The set $(A\cap B)\setminus C$ 
The set $B \setminus (C \cup A)$ 

The set $A\cup B \cup C$ 
The set $A\cap B \cap C$ 

$\bigl((A \cap B) \setminus C \bigr) \cup \bigl((B \cap C) \setminus A \bigr) \cup \bigl((C \cap A) \setminus B \bigr)$

$\bigl(A \setminus (B \cup C)\bigr) \cup \bigl(B \setminus (C \cup A)\bigr) \cup \bigl(C \setminus (A \cup B)\bigr)$

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$u(t) = t\sin(t)$ in navy blue and its inverse $u^{1}$ in maroon 

The cardioid from Problem 6 
The cardioid from Problem 6 scaled by 2 
The cardioid from Problem 6 scaled by 2 and translated. 

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RegionPlot3D[
And[y^2 + z^2 < 1, z^2 + x^2 < 1], {x, 1, 1}, {y, 1, 1}, {z, 1,
1}, PlotPoints > {200, 200, 200}, Mesh > None,
PlotStyle > {Opacity[1]},
Ticks > {Range[2, 2, 1], Range[2, 2, 1], Range[2, 2, 1]},
ImageSize > 500
]
RegionPlot3D[
And[x^2+y^2 < 1, y^2 + z^2 < 1, z^2 + x^2 < 1], {x, 1, 1}, {y, 1, 1}, {z, 1,
1}, PlotPoints > {200, 200, 200}, Mesh > None,
PlotStyle > {Opacity[1]},
Ticks > {Range[2, 2, 1], Range[2, 2, 1], Range[2, 2, 1]},
ImageSize > 500
]
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A crosssection is an equilateral triangle.
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A crosssection is a square.
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A crosssection is a regular pentagon.
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A crosssection is a regular hexagon.
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A crosssection is a regular heptagon.
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A crosssection is a regular octagon.
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Folium of Descartes $x^3+y^3 = 6xy$ 

the smallest osculating circle in teal 

yellow points correspond to no real solutions of (qe)


yellow points correspond to a unique real solution of (ce)


yellow points correspond to no real solutions of (ep)


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$ X = [1,+\infty), \ Y = [1/e,+\infty), \ W_0 = \bigl\{ \bigl(x e^x,x \bigr) \ \bigl \bigr. \ x \in X \bigr\} $ The Lambert $W_0$ function 
$X = (\infty,1), \ Y = (1/e,0), \ W_{1} = \bigl\{ \bigl(x e^x,x \bigr) \ \bigl \bigr. \ x \in X \bigr\}$ The Lambert $W_{1}$ function 

$f(x)$ in navy blue and its inverse $f^{1}(x)$ in maroon 

$ X = [1,+\infty), \ Y = [1/e,+\infty), \ W_0 = \bigl\{ \bigl(x e^x,x \bigr) \ \bigl \bigr. \ x \in X \bigr\} $ The Lambert $W_0$ function 
$X = (\infty,1), \ Y = (1/e,0), \ W_{1} = \bigl\{ \bigl(x e^x,x \bigr) \ \bigl \bigr. \ x \in X \bigr\}$ The Lambert $W_{1}$ function 

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the navy blue function is $e^x$, the purple circle is a unit circle
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the sine function in navy blue and its many tangents in gray
$x  0 \lt \epsilon^2$ implies $\sqrt{x}  0  \lt \epsilon$.
Fact B. For all $a \gt 0$ and all $x \geq 0$ we have $\displaystyle \biggl\sqrt{x} \sqrt{a} \biggr \leq \frac{1}{\sqrt{a}} xa$.
To toggle the proof of Fact B click$x  a \lt \delta(\epsilon)$ implies $\sqrt{x}  \sqrt{a}  \lt \epsilon$.
$x  a \lt \sqrt{a} \, \epsilon$ implies $\sqrt{x}  \sqrt{a}  \lt \epsilon$.
for every real number $y$ between $f(a)$ and $f(b)$ there exists $x \in [a,b]$ such that $y = f(x)$.
there exist $c, d \in [a,b]$ such that $f(c) \leq f(x) \leq f(d)$ for all $x \in [a,b]$.
If $xa \lt d$, then $f(x)  f(a)  \lt K xa$.
Fact A. Let $a\in\mathbb R$ be such that $a \gt 0$. If $xa \lt a/2$, then $\displaystyle \biggl\frac{1}{x} \frac{1}{a} \biggr \lt \frac{2}{a^2} xa$.
To toggle the proof of Fact A click$ \displaystyle xa \lt \min\left\{ \frac{a}{2}, \frac{a^2}{2} \epsilon \right\}$ implies $\displaystyle \biggl\frac{1}{x} \frac{1}{a} \biggr \lt \epsilon$.
$xa \lt \delta(\epsilon)$ implies $f(x)  f(a)  \lt \epsilon$.
If $0 \lt xa \lt d$, then $f(x)  L  \lt K xa$.
If $0 \lt xa \lt d$, then $f(x)  f(a)  \lt K xa$.
$0 \lt \delta(\epsilon) \leq 1$ and $0 \lt x \lt \delta(\epsilon)$ implies $\cos(x)  1  \lt \epsilon$.
Fact B. If $x \lt \pi/3$, then $\cos(x)  1 \leq x$.
To toggle the proof of Fact B click
Figure for $\displaystyle \lim_{x\to 0} \cos(x) = 1$ 

