From 3dcc07703a9c73df4ba89770849787965c4aef31 Mon Sep 17 00:00:00 2001 From: Kai Stevenson Date: Wed, 2 Aug 2023 07:08:47 -0700 Subject: wrote more calculus, fixed latex positioning problem --- src/style.php | 1 + src/writing/calculus.php | 115 +++++++++++++++++++++++++++++++++++++++++++++ src/writing/derivative.php | 99 -------------------------------------- src/writing/index.php | 2 +- 4 files changed, 117 insertions(+), 100 deletions(-) create mode 100644 src/writing/calculus.php delete mode 100644 src/writing/derivative.php diff --git a/src/style.php b/src/style.php index a04d94d..a3a8dcb 100644 --- a/src/style.php +++ b/src/style.php @@ -135,5 +135,6 @@ hr { } .centermath { text-align: center; + text-indent: 0px; margin: 14px 40px 14px 40px; } diff --git a/src/writing/calculus.php b/src/writing/calculus.php new file mode 100644 index 0000000..ddfdd0c --- /dev/null +++ b/src/writing/calculus.php @@ -0,0 +1,115 @@ + +

+ Calculus is often used as the complete name of a branch of mathematics dealing with + rates of change and very small numbers. In fact, a calculus is a specific discipline + or method for the analysis of some set of problems. The fact that the calculus of infinitesimals + has come to be known in this way is a testament to its importance. +

+

Functions

+

+ To understand calculus, it is critical to understand the derivative. Let us begin with + the definition of a function. +

+

+

+ This expression defines a relation between two quantities x and y. It can be said that + y is given in terms of x. Specifically, we have defined a function f that provides + a mapping for values of x. The actual mapping of a given function varies; what follows is + one example. +

+

+

+ In this example, the value of the function is said to be equal to the product of its argument and + a constant a. Take note of the fact that this is a constant multiple, as it means that + the function is said to be linear--an important property. Evidently, + by the transitive property. + Here, the relation between y and x becomes clear; y is greater than x by a multiple a. + Observe the pattern that forms when evaluating f at various x. +

+

+

+

+

+

+ As x increases by increments of 1, y increases by increments of a. This gives rise to + the familiar slope-intercept form of a line, + . m, here, is the slope of the line, + the ratio between the growth of y and x, or the rate of change of the function mapping + the two. Since b is a constant term, it has no impact on the rate of change of the function. + It is this understanding that we will build upon. Consider two points in the form (x, y): + (0, 0) & (3, 9). As x increases by 3, y increases by 9. We refer to these as delta x + and delta y. We obtain the rate of change as follows: +

+

+

+

+ The implication of this is that a line drawn between the two points would have a slope + of 3. We note two functions upon whose graphs these points fall: + , and . If graphed, + the path of would appear identical to a straight line drawn + between the points, except + that its line would continue infinitely in either direction. Recall that it is a linear + function, and so every segment of its path has the same inclination as the whole. Compare + this to , a parabolic curve. Though both points fall upon + its graph, the path it takes between them is not linear, but curved. It appears to follow a + much less direct route. However, as each function has a slope of 3 over the interval [0, 3], + we say that they share the same average rate of change over that interval. The linear + function appears to move towards its destination faster in the start, but as the parabola + steepens, it catches up, such that both functions reach (3, 9). If we continued observing + this competition, the parabola would continue accelerating more and more rapidly, its value + rapidly eclipsing the straight line. +

+

+ Suppose we wished to know the slope of a function not over an interval but at a point. + This is easily accomplished with a linear function--it is observed that the slope of a + linear function is the same over any interval, and so it is reasonable to extend that + to a single point. It is more difficult, however, with non-linear functions. + Take on [0, 1]: +

+

+

+

+

Compare to the interval [0, 0.5]:

+

+

+

+

Or to [0, 0.25]:

+

+

+

+

+ As the interval becomes smaller, m decreases. One could use a computer to perform this + calculation with increasingly small values, and would find that m gets very close to 2. + As we can see, when moving from [0, 1] to [0, 0.5], m moved half of the way to 3. + On the next interval, it moved half of the distance that remains, again. + This would continue infinitely, with m never reaching 2. A new calculus is needed + to precisely describe the slope of non-linear functions at a point. This value, + the rate that the function will change over the next infinitesimally small interval, + is known as its instantantaneous rate of change at a point. +

+

Limits

+

+ The limit of a function is the value L it approaches as its argument approaches + c. +

+

+

+ Limits enable us to make statements about the behaviour of functions near a point, + even when the behaviour differs at that point. For example, 1/0 is undefined, + but . This is fairly + self-evident--division by an infinitely small number will produce an infinitely large one. + The limit is formally defined as follows: +

+

0\\ \\exists\\delta\\in\\mathbb{R},\\newline|f(x)-L|<\\epsilon\\ \\textup{whenever}\\ 0 < |x-c| < \\delta\$")); ?>

+

+ diff --git a/src/writing/derivative.php b/src/writing/derivative.php deleted file mode 100644 index aad4332..0000000 --- a/src/writing/derivative.php +++ /dev/null @@ -1,99 +0,0 @@ - -

- Calculus is often used as the complete name of a branch of mathematics dealing with - rates of change and very small numbers. In fact, a calculus is a specific discipline - or method for the analysis of some set of problems. The fact that the calculus of infinitesimals - has come to be known in this way is a testament to its importance. -

-

- To understand calculus, it is critical to understand the derivative. Let us begin with - the definition of a function. -

-

-

- This expression defines a relation between two quantities x and y. It can be said that - y is given in terms of x. Specifically, we have defined a function f that provides - a mapping for values of x. The actual mapping of a given function varies; what follows is - one example. -

-

-

- In this example, the value of the function is said to be equal to the product of its argument and - a constant a. Take note of the fact that this is a constant multiple, as it means that - the function is said to be linear--an important property. Evidently, - by the transitive property. - Here, the relation between y and x becomes clear; y is greater than x by a multiple a. - Observe the pattern that forms when evaluating f at various x. -

-

-

-

-

-

- As x increases by increments of 1, y increases by increments of a. This gives rise to - the familiar slope-intercept form of a line, - . m, here, is the slope of the line, - the ratio between the growth of y and x, or the rate of change of the function mapping - the two. Since b is a constant term, it has no impact on the rate of change of the function. - It is this understanding that we will build upon. Consider two points in the form (x, y): - (0, 0) & (3, 9). As x increases by 3, y increases by 9. We refer to these as delta x - and delta y. We obtain the rate of change as follows: -

-

-

-

- The implication of this is that a line drawn between the two points would have a slope - of 3. We note two functions upon whose graphs these points fall: - , and . If graphed, - the path of would appear identical to a straight line drawn - between the points, except - that its line would continue infinitely in either direction. Recall that it is a linear - function, and so every segment of its path has the same inclination as the whole. Compare - this to , a parabolic curve. Though both points fall upon - its graph, the path it takes between them is not linear, but curved. It appears to follow a - much less direct route. However, as each function has a slope of 3 over the interval [0, 3], - we say that they share the same average rate of change over that interval. The linear - function appears to move towards its destination faster in the start, but as the parabola - steepens, it catches up, such that both functions reach (3, 9). If we continued observing - this competition, the parabola would continue accelerating more and more rapidly, its value - rapidly eclipsing the straight line. -

-

- Suppose we wish to know the slope of a function not over an interval but at a point. - This is easily accomplished with a linear function--it is observed that the slope of a - linear function is the same over any interval, and so it is reasonable to extend that - to a single point. It is more difficult, however, with non-linear functions. - Take on [0, 1]: -

-

-

-

-

Compare to the interval [0, 0.5]:

-

-

-

-

Or to [0, 0.25]:

-

-

-

-

- As the interval becomes smaller, m decreases. One could use a computer to perform this - calculation with increasingly small values, and would find that m gets very close to 2. - As we can see, when moving from [0, 1] to [0, 0.5], m moved half of the way to 3. - On the next interval, it moved half of the distance that remains, again. - This would continue infinitely, with m never reaching 2. A new calculus is needed - to precisely describe the slope of non-linear functions at a point. This value, - the rate that the function will change over the next infinitesimally small interval, - is known as its instant rate of change at a point. -

- diff --git a/src/writing/index.php b/src/writing/index.php index c354f70..f138822 100644 --- a/src/writing/index.php +++ b/src/writing/index.php @@ -22,7 +22,7 @@ require($_SERVER["DOCUMENT_ROOT"] . "/header.php"); My writing in Greek.

Manifestos

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