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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 \(c \neq 1\): \[\log_c(a) = b \text{, means } a = c^b\] \[\log_c(a) + \log_c(b) = \log_c(ab)\] \[\log_c(a) - \log_c(b) = \log_c\left(\frac{a}{b}\right)\] \[n\log_c(a) = \log_c\left(a^n\right)\] \[\log_c(1) = 0\] \[\log_c(c) = 1\] 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\(\,x\) | \(\ln(\sin(x))+c\) | \(0< x< \pi\) |
| \(\cosh(x)\) | \(\sinh(x)+c\) | |
| \(\sinh(x)\) | \(\cosh(x) + c\) | |
| \(\tanh(x)\) | \(\ln(\cosh(x))+c\) | |
| \(\text{coth}(x)\) | \(\ln(\sinh(x))+c \) | \(x>0\) |
| \({1\over x^2+a^2}\) | \({1\over a}\tan^{-1}{x\over a}+c\) | \(a>0\) |
| \({1\over x^2-a^2}\) | \({1\over 2a}\ln{x-a\over x+a}+c\) | \(|x|>a>0\) |
| \({1\over a^2-x^2}\) | \({1\over 2a}\ln{a+x\over a-x}+c\) | \(|x|\) |
| \(\frac{1}{\sqrt{x^2+a^2}}\) | \(\sinh^{-1}\left(\frac{x}{a}\right) + c\) | \(a>0\) |
| \({1\over \sqrt{x^2-a^2}}\) | \(\cosh^{-1}\left(\frac{x}{a}\right) + c\) | \(x\geq a > 0\) |
| \({1\over \sqrt{x^2+k}}\) | \(\ln (x+\sqrt{x^2+k})+c\) | |
| \({1\over \sqrt{a^2-x^2}}\) | \(\sin^{-1}\left(\frac{x}{a}\right)+c\) | \(-a\leq x\leq a\) |
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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