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 $c \neq 1$: $$\log_c(a) = b \text{, means } a = c^b$$ $$\log_c(a) + \log_c(b) = \log_c(ab)$$ $$\log_c(a) - \log_c(b) = \log_c\left(\frac{a}{b}\right)$$ $$n\log_c(a) = \log_c\left(a^n\right)$$ $$\log_c(1) = 0$$ $$\log_c(c) = 1$$ 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 $2.718$.

The Quadratic Formula

[[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. $Q(x)$).

Degrees and Radians

[[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]]

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

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]]

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

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

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