title: Advanced Mathematics
subtitle: DLMDSAM01
Author: Dr. Robert Graf
publisher: IU International University of Applied Sciences
year: 2023

Calculus

https://en.wikibooks.org/wiki/LaTeX/Mathematics Wiki List of important Mathematical Series…

Also, some useful calculus

For LaTeX symbols, I use This pdf. KaTeX Support

Side note

Certainly! Here’s a quick cheatsheet of commonly used set notation in LaTeX markdown:

1. Set Membership: a \in A means “a belongs to set A.”
2. Set Non-Membership: b \notin B means “b does not belong to set B.”
3. Set Cardinality: |A| denotes the cardinality (number of elements) of set A.
4. Set Union: A \cup B represents the union of sets A and B.
5. Set Intersection: A \cap B represents the intersection of sets A and B.
6. Set Difference: A \setminus B or A - B represents the set difference of A and B.
7. Subset: A \subset B or A \subseteq B means that set A is a subset (or a proper subset) of set B.
8. Superset: A \supset B or A \supseteq B means that set A is a superset (or a proper superset) of set B.
9. Empty Set: \emptyset denotes the empty set.
10. Universal Set: \mathbb{U} or \mathbb{V} represents the universal set.
11. Complement: A^c or A' represents the complement of set A.
12. Cartesian Product: A \times B represents the Cartesian product of sets A and B.
13. Power Set: \mathcal{P}(A) represents the power set of set A.

Feel free to use these notations as needed in your LaTeX markdown!

LaTeX Cheat Sheet & Quick Reference short-math-guide.pdf (ctan.org) Aligning equations with amsmath - Overleaf, Online LaTeX Editor Seems to have basics of using LaTeX.

$y=f(x)$ actually has a lot to it

• Expresses relationship between variables.
• $y \rightarrow$ the dependent variable
• $x \rightarrow$ the independent variable
• $f \rightarrow$ the name of the function

Rate of change is very useful (eg. velocity is distance w/respect to time). Differentiation is the method of finding the rate of change given a function. And, given multiple independent variables, partial differentiation allows us to explore the rate of change with respect to each independent variable.

Integration can be seen as the inverse of differentiation, hence the alias anti-derivative.

Suggested Readings: …

Differentiation and Integration

https://www.malinc.se/math/latex/basiccodeen.php https://kapeli.com/cheat_sheets/LaTeX_Math_Symbols.docset/Contents/Resources/Documents/index Math Symbols and meaning MicroSoft MathSolver

Derivatives and Functions of a Single Variable

Considering the example $v(t)=\dfrac{\Delta s}{\Delta t}$ just to see that by $\Delta t$ we mean an interval of time. It’s technically the average velocity over that unit of time. A Function is a relation between two sets that associates every element of one set to exactly one element of the other set.

We kind of brush over the concept of a limit, which is quite critical to the fundamentals of Calculus, but typically not invoked directly (so understandable). A limit is a self describing term for the result of a function, the dependent variable, as the independent variable(s) approach their limit’s value.

The slope of a curve is typically denoted as:

definition 1.1

However, for actual curved shaped curves, non-linear relationships typically, the slope is not exact. We can approximate the slope by decreasing the size of $\Delta x$, until it is indistinguishable from $0$. This gives us the change in $y$ per infinitesimal change in $x$. The slope which would intercept the curve at both $x$ points now only touches the curve in the one spot. Geometrically, we refer to it as the tangent line, the slope of the line tangent to the function $f(x)$ at the point $x$.

Additionally, In math, equivalent is different from equal. Equal means same in all aspects, whereas equivalent means similar but not identical. Use the equal sign to express an identity, and equivalent to express, say, same truth values.

Mathematically, it looks more like:

definition 1.2

Note that (the importance of limits) a function is differentiable at $x_{t}$ if, and only if (IFF), the limit exists at point $x = x_t$. For the limit to exist, the definition of the limit requires that the quotient $\frac{f(x+\Delta x)-f(x)}{\Delta x}$ approaches the same value, $f'(x)$, from both the left and right. That’s more limit talk.

EXAMPLE

Lets find $f'(x)$ for $f(x) = x^2$.

