Skip to content

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, lower case, name
alpha
beta
gamma
delta
epsilon
zeta
eta
theta
kappa
mu
nu
xi
omicron
pi
iota
rho
sigma
lambda
tau
upsilon
phi
chi
psi
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]]

Proper fractions occur with when and are polynomials with the degree of less than the degree of . This this case, we proceed as follows: write in factored form,

  • a linear factor in the denominator produces a partial fraction of the form
  • a repeated linear factors in the denominator produce partial fractions of the form
  • a quadratic factor in the denominator produces a partial fraction of the form
  • Improper fractions require an additional term which is a polynomial of degree where is the degree of the numerator (i.e. ) and is the degree of the denominator (i.e. ).

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.

, constant
, any constant

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]]

cot
(
(

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: