## STACK Documentation

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# Hints

STACK contains a "formula sheet" of useful fragments which a teacher may wish to include in a consistent way. This is achieved through the "hints" system.

Hints can be included in any CASText.

To include a hint, use the syntax

[[facts:tag]]


The "tag" is chosen from the list below.

## All supported fact sheets

### The Greek Alphabet

[[facts:greek_alphabet]]

Upper case, $$\quad$$ lower case, $$\quad$$ name
$$A$$ $$\alpha$$ alpha
$$B$$ $$\beta$$ beta
$$\Gamma$$ $$\gamma$$ gamma
$$\Delta$$ $$\delta$$ delta
$$E$$ $$\epsilon$$ epsilon
$$Z$$ $$\zeta$$ zeta
$$H$$ $$\eta$$ eta
$$\Theta$$ $$\theta$$ theta
$$K$$ $$\kappa$$ kappa
$$M$$ $$\mu$$ mu
$$N$$ $$u$$ nu
$$\Xi$$ $$\xi$$ xi
$$O$$ $$o$$ omicron
$$\Pi$$ $$\pi$$ pi
$$I$$ $$\iota$$ iota
$$P$$ $$\rho$$ rho
$$\Sigma$$ $$\sigma$$ sigma
$$\Lambda$$ $$\lambda$$ lambda
$$T$$ $$\tau$$ tau
$$\Upsilon$$ $$\upsilon$$ upsilon
$$\Phi$$ $$\phi$$ phi
$$X$$ $$\chi$$ chi
$$\Psi$$ $$\psi$$ psi
$$\Omega$$ $$\omega$$ omega

### Inequalities

[[facts:alg_inequalities]]

$a>b \hbox{ means } a \hbox{ is greater than } b.$ $a < b \hbox{ means } a \hbox{ is less than } b.$ $a\geq b \hbox{ means } a \hbox{ is greater than or equal to } b.$ $a\leq b \hbox{ means } a \hbox{ is less than or equal to } b.$

### The Laws of Indices

[[facts:alg_indices]]

The following laws govern index manipulation: $a^ma^n = a^{m+n}$ $\frac{a^m}{a^n} = a^{m-n}$ $(a^m)^n = a^{mn}$ $a^0 = 1$ $a^{-m} = \frac{1}{a^m}$ $a^{\frac{1}{n}} = \sqrt[n]{a}$ $a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m$

### The Laws of Logarithms

[[facts:alg_logarithms]]

For any base $$c>0$$ with $$c \neq 1$$: $\log_c(a) = b \mbox{, means } a = c^b$ $\log_c(a) + \log_c(b) = \log_c(ab)$ $\log_c(a) - \log_c(b) = \log_c\left(\frac{a}{c}\right)$ $n\log_c(a) = \log_c\left(a^n\right)$ $\log_c(1) = 0$ $\log_c(b) = 1$ The formula for a change of base is: $\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$ Logarithms to base $$e$$, denoted $$\log_e$$ or alternatively $$\ln$$ are called natural logarithms. The letter $$e$$ represents the exponential constant which is approximately $$2.718$$.

[[facts:alg_quadratic_formula]]

If we have a quadratic equation of the form: $ax^2 + bx + c = 0,$ then the solution(s) to that equation given by the quadratic formula are: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$

### Partial Fractions

[[facts:alg_partial_fractions]]

Proper fractions occur with ${\frac{P(x)}{Q(x)}}$ when $$P$$ and $$Q$$ are polynomials with the degree of $$P$$ less than the degree of $$Q$$. This this case, we proceed as follows: write $$Q(x)$$ in factored form,

• a linear factor $$ax+b$$ in the denominator produces a partial fraction of the form ${\frac{A}{ax+b}}.$
• a repeated linear factors $$(ax+b)^2$$ in the denominator produce partial fractions of the form ${A\over ax+b}+{B\over (ax+b)^2}.$
• a quadratic factor $$ax^2+bx+c$$ in the denominator produces a partial fraction of the form ${Ax+B\over ax^2+bx+c}$
• Improper fractions require an additional term which is a polynomial of degree $$n-d$$ where $$n$$ is the degree of the numerator (i.e. $$P(x)$$) and $$d$$ is the degree of the denominator (i.e. $$Q(x)$$).