1. Define a derivative.

2. Substitute value of $x$.

3. Expand Polynomial.

4. $x^{2} - x^{2} = 0$.

5. Divide by $\Delta x$.

6. Let $\Delta x \to 0$ (basically substitution).

The trick is that $\Delta x$ becomes infintessimally small as it approaches zero, but is always non-zero. Hence, does not cause function to become undefined.

Additionally, for a function to be differentiable at $x_t$, a function must be continuous at $x_t$. If the function were not continuous at that point, it would also not have a limit. However, being continuous is a weak criteria. Take $f(x) = |x|$ for example. The derivative taken at $x=0$ from the left is $-1$, but from the right is $+1$.

EXAMPLE

From definition, calculate $f'(x)$ for $f(x)=2x+1$.

$\Box$

Using definition 1.2 and the laws of limits, we can find derivatives of many fundamental functions. Let $n>0$ be a natural number and $a$ be a real-valued constant.

What we have been discussing is just the first derivative of a function. The derivative goes by other name, such as slope or gradient.

Higher Order Derivatives

A derivative of a derivative is a higher order derivative. The results are obtained by applying the definition of a derivative onto the result of itself.

definition 1.3

More generally, for the $n^{th}$ derivative;

definition 1.4

… whenever the limit exists.

EXAMPLE

Using the definition for $n^{th}$ derivative, calculate the $f''(x)$ for $f(x)=ax^2+bx+c$.

And then

$\Box$

Stationary Points

Imagine the parabola of $f(x)=x^2$. At $x=0$, the function achieves a local minimum. Graphically, the line tangent to the graph at this point is horizontal, with a slope $m=0$.

For $f(x)=|x|$, $x=0$ is a local extrema, a critical point defined to be a place where the derivative is zero or does not exist.

There are 3 different stationary points:

• Function has maximum at stationary point, $x=a$ if $f'(a) = 0$ and $f''(a) < 0$.
• Function has minimum at stationary point, $x=a$ if $f'(a) = 0$ and $f''(a) > 0$.
• Stationary point at $x=a$ is called a saddle point if $f'(a) = 0$ and $f''(a)$ changes sing at the point.

Worth noting that these are only for local min-max, and not the functions’ global min-max point.

EXAMPLE

Find the stationary points for $f(x) = 3x^2 - 6x$. Determine if the points are minimum, maximum, or saddle.

The points are found in the first derivative, and described in the second derivative.

We will use techniques over the general derivative definition

Now, you can solve for $f'(x) = 0$ and see the stationary point is at $x=1$.

Also, since $f''(x)$ is positive, the rate of change will be increasing at this point. We don’t need to check for saddle because it’s a constant. Instead, you can envision this as a bowl shape, meaning this point is a minimum.

EXAMPLE

Suppose we have 6m of framing material and we want to build a rectangular window with sides $x$ and $y$. Choose the values of $x$ and $y$ to maximize the area because we want a lot of sunlight.

We know that the area is $A=xy$, and we have $C=2x+2y$ amount of material. We want to maximize the output, so we need a function to solve for the derivative at 0.

Rearrange circumference formula

Pop that into the Area

Then, go for the derivative

And solve $x$ for $A'(x) = 0$

Solve for $y=3-3/2=6/2-3/2=3/2$.

It shouldn’t be too much of a surprise that the largest area comes from a square.

$\Box$

Rules of Differentiation

Differentiation of Functions with a Constant

Let $a \in \mathbb{R}$ and be an arbitrary constant and $g(x)$ be some function.

Differentiation of Products

Suppose our function is the product of multiple functions (eg. $f(x) = u(x) \times v(x)$). Decomposing a function into product parts may ease the strain of applying the definition of a derivative to the entire function. The idea is to choose a $u(x)$ and $v(x)$ that are easy to differentiate. Basically, we are about to derive the product rule of differentiation…

1. Define a derivative:

2. Substitution of $u(x) \times v(x)$

3. Since $u(x+\Delta x) \times v(x) - u(x+\Delta x) \times v(x) = 0$, we add that in (tricky substitution):

4. Reorganize for show:

5. Factor out what we need:

6. Distribute $\Delta x$

7. As $\Delta x \to 0$, $u(x + \Delta x) \to u(x)$ and we have something like:

definition 1.5

We can give a try at something like $f(x)=u(x)v(x)w(x)$, applying the rule on itself, but to spare the complexity:

Q: Find the derivative of $f(x) = x^2 \sin(x)$

The Chain Rule

It might be hard to imagine right now, but sometimes it is easier to think of functions as functions of functions! Consider $f(x) = (x-1)^2$. If we let $u(x)=x-1$, then $f(x)=u(x)^2$.

definition 1.6

Let’s walk through an example.

