us.

Brief Summary

y = log_ax if a^y = x

Points Relative to Deleuze

Deleuze writes:

Since intensity is already difference, it refers to a series of other differences that it affirms by affirming itself. It is said that in general there are no reports of null frequencies, no effectively null potentials, no absolutely null pressure, as though on a line with logarithmic gradations where zero lies at the end of an infinite series of smaller and smaller fractions. (Difference and Repetition 234c)

As a sound tapers off seemingly to nothing, it is perhaps undergoing an infinite series of lowering variations. The diminishing wave is not so much being negated as it is being constituted by infinitesimal differential relations.

Summary of

Edwards & Penney
Calculus

Chapter 1: Functions, Graphs, and Models
Section 1.4: Transcendental Functions

Subsection 4: Logarithmic Functions

Before we examine logarithmic functions, first recall exponential functions. In an exponential function, a constant base is raised to a variable power. A logarithmic function is an inverse to an exponential function. So consider this formulation for logarithmic functions:

y = log_ax if a^y = x

Here we have a, y, and x. a is the base, y is an exponential power of a, and x is the value of a raised to the power of y. "The base a logarithm of the positive number x is the power to which a must be raised to get x." (p.39c)

Base 10 is the common logarithm: log₁₀x

Later we discuss the natural logarithm e, which is a special irrational number.

lnx = log_ex

e = 2.71828182845904523536 . . . .

Edwarts and Penny graph y = lnx and y = log₁₀x.


Both graphs rise slowly and steadily as they move to the right, and also, both graphs cross through point (1, 0). Logarithmic functions cannot have negative numbers or zero in their domains, because exponential functions cannot take on zero or negative values. The function log x = log₁₀x increases slowly as x increases. We see this in the graph and also in the fact that log₁₀100,000 = 5 and log₁₀1,000,000 = 6. Consider how on the contrary in exponential function a^x (with a > 1) growth increases more rapidly than any other powerfunction as x → ∞. The f0llowing example shows how logarithmic functions increase slower than power functions.


Example



This table compares the rate of growth in the power function


with the rate of growth for the logarithmic function g(x) = logx. Here are graphs displaying these values.



As we can see in both the table and the graph, when x > 100,000, logx is smaller than



The graph below shows us the values when x is much smaller.


Here we see that logx at the lower values begins smaller than




but when x nears 5, the growth of logx overtakes


Then much later in their development,


overtakes logx when x = 100,000. Then, when x = 10⁵⁰,


equals 5,000,000,000, while logx only equals 50.



Deleuze, Gilles. Difference & Repetition. Transl. Paul Patton. New York: Columbia University Press, 1994.

Text summary and images from:
Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, pp.39-40.

Shell image from:
http://en.wikipedia.org/wiki/File:NautilusCutawayLogarithmicSpiral.jpg

Powers of Transformation. Exponential Functions in Edwards & Penney's Calculus

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Powers of Transformation
Exponential Functions in Edwards & Penney's Calculus

What do exponential functions got to do with you?

As we are learning how to swim, there is a whole host of powers/abilities/capacities that we are acquiring. For example, we are becoming able to enjoy swimming at the beach, to learn watersports like waterpolo or competitive swimming, to be capable of saving a drowning person, to be fit for sailing, and so on. So in a sense, becoming a swimmer does not merely add a little to our powers, but raises our powers to a whole new level of expression. Then consider if we do take up sailing. As we are becoming a sailor, we then are rising to yet another even higher level; for, we can move great distances on the water, compete in races, have new experiences out at sea, and so forth. If we continue through a sequence of changes in which one increased power raises us up to a far greater level from which we can yet rise yet remarkably further, this is something like growing exponentially. Each increase gives us more powers. But also, each increase gives us more power to increase. So one way we might understand our continuous changing is that we go through a series of states. But maybe we can also judge the changes that we go through and that we put ourselves through on the basis of how they increase our ability to increase our abilities. Who we are changes over time, but there is a sort of 'constant', which is our constantly changing in power.