$0 \lt \delta(\epsilon) \leq \pi/3$ and $0 \lt x \lt \delta(\epsilon)$ implies $\cos(x)  1  \lt \epsilon$.
$0 \lt \delta(\epsilon) \leq 1$ and $0 \lt x \lt \delta(\epsilon)$ implies $\displaystyle \left \frac{\sin x}{x}  1 \right \lt \epsilon$.
Fact C. If $0 \lt x \lt \pi/3$, then $\displaystyle \biggl\frac{\sin x}{x} 1 \biggr \lt x$.
To toggle the proof of Fact C clickFact C1. If $0 \lt x \lt \pi/3$, then $\displaystyle \cos(x) \lt \frac{\sin x}{x} \lt 1$.
Figure for $\displaystyle \lim_{x\to 0} \frac{\sin x}{x} = 1$ 

$0 \lt \delta(\epsilon) \leq \pi/3$ and $0 \lt x \lt \delta(\epsilon)$ implies $\displaystyle \biggl\frac{\sin x}{x} 1 \biggr \lt \epsilon$.
$0 \lt \delta(\epsilon) \leq 1$ and $0 \lt x \lt \delta(\epsilon)$ implies $\displaystyle \left \frac{1\cos(x)}{x^2}  \frac{1}{2} \right \lt \epsilon$.
Fact D. If $0 \lt x \lt 1$, then $\displaystyle \biggl\frac{1\cos(x)}{x^2}  \frac{1}{2} \biggr \lt x$.
To toggle the proof of Fact D click
Figure for $\displaystyle \lim_{x\to 0} \frac{1 \cos x}{x^2} = \frac{1}{2}$ 

$0 \lt \delta(\epsilon) \leq 1$ and $0 \lt x \lt \delta(\epsilon)$ implies $\displaystyle \biggl\frac{1 \cos(x)}{x^2}  \frac{1}{2} \biggr \lt \epsilon$.
$0 \lt \delta(\epsilon) \leq 1$ and $0 \lt x \lt \delta(\epsilon)$ implies $\displaystyle \left \frac{1\cos(x)}{x}  0 \right \lt \epsilon$.
Fact E. If $0 \lt x \lt 1$, then $\displaystyle \biggl\frac{1\cos(x)}{x} \biggr \lt \frac{1}{2} x$.
To toggle the proof of Fact E click$0 \lt \delta(\epsilon) \leq 1$ and $0 \lt x \lt \delta(\epsilon)$ implies $\displaystyle \biggl\frac{1 \cos(x)}{x}  0 \biggr \lt \epsilon$.
$0 \lt xa \lt \delta(\epsilon)$ implies $f(x)  L  \lt \epsilon$.
$0 \lt \delta(\epsilon) \leq 1$ and $0 \lt x2 \lt \delta(\epsilon)$ implies $x^2  4  \lt \epsilon$.
Fact A. If $x2 \lt 1$, then $x^2  4 \leq 5 x  2$.
To toggle the proof of Fact A click$0 \lt \delta(\epsilon) \leq 1$ and $0 \lt x2 \lt \delta(\epsilon)$ implies $x^2  4  \lt \epsilon$.
$x \gt X(\epsilon)$ implies $f(x)  L  \lt \epsilon$.
$\tanh$ in navy blue and its inverse $\tanh^{1}$ in maroon 

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The composition at a generic point $x$ 
The composition at another generic point $x$ 

Finding a minimum of the composition 
Finding a maximum of the composition 
Sine with yarn 

$\color{blue}{\cosh(x)} = \color{green}{\dfrac{1}{2} e^x} + \color{red}{\dfrac{1}{2} e^{x}}, \ x \in \mathbb R $ 
$ \color{blue}{\sinh(x)} = \color{green}{\dfrac{1}{2} e^x} \color{red}{ \dfrac{1}{2} e^{x}}, \ x \in \mathbb R $ 

$ \tanh(x) = \dfrac{\sinh(x)}{\cosh(x)} = \dfrac{e^x  e^{x}}{e^x + e^{x}}, \ x \in \mathbb R $ 

Hyperbola $\bigl\{ (x,y) \in \mathbb R\!\times \!\mathbb R \, \bigl \bigr. \, x^2  y^2 = 1\bigr\}$ 

$ X = [0,+\infty), \ Y = [0,+\infty), \ f_1 = \bigl\{ (x,x^2) \ \bigl \bigr. \ x \in X \bigr\} $ 
$X = \mathbb R, \ Y = \mathbb R, \ f_2 = \bigl\{ (x,x^3) \ \bigl \bigr. \ x \in X \bigr\}$ 

$X= \mathbb R, \ Y = (0,+\infty), \ f_3 = \bigl\{ (x,e^x) \ \bigl \bigr. \ x \in X \bigr\}$ 
$X = (\pi/2,\pi/2), \ Y = \mathbb R, \ f_4 = \bigl\{ \bigl( x,\tan(x) \bigr) \ \bigl \bigr. \ x \in X \bigr\} $ 
$ X = [\pi/2,\pi/2], \ Y = [1,1], \ f_5 = \bigl\{ \bigl(x,\sin(x)\bigr) \ \bigl \bigr. \ x \in X \bigr\} $ 
$ X = [0,\pi], \ Y = [1,1], \ f_6 = \bigl\{ \bigl(x,\cos(x)\bigr) \ \bigl \bigr. \ x \in X \bigr\} $ 
The Unit Circle $\bigl\{ (x,y) \in {\mathbb R}\times {\mathbb R} \,  \, x^2+y^2=1 \bigr\}$ 

The sign function 
The unit step function 

The floor function 
The ceiling function 
The number $e$ is the value of the account in which 1 dollar has been invested for 1 year in a savings account paying 100% annual interest compounded continuously.
These four real numbers together with the imaginary unit $i$ form the set of numbers which I call "the Hall of Fame of Numbers": \[ \bigl\{ 1, 0, \pi, e, i \bigr\}. \]