1. define our function

2. Substitute

3. Define derivative

4. What is $\dfrac {dy} {du}$?

5. What is $\dfrac {du} {dx}$?

6. And therefore:

7. Substitute $u(x) = x-1$

Differentiation of Quotients

Similar to the product rule, the quotient rule finds $f'(x)$ for $f(x)= \dfrac {u(x)} {v(x)}$. Basically, you can consider applying the product rule to $f'(x)$ for $f(x)= u(x)v(x)^{-1}$.

definition 1.7

EXAMPLE

Find $f'(x)$ for

You can do this using the product rule or the Quotient rule. I like only remembering one formula so this is how I would do it… note that the $v'(x)$ will also require using the chain rule, which is the derivative of the function times the derivative of the inside.

You can continue to factor this pulling out a $2(\sin(3x)-1)$ from each side, but doesn’t make a huge difference.

$\Box$

Integrals of Functions of a Single Variable

Integrals as Area Under the Curve

So far, we have been using the definition of slope as $m= \dfrac {\Delta y} {\Delta x}$, which defines an average rate of change. We then consider progressively shorter $x$ intervals, $\Delta x \to 0$ to determine the instantaneous rate of change $f'(x)= \frac {dy} {dx}$.

But given the rate of change, or the function of it, can we determine the coordinate values? For example, if given the function of velocity, can we determine an objects position at a given point in time? If velocity is change of position per unit time, expressed as $v(t)= \dfrac {\Delta s} {\Delta t}$, then we solve for position by $\Delta s = \Delta t \times v(t)$. Graphically, this looks like base time height, which thus becomes the area under the curve.

Informally, we can approximate this value summing sub-intervals between $t_0$ and $t_n$. However, we will technically be summing rectangles, which will produce an error for a smooth curve. To increase accuracy, we shorten the interval, increasing the total number of intervals. And, if we let the interval length approach 0, we start our journey into integration.

Below is a function summing the intervals. Let $f(x)$ be a function defined over the interval $a<x<b$. We divide our interval $[a,b]$ into $n$ sub-intervals such at $a=\xi_1<\xi_2<...<\xi_n =b$. That means $f(x_i) \times (\xi_i-\xi_{i-1})$ is like our base times height.

definition 1.8

Note that the area under some curves over certain intervals is not finite (eg. $f(x)= \frac {1}{x}$). However, if the limit does exist, as $n \to \infty$ we define an integral as… realizing that $n$ may not be defined correctly above…

definition 1.9

The integral is undefined if the limit does not exist. For closed, finite intervals, the limit can exist if the function is continuous over that interval. It is both interesting and convenient to consider integration to be the summation of infinite parts.

The function to be integrated, $f(x)$ is called the integrand. The process of integration kind of involves summing rectangles of area on the interval, and letting the width of the intervals approach 0. Certain functions in a summation can be expressed as an ordinary function, making the process easier. For example

Let’s look at some interesting integral properties or identities:

Integrals as inverse of differentiation

A formal definition is:

(1.14)

We are going to prove quick the Fundamental Theorem of Calculus, that states the derivative of the integral gives back the original integrand.

Consider:

We split integral into workable pieces:

Then a quick substitute:

Now, divide both sides by $\Delta x$.

Then, we consider the limit as $\Delta x \to 0$

You may also see it written with definition of $F(x)$ substituted back in as:

Typically, the definition is written as:

Where $c$ is a constant. This is because the derivative of a constant $c$ will be $0$ and therefore the value is lost. However, when going backwards, we account for the lost value with a placeholder. We refer to $c$ as the constant of integration.

