Commit 5fa72216 by Soren

### Catch up on some calc p1

parent 103fc5e9
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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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