Verified Commit 5fa72216 authored by Soren's avatar Soren

Catch up on some calc p1

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#+TITLE: Optimization
#+DATE: <2018-11-30 Fri>
#+AUTHOR: Soren
#+EMAIL: soren@disroot.org
#+LANGUAGE: en
#+SELECT_TAGS: export
#+EXCLUDE_TAGS: noexport
#+CREATOR: Emacs 25.2.2 (Org mode 9.1.14)
#+LaTeX_HEADER: \usepackage{siunitx}
* Optimization
Since first derivatives' zeroes can be used to calculate
relative minima and maxima of a function, optimizing
a physical problem or a related rate is done that way.
** Cutting Corners for a Box
Given a box that is $24 \si{in} \times 24 \si{in}$,
find the $x$ that makes the largest volume possible if
the corners are cut out in squares of $x \times x$.
\begin{align}
V &= l \cdot w \cdot h \\
V &= \left(24 - 2x\right)\left(24 - 2x\right)x \\
V &= 4x^{3} - 96x^{2} + 576x \\
\frac{dV}{dx} &= 12x^{2} - 192x + 576 \\
\frac{dV}{dx} &= 12\left(x^{2} - 16x + 48\right) \\
\frac{dV}{dx} &= 12\left(x - 12\right)\left(x - 4\right) \\
\frac{dV}{dx} = 0, x &= \cancel{12}, 4 \\
V &= \left(24 - 2\left(4\right)\right)
\left(24 - 2\left(4\right)\right)4 \\
V &= 16 \cdot 16 \cdot 4 \\
V &= 1024 \si{in^{3}}
\end{align}
#+TITLE: Antiderivatives
#+DATE: <2018-11-30 Fri>
#+AUTHOR: Soren
#+EMAIL: soren@disroot.org
#+LANGUAGE: en
#+SELECT_TAGS: export
#+EXCLUDE_TAGS: noexport
#+CREATOR: Emacs 25.2.2 (Org mode 9.1.14)
* Antiderivatives
Definition:
If $F\left(x\right)$ is the antiderivative of $f\left(x\right)$
then $F\left(x\right) = \int{f\left(x\right)}{dx}$.
** Rules
*** Constants
For any constant $C$:
\begin{align}
\frac{d}{dx} C &= 0 \\
\int{0}{dx} &= C
\end{align}
*** Linears
For any constant $k$:
\begin{align}
\frac{d}{dx} kx &= k \\
\int{k}{dx} &= kx + C
\end{align}
*** Constant Multiples
For any function $f\left(x\right)$ and ant constant $k$:
\begin{align}
\frac{d}{dx} kf\left(x\right) &= kf'\left(x\right) \\
\int{kf'\left(x\right)}{dx} &= kf\left(x\right) + C
\end{align}
*** Powers
\begin{align}
\frac{d}{dx} x^{n} &= nx^{n - 1} \\
\int{\left(n + 1\right)x^{n}} &= x^{n + 1} \\
\int{x^{n}} &= \frac{x^{n + 1}}{n + 1}
\end{align}
#+TITLE: Riemann Sums
#+DATE: <2018-11-30 Fri>
#+AUTHOR: Soren
#+EMAIL: soren@disroot.org
#+LANGUAGE: en
#+SELECT_TAGS: export
#+EXCLUDE_TAGS: noexport
#+CREATOR: Emacs 25.2.2 (Org mode 9.1.14)
* Riemann Sums
A Riemann Sum is a way of representing area under a curve, A.K.A
a definite integral.
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