And an interesting pattern that appears a bit in probability theory is, let $x_0$ be an arbitrary fixed point such that $x_0 \in (a,b)$. Then

The above is nothing new. However, we are going to do a switch-and-flip! New line in Latex Equation - TeX - LaTeX Stack Exchange = “align”, “flalign”, “align*”, “aligned”. Art of Problem Solving

call this equation 1.21.

Integrals with Infinite Bounds of Integration

Previous definitions expected the bounds of integration to be finite. However, it is often the case that one or both bounds are infinite. We can extend the definition:

Where the limit as $b$ approaches $\infty$ is evaluated after the integral is calculated. That is, integrate and then evaluate.

Evaluation of integrals

Unlike derivatives, integrals usually cannot be evaluated as easily. As such, we have an extend list of recipes. Note, that $u$ is typically a function $u(x)$, and that $du$ is the derivative such that $du=u'(x)dx$. The notation can be a little confusing. Khan Academy has a huge section on integration techniques, and a u-substitution.

Page 28… Needs to be wrapped up. Even the book says that large number of integrals can be found in tables of integrals.

To evaluate unknown integrals, we try to transform them into forms that are easier to evaluate. Here’s a quick reference of some techniques:

• Logarithmic integration
• Decomposition = When integrand is a linear combination of integrable functions. You can split integral of sum into sum of simpler integrals:
• Substitution = Essentially reverses the chain rule for derivatives. Helps to integrate composite functions. Brings back memories. You need to substitute not just the $u$ but also find a $du$.
  • Derivative of chain rule $w(u(x)) \to w'(u(x))u'(x)$
  • Think, we are going backwards.
• Integration by parts = Similar to substitution reversing the chain rule, integration by parts reverse the product rule.
  • $\int uv'dx = uv - \int vu' dx$

We probably need many examples.

Try to evaluate $\int_a^b x * \cos{(x)} dx$. Hint: look at the the above list… should make skip over all but integration by parts.

Try to evaluate the following:

From page 30. It’s a cool situation where you factor the bottom, use partial fraction decomposition, and notice things being to look a bit logarithmic.

EXAMPLE

Evaluate the following integral

The best way to probably handle this is integration by parts.

I say this because in the derivative case it would be a power rule.

At this point the reader may become annoyed realizing they, again, have to perform another integration by parts… We take out the negative from the cosine to make things easier.

And we put this back into the former equation

$\Box$

EXAMPLE

p.17

Evaluate the following

You can probably tell, based on it being a product, that we will require integration by parts.

To be honest, if we let $v(x)=e^{3x}$, I think we will be integrating forever. So…

also,

just needed to write my notes,

We now calculate $I(x)$

Funny enough, you can see how we have recreated our integral in question. We can stir to combine…

$\Box$

EXAMPLE

You are in a car moving at speed $v(t)$. How far has the vehicle travelled between $t_0=0s$ and $t_1=5s$?

The anti-derivative of velocity is distance. So, it’s a bounded integration problem. I am going to suggest $u$ substitution because I don’t see integration by parts…

Luckily, the bounds can be left where they are because they are just constant values.

Nice $\Box$

Taylor Approximation

Taylor’s Theorem, named after Brook Taylor who expressed this relationship in 1712, provides an approximation to a function in the vicinity of a given point $x_0$ as a sum. The theorem requires the function $f(x)$ be continuous and that all derivatives up to order $f^n(x)$ exist in order to generate the $n^{th}$ degree polynomial approximation of $f(x)$ near $x_0$. You can always refer to Wikipedia for more information. However, per equation 1.21, which is

We say that $x$ and $x-\epsilon$ are in the vicinity of $a$, and rewrite the fecker as:

Call this equation 1.24

Now comes the magic. We must assume that $\epsilon$ is very small. So small that we can assume $f'(x) \approx f'(a)$. A side-effect of this assumption is that

We are kind of saying that $f(a+\epsilon)$ is $f(a)$ plus the tiny increment multiplied by the rate of change. After all, that is actually what we kind of assume integration is actually doing under the hood. So, then we express in terms of $x$ and $a$, assuming we stay close to the point $a$, to get the approximation

call this equation 1.26. Sorry the numbering is a bit off.

My example: Let $f(x) = 3x^2$. Compare the actual and approximate values if $a=6$ and $\epsilon = 0.1$.