[[facts:trig_degrees_radians]]

$360^\circ= 2\pi \hbox{ radians},\quad 1^\circ={2\pi\over 360}={\pi\over 180}\hbox{ radians}$ $1 \hbox{ radian} = {180\over \pi} \hbox{ degrees} \approx 57.3^\circ$

### Standard Trigonometric Values

[[facts:trig_standard_values]]

$\sin(45^\circ)={1\over \sqrt{2}}, \qquad \cos(45^\circ) = {1\over \sqrt{2}},\qquad \tan( 45^\circ)=1$ $\sin (30^\circ)={1\over 2}, \qquad \cos (30^\circ)={\sqrt{3}\over 2},\qquad \tan (30^\circ)={1\over \sqrt{3}}$ $\sin (60^\circ)={\sqrt{3}\over 2}, \qquad \cos (60^\circ)={1\over 2},\qquad \tan (60^\circ)={ \sqrt{3}}$

### Standard Trigonometric Identities

[[facts:trig_standard_identities]]

$\sin(a\pm b)\ = \ \sin(a)\cos(b)\ \pm\ \cos(a)\sin(b)$ $\cos(a\ \pm\ b)\ = \ \cos(a)\cos(b)\ \mp \sin(a)\sin(b)$ $\tan (a\ \pm\ b)\ = \ {\tan (a)\ \pm\ \tan (b)\over1\ \mp\ \tan (a)\tan (b)}$ $2\sin(a)\cos(b)\ = \ \sin(a+b)\ +\ \sin(a-b)$ $2\cos(a)\cos(b)\ = \ \cos(a-b)\ +\ \cos(a+b)$ $2\sin(a)\sin(b) \ = \ \cos(a-b)\ -\ \cos(a+b)$ $\sin^2(a)+\cos^2(a)\ = \ 1$ $1+{\rm cot}^2(a)\ = \ {\rm cosec}^2(a),\quad \tan^2(a) +1 \ = \ \sec^2(a)$ $\cos(2a)\ = \ \cos^2(a)-\sin^2(a)\ = \ 2\cos^2(a)-1\ = \ 1-2\sin^2(a)$ $\sin(2a)\ = \ 2\sin(a)\cos(a)$ $\sin^2(a) \ = \ {1-\cos (2a)\over 2}, \qquad \cos^2(a)\ = \ {1+\cos(2a)\over 2}$

### Hyperbolic Functions

[[facts:hyp_functions]]

Hyperbolic functions have similar properties to trigonometric functions but can be represented in exponential form as follows: $\cosh(x) = \frac{e^x+e^{-x}}{2}, \qquad \sinh(x)=\frac{e^x-e^{-x}}{2}$ $\tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{{e^x-e^{-x}}}{e^x+e^{-x}}$ ${\rm sech}(x) ={1\over \cosh(x)}={2\over {\rm e}^x+{\rm e}^{-x}}, \qquad {\rm cosech}(x)= {1\over \sinh(x)}={2\over {\rm e}^x-{\rm e}^{-x}}$ ${\rm coth}(x) ={\cosh(x)\over \sinh(x)} = {1\over {\rm tanh}(x)} ={{\rm e}^x+{\rm e}^{-x}\over {\rm e}^x-{\rm e}^{-x}}$

### Hyperbolic Identities

[[facts:hyp_identities]]

The similarity between the way hyperbolic and trigonometric functions behave is apparent when observing some basic hyperbolic identities: ${\rm e}^x=\cosh(x)+\sinh(x), \quad {\rm e}^{-x}=\cosh(x)-\sinh(x)$ $\cosh^2(x) -\sinh^2(x) = 1$ $1-{\rm tanh}^2(x)={\rm sech}^2(x)$ ${\rm coth}^2(x)-1={\rm cosech}^2(x)$ $\sinh(x\pm y)=\sinh(x)\ \cosh(y)\ \pm\ \cosh(x)\ \sinh(y)$ $\cosh(x\pm y)=\cosh(x)\ \cosh(y)\ \pm\ \sinh(x)\ \sinh(y)$ $\sinh(2x)=2\,\sinh(x)\cosh(x)$ $\cosh(2x)=\cosh^2(x)+\sinh^2(x)$ $\cosh^2(x)={\cosh(2x)+1\over 2}$ $\sinh^2(x)={\cosh(2x)-1\over 2}$

### Inverse Hyperbolic Functions

[[facts:hyp_inverse_functions]]

$\cosh^{-1}(x)=\ln\left(x+\sqrt{x^2-1}\right) \quad \mbox{ for } x\geq 1$ $\sinh^{-1}(x)=\ln\left(x+\sqrt{x^2+1}\right)$ $\tanh^{-1}(x) = \frac{1}{2}\ln\left({1+x\over 1-x}\right) \quad \mbox{ for } -1< x < 1$

### Standard Derivatives

[[facts:calc_diff_standard_derivatives]]

The following table displays the derivatives of some standard functions. It is useful to learn these standard derivatives as they are used frequently in calculus.

