Fact sheets
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. Note, these hints are basic HTML strings and are stored in the language files.
All supported fact sheets
The Greek Alphabet
[[facts:greek_alphabet]]
Upper case | lower case | 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]]
The Laws of Indices
[[facts:alg_indices]]
The following laws govern index manipulation:
The Laws of Logarithms
[[facts:alg_logarithms]]
For any base with : The formula for a change of base is: Logarithms to base , denoted or alternatively are called natural logarithms. The letter represents the exponential constant which is approximately .
The Quadratic Formula
[[facts:alg_quadratic_formula]]
If we have a quadratic equation of the form: then the solution(s) to that equation given by the quadratic formula are:
Partial Fractions
[[facts:alg_partial_fractions]]
Fractions when and are polynomials with the degree of less than the degree of are called proper algebraic fractions. To re-write this as partial fractions write 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]]
Standard Trigonometric Values
[[facts:trig_standard_values]]
Standard Trigonometric Identities
[[facts:trig_standard_identities]]
Hyperbolic Functions
[[facts:hyp_functions]]
Hyperbolic functions have similar properties to trigonometric functions but can be represented in exponential form as follows:
Hyperbolic Identities
[[facts:hyp_identities]]
The similarity between the way hyperbolic and trigonometric functions behave is apparent when observing some basic hyperbolic identities:
Inverse Hyperbolic Functions
[[facts:hyp_inverse_functions]]
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 \(n\) | \(nx^{n-1}\) |
\(e^x\) | \(e^x\) |
\(\ln(x)=\log_{\rm e}(x)\) | \(\frac{1}{x}\) |
\(\sin(x)\) | \(\cos(x)\) |
\(\cos(x)\) | \(-\sin(x)\) |
\(\tan(x) = \frac{\sin(x)}{\cos(x)}\) | \(\sec^2(x)\) |
\(cosec(x)=\frac{1}{\sin(x)}\) | \(-cosec(x)\cot(x)\) |
\(\sec(x)=\frac{1}{\cos(x)}\) | \(\sec(x)\tan(x)\) |
\(\cot(x)=\frac{\cos(x)}{\sin(x)}\) | \(-cosec^2(x)\) |
\(\cosh(x)\) | \(\sinh(x)\) |
\(\sinh(x)\) | \(\cosh(x)\) |
\(\tanh(x)\) | \(sech^2(x)\) |
\(sech(x)\) | \(-sech(x)\tanh(x)\) |
\(cosech(x)\) | \(-cosech(x)\coth(x)\) |
\(coth(x)\) | \(-cosech^2(x)\) |
\[ \frac{\mathrm{d}}{\mathrm{d}x}\left(\sin^{-1}(x)\right) = \frac{1}{\sqrt{1-x^2}}\] \[ \frac{\mathrm{d}}{\mathrm{d}x}\left(\cos^{-1}(x)\right) = \frac{-1}{\sqrt{1-x^2}}\] \[ \frac{\mathrm{d}}{\mathrm{d}x}\left(\tan^{-1}(x)\right) = \frac{1}{1+x^2}\] \[ \frac{\mathrm{d}}{\mathrm{d}x}\left(\cosh^{-1}(x)\right) = \frac{1}{\sqrt{x^2-1}}\] \[ \frac{\mathrm{d}}{\mathrm{d}x}\left(\sinh^{-1}(x)\right) = \frac{1}{\sqrt{x^2+1}}\] \[ \frac{\mathrm{d}}{\mathrm{d}x}\left(\tanh^{-1}(x)\right) = \frac{1}{1-x^2}\]
The Linearity Rule for Differentiation
[[facts:calc_diff_linearity_rule]]
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 with respect to . or, using alternative notation,
The Quotient Rule
[[facts:calc_quotient_rule]]
The quotient rule for differentiation states that for any two differentiable functions and ,
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 , then defining , the derivative with respect to is given by: Alternatively, we can write:
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 with respect to .
or, using alternative notation,
The Quotient Rule
The quotient rule for differentiation states that for any two differentiable functions and ,
The Chain Rule
The following rule allows one to find the derivative of a composition of two functions.
Assume we have a function , then defining , the derivative with respect to is given by:
Alternatively, we can write:
Standard Integrals
[[facts:calc_int_standard_integrals]]
\[\int k\ \mathrm{d}x = kx +c, \text{ where } k \text{ is constant.}\] \[\int x^n\ \mathrm{d}x = \frac{x^{n+1}}{n+1}+c, \quad (n\ne -1)\] \[\int x^{-1}\ \mathrm{d}x = \int {\frac{1}{x}}\ \mathrm{d}x = \ln(|x|)+c = \ln(k|x|)\]
\(f(x)\) | \(\int f(x)\ \mathrm{d}x\) | |
---|---|---|
\(e^x\) | \(e^x+c\) | |
\(\cos(x)\) | \(\sin(x)+c\) | |
\(\sin(x)\) | \(-\cos(x)+c\) | |
\(\tan(x)\) | \(\ln(\sec(x))+c\) | \(-\frac{\pi}{2} < x < \frac{\pi}{2}\) |
\(\sec x\) | \(\ln (\sec(x)+\tan(x))+c\) | \( -{\pi\over 2}< x < {\frac{\pi}{2}}\) |
\(\text{cosec}(x)\) | \(\ln (\text{cose}c(x)-\cot(x))+c\quad\) | \(0 < x < \pi\) |
cot(\x\) | \(\ln(\sin(x))+c\) | \(0< x< \pi\) |
\(\cosh(x)\) | \(\sinh(x)+c\) | |
\(\sinh(x)\) | \(\cosh(x) + c\) | |
\(\tanh(x)\) | \(\ln(\cosh(x))+c\) | |
\(\text{coth}(x)\) | \(\ln(\sinh(x))+c \) | \(x>0\) |
\({1\over x^2+a^2}\) | \({1\over a}\tan^{-1}{x\over a}+c\) | \(a>0\) |
\({1\over x^2-a^2}\) | \({1\over 2a}\ln{x-a\over x+a}+c\) | \(x > a >0\) |
\({1\over a^2-x^2}\) | \({1\over 2a}\ln{a+x\over a-x}+c\) | \(a > x >0\) |
\(\frac{1}{\sqrt{x^2+a^2}}\) | \(\sinh^{-1}\left(\frac{x}{a}\right) + c\) | \(a>0\) |
\({1\over \sqrt{x^2-a^2}}\) | \(\cosh^{-1}\left(\frac{x}{a}\right) + c\) | \(x\geq a > 0\) |
\({1\over \sqrt{x^2+k}}\) | \(\ln (x+\sqrt{x^2+k})+c\) | |
\({1\over \sqrt{a^2-x^2}}\) | \(\sin^{-1}\left(\frac{x}{a}\right)+c\) | \(-a\leq x\leq a\) |
The Linearity Rule for Integration
[[facts:calc_int_linearity_rule]]
Integration by Substitution
[[facts:calc_int_methods_substitution]]
Integration by Parts
[[facts:calc_int_methods_parts]]
or alternatively:
Integration by Parts
[[facts:calc_int_methods_parts_indefinite]]
or alternatively: