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Introduction to the Calculus and Physics Tips Series

Welcome to Qubit Factory!

The Repository of Knowledge (see RoK information page) is a database of academic content that currently holds my personal intellectual items. This website hosts the part of the RoK called “(2011-2012) AP Calculus AB” and presents a chronological viewpoint of my adventures in calculus over the past year. Since 2011, I have been creating short “tips” about limits, derivatives, and integrals, and these tips are now posted online for all to see. After all, the RoK is eventually slated to become public in its entirety.

The “Tips” series constitutes a large portion of the sub-repository for calculus knowledge, and its primary purpose is to provide an outlet for the quick-but-formal review of calculus topics. Click the buttons below to go to the individual tip sections (Note: articles may emerge in reverse chronological order):

Limits Tip Series

Derivatives Tip Series

Integration Tip Series

Physics Tip Series

Additionally, here are some highlights from the tip series that you may enjoy:

Derivatives #15: A generalization of the Chain Rule in one variable.

Derivatives #33: A proof that cubic functions only have one point of inflection.

Derivatives #48: The visual pitfalls of L’Hopital’s rule.

Integration #9: A clear definition of the area problem and why integrals are needed to solve it.

Integration #21: The process of u-substitution.

Integration #66: Derivation of the Rate Law equations from chemistry.

Integration #72: Infinite integrations of an integrand product and of kinetic energy.

Physics #47: Derivation of the Master Equation for the Simple Harmonic Oscillator

Integration #93: The Exclusion Integral

Integration #96: Discretization of the Exclusion Integral to form the Discrete Exclusion Summation

Recent Excursions in Calculus and Physics

In this section I’ll create hyperlinks to my favorite new posts that are not part of the original tip series. Enjoy!

20120829: Multiple Integrations of Sine

20120829: Generalization of the Multiple Integral

20120824: Generalization of the Dirac Delta Function

20120823: On the Calculus of Cis

On a final note, it is important to note that the chronological progression of these calculus tips reflects my thought patterns. As a result, the mathematics in some tips may be incorrect. I will strive to correct any errors that I realize after posting a given calculus tip.

I hope you enjoy the Calculus and Physics Tip Series!

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Image | April 14, 2013

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April 14, 2013

Calculus #169: The Logistic Equation

The logistic model is given by

$\frac{\mathrm{d}P}{\mathrm{d}t} \approx kP$

if $P$ is small.

The logistic differential equation is given by

$\frac{\mathrm{d}P}{\mathrm{d}t} = kP\left(1-\frac{P}{K}\right)$

where

$P$ is population

$K$ is carrying capacity

$k$ is relative growth rate

Note that

$\lim\limits_{t\to\infty}\left(\frac{\mathrm{d}P}{\mathrm{d}t}\right)=K.$

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April 14, 2013

Calculus #168: Euler’s Method

Euler’s Method is an iterative method that approximates a function using the derivative $y' = F(x,y)$ .

The algorithm is carried out in the following manner, using a step size $h$ :

$y_0 := y_0$

$y_1 = y_0 + hF(x_0, y_0)$

$y_2 = y_1 + hF(x_1,y_1)$

$\vdots$

$y_n\equiv y_{n-1}+hF(x_{n-1},y_{n-1})$

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March 26, 2013

Calculus #167: Taylor Series

A Taylor series around a is given by the formula

$f(x) \equiv \sum\limits_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n$ .

A Maclaurin series is centered around 0, and is thus given as:

$f(x) \equiv \sum\limits_{n=0}^\infty \frac{f^{(n)}(0)}{n!}(x)^n$ .

The Taylor’s Inequality states that if $|f^{(n+1)}(x)|\leq M$ for $|x-a|\leq d$ , then the remainder $R_n(x)$ of the Taylor series satisfies the inequality

$|R_n(x)|\leq\frac{M}{(n+1)!}|x-a|^{n+1}$

for

$|x-a|\leq d.$

The Five Major Series

$\frac{1}{1-x}\equiv\sum\limits_{n=0}^\infty x^n = 1 + x + x^2 + \cdots$

$e^x\equiv\sum\limits_{n=0}^\infty \frac{x^n}{n!} 1 + x + \frac{x^2}{2!} + \cdots$

$\sin x\equiv\sum\limits_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!}+\cdots$

$\cos x\equiv\sum\limits_{n=0}^\infty (-1)^n\frac{x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots$

$\arctan x\equiv\sum\limits_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{2n+1}=x - \frac{x^3}{3} + \frac{x^5}{5}+\cdots$

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March 26, 2013

Calculus #166: Power Series

A power series is of the form $\sum\limits_{n=0}^\infty c_n x^n = c_0 + c_1 x + c_2 x^2 + \cdots$ .