$$f(x)$$ $$f'(x)$$
$$k$$, constant $$0$$
$$x^n$$, any constant $$n$$ $$nx^{n-1}$$
$$e^x$$ $$e^x$$
$$\ln(x)=\log_{\rm e}(x)$$ $$\frac{1}{x}$$
$$\sin(x)$$ $$\cos(x)$$
$$\cos(x)$$ $$-\sin(x)$$
$$\tan(x) = \frac{\sin(x)}{\cos(x)}$$ $$\sec^2(x)$$
$$cosec(x)=\frac{1}{\sin(x)}$$ $$-cosec(x)\cot(x)$$
$$\sec(x)=\frac{1}{\cos(x)}$$ $$\sec(x)\tan(x)$$
$$\cot(x)=\frac{\cos(x)}{\sin(x)}$$ $$-cosec^2(x)$$
$$\cosh(x)$$ $$\sinh(x)$$
$$\sinh(x)$$ $$\cosh(x)$$
$$\tanh(x)$$ $$sech^2(x)$$
$$sech(x)$$ $$-sech(x)\tanh(x)$$
$$cosech(x)$$ $$-cosech(x)\coth(x)$$
$$coth(x)$$ $$-cosech^2(x)$$

$\frac{d}{dx}\left(\sin^{-1}(x)\right) = \frac{1}{\sqrt{1-x^2}}$ $\frac{d}{dx}\left(\cos^{-1}(x)\right) = \frac{-1}{\sqrt{1-x^2}}$ $\frac{d}{dx}\left(\tan^{-1}(x)\right) = \frac{1}{1+x^2}$ $\frac{d}{dx}\left(\cosh^{-1}(x)\right) = \frac{1}{\sqrt{x^2-1}}$ $\frac{d}{dx}\left(\sinh^{-1}(x)\right) = \frac{1}{\sqrt{x^2+1}}$ $\frac{d}{dx}\left(\tanh^{-1}(x)\right) = \frac{1}{1-x^2}$

### The Linearity Rule for Differentiation

[[facts:calc_diff_linearity_rule]]

${{\rm d}\,\over {\rm d}x}\big(af(x)+bg(x)\big)=a{{\rm d}f(x)\over {\rm d}x}+b{{\rm d}g(x)\over {\rm d}x}\quad a,b {\rm\ constant.}$

### The Product Rule

[[facts:calc_product_rule]]

The following rule allows one to differentiate functions which are multiplied together. Assume that we wish to differentiate $$f(x)g(x)$$ with respect to $$x$$. $\frac{\mathrm{d}}{\mathrm{d}{x}} \big(f(x)g(x)\big) = f(x) \cdot \frac{\mathrm{d} g(x)}{\mathrm{d}{x}} + g(x)\cdot \frac{\mathrm{d} f(x)}{\mathrm{d}{x}},$ or, using alternative notation, $(f(x)g(x))' = f'(x)g(x)+f(x)g'(x).$

### The Quotient Rule

[[facts:calc_quotient_rule]]

The quotient rule for differentiation states that for any two differentiable functions $$f(x)$$ and $$g(x)$$, $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)=\frac{g(x)\cdot\frac{df(x)}{dx}\ \ - \ \ f(x)\cdot \frac{dg(x)}{dx}}{g(x)^2}.$

### The Chain Rule

[[facts:calc_chain_rule]]

The following rule allows one to find the derivative of a composition of two functions. Assume we have a function $$f(g(x))$$, then defining $$u=g(x)$$, the derivative with respect to $$x$$ is given by: $\frac{df(g(x))}{dx} = \frac{dg(x)}{dx}\cdot\frac{df(u)}{du}.$ Alternatively, we can write: $\frac{df(x)}{dx} = f'(g(x))\cdot g'(x).$

### Calculus rules

[[facts:calc_rules]]

The Product Rule
The following rule allows one to differentiate functions which are multiplied together. Assume that we wish to differentiate $$f(x)g(x)$$ with respect to $$x$$. $\frac{\mathrm{d}}{\mathrm{d}{x}} \big(f(x)g(x)\big) = f(x) \cdot \frac{\mathrm{d} g(x)}{\mathrm{d}{x}} + g(x)\cdot \frac{\mathrm{d} f(x)}{\mathrm{d}{x}},$ or, using alternative notation, $(f(x)g(x))' = f'(x)g(x)+f(x)g'(x).$ The Quotient Rule
The quotient rule for differentiation states that for any two differentiable functions $$f(x)$$ and $$g(x)$$, $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)=\frac{g(x)\cdot\frac{df(x)}{dx}\ \ - \ \ f(x)\cdot \frac{dg(x)}{dx}}{g(x)^2}.$ The Chain Rule
The following rule allows one to find the derivative of a composition of two functions. Assume we have a function $$f(g(x))$$, then defining $$u=g(x)$$, the derivative with respect to $$x$$ is given by: $\frac{df(g(x))}{dx} = \frac{dg(x)}{dx}\cdot\frac{df(u)}{du}.$ Alternatively, we can write: $\frac{df(x)}{dx} = f'(g(x))\cdot g'(x).$

