4.5.1: Equation Formatting Conventions

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\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

For each of the following exercises, fix the LaTeX and formatting as described in the Tutorial.

1. Equation numbering

Note that this equation is not automatically numbered since it is inline.

\(f(x) = x^2 + y^2\)

Remove the equation number from this equation while keeping it in display mode.

\[f(x) = x^2 + y^2\]

Remove the aut\nonumbering on this aligned equation by adding asterisks.

\[\begin{align} f'(x) &= 3\cdot(x^2 + 1)^2\cdot 2x \\[4pt] &=6x(x^2+1)^2 \end{align}\]

2. Aligning equal signs over multiple lines using begin{align} and end{align}

Format this chain equality using begin(align*} and end{align*} commands. Also fix other issues with the trig functions and powers.

Example 1:

\(\displaystyle \frac{1}{12}(cos(3x)−9cosx)=\frac{1}{12}(cos(x+2x)−9cosx)\)

\(\displaystyle =\frac{1}{12}(cos(x)cos(2x)−sin(x)sin(2x)−9cosx)\)

\(\displaystyle =\frac{1}{12}(cosx(2cos^2x−1)−sinx(2sinxcosx)−9cosx)\)

\(\displaystyle =\frac{1}{12}(2cos_3x−cosx−2cosx(1−cos^2x)−9cosx)\)

\(\displaystyle =\frac{1}{12}(4cos_3x−12cosx)\)

\(\displaystyle =\frac{1}{3}cos_3x−cosx.\)

Format this chain equality with explanations using begin(align*} and end{align*} commands.

Example 2:

Thus,

\(∫\dfrac{\sqrt{4−x^2}}{x}dx=∫\dfrac{\sqrt{4−(2\sin θ)^2}}{2\sin θ}2\cos θ \, dθ\) Substitute \(x=2\sin θ\) and \(=2\cos θ\,dθ.\)

\(=∫\dfrac{2\cos^2θ}{\sin θ}\,dθ\) Substitute \(\cos^2θ=1−\sin^2θ\) and simplify.

\(=∫\dfrac{2(1−\sin^2θ)}{\sin θ}\,dθ\) Substitute \(\sin 2θ=1−\cos^2θ\).

\(=∫ (2\csc θ−2\sin θ)\,dθ\) Separate the numerator, simplify, and use \(\csc θ=\dfrac{1}{\sin θ}\).

\(=2 ln |\csc θ−\cot θ|+2\cos θ+C\) Evaluate the integral.

\(=2 ln \left|\dfrac{2}{x}−\dfrac{\sqrt{4−x^2}}{x}\right|+\sqrt{4−x^2}+C.\) Use the reference triangle to rewrite the expression in terms of \(x\) and simplify.

3. Formatting functions correctly: sin x, cos x, ln x, arctan x, etc.

If you have not already done so, fix the functions in part 2 above.

4. Proper use of displaystyle

First consider #2 above, to determine whether \displaystyle is needed there.

Add \(\displaystyle\) where it is needed in the examples below and remove it where it is not. Also consider spacing of the differential in integrals.

\[(\displaystyle ∫^5_0(5+2t)\,dt. \nonumber\]

\(\displaystyle (\int_0^2 (x^2+1)dx\)

\(\displaystyle \sum_{n=1}^\infty a_n\)

\(\displaystyle \lim_{x\to 3} f(x) = 9\)

Note that \( \lim_{n\to \infty} \sum_{i=1}^n f(x_i)\,\Delta x_i = \int_a^b f(x) \, dx\).

5. Using \left and \right, \big, \Big, \bigg, and \Bigg to size grouping symbols appropriately.

Fix the alignment in the following problem, and also address the size of the grouping symbols and the evaluation bar.

\[ \begin{align*} ∫^5_010+\cos(\frac{π}{2}t)\,dt = (10t+\frac{2}{π} \sin (\frac{π}{2}t))|^5_0 \\[4pt] =(50+\frac{2}{π})−(0−\frac{2}{π} \sin 0)≈50.6. \end{align*}\]