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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 \(2.718\).

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]]

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

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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