### Standard Integrals

[[facts:calc_int_standard_integrals]]

$\int k\ dx = kx +c, \mbox{ where k is constant.}$ $\int x^n\ dx = \frac{x^{n+1}}{n+1}+c, \quad (n\ne -1)$ $\int x^{-1}\ dx = \int {\frac{1}{x}}\ dx = \ln(|x|)+c = \ln(k|x|) = \left\{\matrix{\ln(x)+c & x>0\cr \ln(-x)+c & x<0\cr}\right.$

$$f(x)$$ $$\int f(x)\ dx$$
$$e^x$$ $$e^x+c$$
$$\cos(x)$$ $$\sin(x)+c$$
$$\sin(x)$$ $$-\cos(x)+c$$
$$\tan(x)$$ $$\ln(\sec(x))+c$$ $$-\frac{\pi}{2} < x < \frac{\pi}{2}$$
$$\sec x$$ $$\ln (\sec(x)+\tan(x))+c$$ $$-{\pi\over 2}< x < {\frac{\pi}{2}}$$
$$\mbox{cosec}(x)$$ $$\ln (\mbox{cose}c(x)-\cot(x))+c\quad$$ $$0 < x < \pi$$
cot$$\,x$$ $$\ln(\sin(x))+c$$ $$0< x< \pi$$
$$\cosh(x)$$ $$\sinh(x)+c$$
$$\sinh(x)$$ $$\cosh(x) + c$$
$$\tanh(x)$$ $$\ln(\cosh(x))+c$$
$$\mbox{coth}(x)$$ $$\ln(\sinh(x))+c$$ $$x>0$$
$${1\over x^2+a^2}$$ $${1\over a}\tan^{-1}{x\over a}+c$$ $$a>0$$
$${1\over x^2-a^2}$$ $${1\over 2a}\ln{x-a\over x+a}+c$$ $$|x|>a>0$$
$${1\over a^2-x^2}$$ $${1\over 2a}\ln{a+x\over a-x}+c$$ $$|x|$$
$$\frac{1}{\sqrt{x^2+a^2}}$$ $$\sinh^{-1}\left(\frac{x}{a}\right) + c$$ $$a>0$$
$${1\over \sqrt{x^2-a^2}}$$ $$\cosh^{-1}\left(\frac{x}{a}\right) + c$$ $$x\geq a > 0$$
$${1\over \sqrt{x^2+k}}$$ $$\ln (x+\sqrt{x^2+k})+c$$
$${1\over \sqrt{a^2-x^2}}$$ $$\sin^{-1}\left(\frac{x}{a}\right)+c$$ $$-a\leq x\leq a$$

### The Linearity Rule for Integration

[[facts:calc_int_linearity_rule]]

$\int \left(af(x)+bg(x)\right){\rm d}x = a\int\!\!f(x)\,{\rm d}x \,+\,b\int \!\!g(x)\,{\rm d}x, \quad (a,b \, \, {\rm constant.})$

### Integration by Substitution

[[facts:calc_int_methods_substitution]]

$\int f(u){{\rm d}u\over {\rm d}x}{\rm d}x=\int f(u){\rm d}u \quad\hbox{and}\quad \int_a^bf(u){{\rm d}u\over {\rm d}x}\,{\rm d}x = \int_{u(a)}^{u(b)}f(u){\rm d}u.$

### Integration by Parts

[[facts:calc_int_methods_parts]]

$\int_a^b u{{\rm d}v\over {\rm d}x}{\rm d}x=\left[uv\right]_a^b- \int_a^b{{\rm d}u\over {\rm d}x}v\,{\rm d}x$ or alternatively: $\int_a^bf(x)g(x)\,{\rm d}x=\left[f(x)\,\int g(x){\rm d}x\right]_a^b -\int_a^b{{\rm d}f\over {\rm d}x}\left\{\int g(x){\rm d}x\right\}{\rm d}x.$

### Integration by Parts

[[facts:calc_int_methods_parts_indefinite]]

$\int u{{\rm d}v\over {\rm d}x}{\rm d}x=uv- \int{{\rm d}u\over {\rm d}x}v\,{\rm d}x$ or alternatively: $\int f(x)g(x)\,{\rm d}x=f(x)\,\int g(x){\rm d}x -\int {{\rm d}f\over {\rm d}x}\left\{\int g(x){\rm d}x\right\}{\rm d}x.$

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