The series $\sum\limits_{n=0}^\infty c_n(x-a)^n$ is a power series in $(x-a)$ or a power series centered at a or a power series about $a$ .

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March 26, 2013

Calculus #165: Absolute Convergence and the Ratio and Root Tests

Definition

A series $\sum a_n$ is called absolutely convergent if the series of absolute values $\sum|a_n|$ is convergent.

Definition

A series $\sum a_n$ is called conditionally convergent if it is convergent but not absolutely convergent.

The Ratio Test

(a) If $\lim\limits_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=L < 1$ , then the series $\sum\limits_{n=1}^\infty a_n$ is absolutely convergent.

(b) If $\lim\limits_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=L > 1$ or $\lim\limits_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$ does not exist, then the series $\sum\limits_{n=1}^\infty a_n$ is divergent.

The Root Test

(a) If $\lim\limits_{n\to\infty}\left[\sqrt[n]{|a_n|}\right]=L<1$ , then the series $\sum\limits_{n=1}^\infty a_n$ is absolutely convergent.

(b) If $\lim\limits_{n\to\infty}\left[\sqrt[n]{|a_n|}\right]=L>1$ or $\lim\limits_{n\to\infty}\left[\sqrt[n]{|a_n|}\right]$ does not exist, then the series $\sum\limits_{n=1}^\infty a_n$ is divergent.

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March 26, 2013

Calculus #164: Alternating Series

The Alternating Series Test

If the alternating series

$\sum\limits_{n=1}^\infty (-1)^{n-1}b_n\,\,\,\,\,\,\,\,\,\{b_n > 0\}$

satisfies

(a) $b_{n+1}\leq b_n\,\,\,\forall\,\,\, n$

(b) $\lim\limits_{n\to\infty} b_n = 0$

then the series is convergent.

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March 26, 2013

Calculus #163: The Comparison Tests

The Comparison Test

Suppose that $\sum a_n$ and $\sum b_n$ are series with positive terms

If $\sum b_n$ is convergent and $a_n \leq b_n\,\,\,\,\forall\,\,\,\, n$ , then $\sum a_n$ is also convergent.

If $\sum b_n$ is divergent and $a_n\geq b_n\,\,\,\,\forall\,\,\,\, n$ , then $\sum a_n$ is also divergent.

The Limit Comparison Test

Suppose that $\sum a_n$ and $\sum b_n$ are series with positive terms. If $\lim\limits_{n\to\infty}\left(\frac{a_n}{b_n}\right)=c$ where $c\in \mathbb{R}$ and $c >0$ , then either both series converge or both series diverge.

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March 26, 2013

Calculus #162: The Integral Test and Estimates of Sums

I. The Integral Test

Suppose $f$ is a continuous, positive, decreasing function on $[1,\infty)$ and let $a_n = f(n)$ . Then the series $\sum\limits_{n=1}^\infty a_n$ is convergent if and only if the improper integral $\int\limits_1^\infty f(x)\,\mathrm{d}x$ is convergent. In other words:

If $\int\limits_1^\infty f(x)\,\mathrm{d}x$ is convergent, then $\sum\limits_{n=1}^\infty a_n$ is convergent.

If $\int\limits_1^\infty f(x)\,\mathrm{d}x$ is divergent, then $\sum\limits_{n=1}^\infty a_n$ is divergent.

II. p-series

The p-series $\sum\limits_{n=1}^\infty\frac{1}{n^p}$ is convergent if $p>1$ and divergent if $p\leq 1$ .

III. Remainder Estimates for the Integral Test

If $\sum a_n$ converges by the integral test and $R_n \equiv s - s_n$ , then

$\int\limits_{n+1}^\infty f(x)\,\mathrm{d}x \leq R_n \leq \int\limits_n^\infty f(x)\,\mathrm{d}x$

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March 26, 2013

Calculus #161: Series

I. Definition

Given a series $\sum\limits_{i=1}^\infty a_i$ , let $s_n$ denote its $n$ th partial sum:

$s_n = \sum\limits_{i=1}^n$

If the sequence $\{s_n\}$ is convergent and $\lim\limits_{n\to\infty}s_n = s \in\mathbb{R}$ , then the series $\sum a_n$ is called convergent and we write