Brief Summary

In an exponential function, a constant base is raised to a variable power.

Points Relative to Deleuze

Our series of self-transformations is for Deleuze and Orson Welles like a series of forgers or fakes of oneself, yet these are self-creative forgeries or fakeries. Each such self-forgery is like an exponential power increase:

It is Welles who, beginning with The Lady from Shanghai, imposes one single character, the forger. But the forger exists only in series of forges who are his metamorphoses, because the power itself exists only in the form of a series of powers which are its exponents. (Deleuze, Cinema 2, 140b)

Summary of

Edwards & Penney
Calculus

Chapter 1: Functions, Graphs, and Models
Section 1.4: Transcendental Functions

Subsection 3: Exponential Functions

Before examining exponential functions, we first will review power functions for contrast. In both cases of power functions and exponential functions, we speak of their 'form', which means we give explicit formulation to the categories of its component parts and of their relations. The form that power functions takes is

f (x) = x^k (where k is a constant)

So in some specific case of a power function, the base is a variable that can take on one from a range of values, while the exponent is specified as some given numerical value.

Exponential functions, however, take this form:

f(x) = a^x

We see that in exponential functions, the base is given as a constant, while the exponent is a variable that may take-on one of a range of values. Edwards and Penney write that in the case of power functions, the variable is raised to a constant power, while in exponential functions, a constant is raised to a variable power. (37c)

Below is a graph (made with geogebra) resembling the diagram in Edwards & Penney, and showing exponential functions y = 2^x (blue) and y = 10^x (red).


Example 6

Consider exponential functions when they have a base that is greater than one (base a > 1). Its value increases quite rapidly when the exponent x is large. Power functions, however, grow more slowly as x increases.


Edwards and Penney then have us consider smaller values for x² and 2^x. Seeing where their graphs overlap tell us the solutions to the equation x² = 2^x.


They also have us consider when the exponent in the exponential function is negative. The graphs for such functions fall from left to right.

The authors then compare the purposes of trigonometric and exponential functions. We use trigonometric functions to describe "periodic phenomena of ebb and flow;" however, we use exponential functions to describe "natural processes of steady growth or steady decline." (p38bc)


from Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, pp.37-38.

Deleuze, Gilles. Cinema 2: The Time Image. Transl. Hugh Tomlinson and Robert Galeta. London & New York: 1989.

The DeComposition of Our Body's Functions. Review and philosophical application of Compound Functions in Edwards & Penney's Calculus (Section 1.4)

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[Other Entries in the Edwards & Penney's Calculus Series]

The DeComposition of Our Body's Functions

Review and philosophical application of
Edwards & Penney's Calculus
Compound Functions
in
Section 1: Functions, Graphs, and Models
Subsection 4: Transcendental Functions

What Do Compound Functions Got to Do With You?

Consider if we are trying to lose weight. So we go on a diet. Over the course of a day, we begin to feel our appetite increase. So this would be one function: as time increases, our appetite increases. But then, as our appetite increases, there is a decrease in our willpower to hold strictly to the diet. So our decrease in willpower is like a function compounded on the first function. Our lives are filled with many such highly complex functional relations.

Brief Summary:

Functions can be compounded one-upon-another as if the output of one machine instantly becomes the input of another.

Points Relative to Deleuze
[Under Ongoing Revision]

We might find a composition of functions in Deleuze's account of sensation, building from his portrayal of Spinoza's affection. Deleuze's example is that we are looking for our glasses in a dark room. Someone enters and turns on the light. This causes us to make a transition from the way we are affected by the dark to the way we are affected by the light. Because we are searching for our glasses, this affection increases our power to act. Consider instead if we are meditating in the dark room. Then when someone turns on the light, this dazzles us and ruins our concentration, and we lose the power to act.

Affection for Spinoza is a matter of our body's composition. What gives our body its power is the differential relations between the speeds of its simple bodies. But its simple bodies are like calculus differentials, infinitely small parts that only obtain a finite value when put into a differential relation with other simple bodies. Consider if we take poison. This will disrupt the ways the simple bodies in our body differentially relate to one another. This is why we die. Our bodies no longer express the range of differential variations that define us. Instead, they eventually express for example dirt.

Now, Deleuze also places a value on our internal decomposition when it comes to the sensations that affect us. The discord of our faculties is what makes sensations more intense [see here and here.] Also consider Spinoza's Ethics Part IV, Proposition 39, Scholium. He writes:

It sometimes happens, that a man undergoes such changes, that I should hardly call him the same. As I have heard tell of a certain Spanish poet, who had been seized with sickness, and though he recovered therefrom yet remained so oblivious of his past life, that he would not believe the plays and tragedies he had written to be his own : indeed, he might have been taken for a grown-up child, if he had also forgotten his native tongue. If this instance seems incredible, what shall we say of infants? A man of ripe age deems their nature so unlike his own, that he can only be persuaded that he too has been an infant by the analogy of other men. (Elwes translation)

From what Spinoza says about the decomposition of the body, it would seem that to change from child to man, or from poet to someone different, would be a decomposition, and thus a loss of power. But could it be that Deleuze takes the opposite view? To change from child to man is not an expression of our body's frailty. It rather shows our body's power to become something new. Imagine we get a sickness like the Spanish poet, and we no longer recognize our past selves. This would seem to be a loss of power. But that would only be so when we take the perspective of the self whom our body no longer expresses. But our bodies are always as perfect as they can be, given the conditions affecting them at any given moment. So to become someone else is not really a deprivation. It could be a testament to the power of our bodies to become new things, to express new essences. The question for Deleuze might not so much be "What can a body do?" as much as it might be "What can a body become?"

So here would be the composition of functions in this case. A certain affection might for Deleuze increase the intensity of sensation. But that means a decrease in the organization of our faculties and our body's compositional relations. Yet this again would be an increase in our body's power to become new things, to express new relations and essences. So it ultimately is an increase in our body's expressive power.

Edwards & Penney's Calculus
Compound Functions
in
Section 1: Functions, Graphs, and Models
Subsection 4: Transcendental Functions

Recall the function machine.

We can think of functions like these 'black box' machines. The machine takes-in a number from one set of values, and it gives-out a value from another set. So consider the squaring function:

The machine takes-in a 2, for example, and then gives-out a 4.

We can also compound our functions. So we can take the result of one function, and then put it into another 'function machine', if you will.

So let's begin with the function g(x). We put in x, and it gives-out u. We then have another function f(u), which is the same as f(g(x)), and we read it as, 'f of g of x'. So we plug the u into the f function, and it gives out the value for f(u). We can then call f(u), or f(g(x)) a new name. We can call it h(x). So,

h(x) = f(g(x))

This requires that all the x values in g's domain correspond (by means of g) to values in f's domain. We call the g function here the 'inner function', and the f function would then be the 'outer function' in this case.

Edwards & Penney now give an example that will illustrate something about notation for composition of functions.

We will first find f(g(x)). First note the g(x). If our number for x is larger than 1, then we will have a negative number. But when we apply it to the f function, we will have the square root of a negative number. So instead we qualify it:

But let's instead consider if we first take f(x), and then after that apply it to the g function. To avoid having the square root of a negative number, we must first begin with a positive. Then after that, there are no restrictions for the g function. So

Edwards and Penney would like then to distinguish two notations:

f(g(x))

and

f ○ g

Our example above shows that the composition of functions f and g is not the same as g and f.

f ○ g ≠ g ○ f

So the notation f(g(x)) might remind us of multiplication notation. In the case of multiplication, it does not matter what order the functions are multiplied. But as we see in the composition of functions, the order does matter sometimes.

from Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, pp. 36-37.

Spinoza. Ethics. Transl. Elwes. available online at:
http://ebooks.adelaide.edu.au/s/spinoza/benedict/ethics/index.html