Formula sheet

$x$

input (independent variable)

$y$

f(x) = output (dependent variable)

you need to name a relationship before working with it, or evaluate it at a specific input.

$C(x)$

total cost

$R(x)$

total revenue

$P(x)$

total profit

$p$

price per unit

you are given a fixed cost, a per-unit cost, and a price, and need any of the three functions.

$g$

the inner function, applied first

$f$

the outer function, applied to g's output

one quantity is defined in terms of a second, which is itself defined in terms of a third, such as cost as a function of labour as a function of the wage.

$f^{-1}$

the inverse of f, which exists only where f is one-to-one

you are given a demand or supply function solved for one variable and need it solved for the other.

$m$

slope, the change in y per unit change in x

$b$

y-intercept, the value of y when x = 0

you know the rate of change and the starting value. In cost functions, m is the variable cost per unit and b is the fixed cost.

$R(x)$

px = price times quantity

$C(x)$

variable cost + fixed cost

revenue and cost are both linear in x. Solve the resulting linear equation for x.

$a$

shapes the curve (a > 0 opens upward, a < 0 opens downward)

$b, c$

shift and position the curve

you are given a quadratic in expanded form and need to identify its shape before finding the vertex.

$x^*$

the input at the vertex

$f(x^*)$

the maximum (a<0) or minimum (a>0) value

you have a quadratic profit, revenue or cost function and need the optimal quantity and the optimal value, without derivatives.

$a, b, c$

the coefficients of ax^2 + bx + c

$b^2 - 4ac$

the discriminant

a quadratic profit or revenue function does not factor easily, or a question directly asks for its roots (break-even points).

$a$

initial value, f(0)

$b$

growth factor per period, b > 0, b \neq 1

a quantity grows or shrinks by the same percentage every period. b > 1 is growth; 0 < b < 1 is decay.

$P$

principal

$r$

annual interest rate

$n$

compounding periods per year

$t$

years

interest compounds a fixed number of times per year, such as monthly (n=12) or quarterly (n=4).

$P$

principal

$r$

annual (continuous) interest rate

$t$

years

$e$

Euler's number, \approx 2.71828

a question says continuously compounded, or you need the theoretical upper limit of compounding more and more often.

$b$

base, b > 0, b \neq 1

$x$

the argument, x > 0

$y$

the exponent that produces x

you need to convert between exponential form and logarithmic form, or as a first step before applying the log laws.

$M, N$

positive numbers

$p$

any real exponent

a variable sits inside an exponent, or a product/quotient/power needs to be broken apart before solving.

$r$

continuous annual growth rate (as a decimal)

$t_double$

approximate doubling time in years

you need a quick mental estimate of doubling time without a calculator, such as for inflation or GDP growth.

$a$

the input value being approached

$L$

the number f(x) approaches

$x \to a$

x gets arbitrarily close to a, from either side, without ever equaling a

you need to state or compute what a function approaches near a point, especially one where the function itself may misbehave.

$x \to a^-$

approach from the left (below a)

$x \to a^+$

approach from the right (above a)

a piecewise function or a graph has a possible jump, and you need to check whether the limit exists at the seam.

$f, g$

the two functions being combined, each with an individually existing limit

a limit is built from sums, differences or products of simpler pieces you already know how to evaluate.

$n, m$

the degrees of the numerator and denominator

$a_n, b_m$

their leading coefficients

a rational function's behaviour as x becomes very large is needed, such as long-run average cost.

$f(a)$

the function's actual value at a

$the limit$

what f approaches near a

you must verify continuity rigorously, not just glance at a picture, especially at the seam of a piecewise function.

$p(x), q(x)$

polynomials

you need to state the interval(s) on which a function is continuous, as a first step before applying the derivative rules of later lessons, which require continuity.

$a, b$

the endpoints of a closed interval

$c$

some point strictly between a and b

a question asks you to argue a root or an equilibrium exists, without necessarily finding its exact value.

$a$

starting input

$h$

change in the input

$f(a+h) - f(a)$

change in the output

a question asks for an average rate over an interval, such as average cost per extra unit between two output levels.

$f'(x)$

derivative of f at x, the instantaneous rate of change

$h \to 0$

the step shrinks toward zero but never equals it

a question says from the definition or using the limit. Otherwise use the rules below, which are much faster.

$k$

any constant

a term has no x. A flat line has slope zero, so fixed costs vanish from marginal cost.

$n$

any real number, including negatives and fractions

you see x to a power. Rewrite roots and fractions first: the square root of x is x to the 1/2, and 1/x is x to the -1.

$k$

constant

$f, g$

differentiable functions

always. Differentiate a polynomial term by term and keep the coefficients.

$C(x)$

total cost

$R(x)$

total revenue

$P(x)$

total profit

$x$

units produced

a question asks for the approximate cost, revenue or profit of the next unit. Use the exact difference only if it asks for the exact change.

$f, g$

differentiable functions

two functions of x are multiplied together, and neither is a plain constant.

$f$

numerator

$g$

denominator, g(x) \neq 0

one function is divided by another, and the denominator is not simply a constant.

$f$

outer function

$g$

inner function

one expression is raised to a power, or otherwise wrapped around another expression in x, such as $(3x^2+1)^5$.

$e$

Euler's number, \approx 2.71828

the base is e and the exponent is exactly x, with nothing more complicated in the exponent.

$g(x)$

any differentiable function in the exponent

the exponent is anything other than plain x, such as kx, x^2, or -0.03x.

$x$

the input, restricted to x > 0

the argument is exactly x. This is also the rule behind $\int \frac{1}{x}dx = \ln|x| + C$ in later courses.

$g(x)$

any differentiable, positive function

the argument of ln is anything other than plain x.

$a$

any positive base, a \neq 1

the base is a number other than e, such as 2^x or log base 10.

$f'(x)/f(x)$

the relative (percentage) rate of change of f at x

a question asks for a growth rate rather than an absolute rate of change — percent per period, not units per period.

$p$

price

$f(p)$

quantity demanded at price p

$f'(p)$

derivative of demand with respect to price

you have a demand function x = f(p) and need the elasticity at a specific price. The minus sign makes E positive, since f'(p) is normally negative (demand curves slope down).

$E(p)$

point elasticity of demand at price p

you need to classify demand at a given price, as a step toward predicting the revenue effect of a price change.

$R(p)$

total revenue as a function of price

you need to predict whether a price increase raises or lowers revenue at the current price.

$f'(x)$

the derivative of f, whose sign on an interval tells you the direction f is moving

you need to describe where a function rises or falls, such as where profit is growing versus shrinking.

$c$

a critical number of f

you are looking for possible local maxima or minima of a function, as the first step before the first-derivative test.

$+ \to -$

f' is positive just before c and negative just after

$- \to +$

the reverse

you have found critical numbers and need to classify each one, or a question directly asks for local extrema.

$f''(x)$

second derivative of f, read 'f double prime'

you need to differentiate a function twice, such as to check concavity or apply the second-derivative test.

$concave up$

graph curves upward, like a cup

$concave down$

graph curves downward, like a frown

you need to describe the curvature of a graph, or confirm whether a critical point is a max or a min via the second-derivative test.

$c$

a candidate inflection point, whose sign change must still be confirmed, exactly as with the first-derivative test

a question asks for the point of diminishing returns, or any point where a curve changes from bending one way to the other.

$if f''(c)$

0, the test is inconclusive and the first-derivative test must be used instead

you already have f'' available (or it is easy to compute) and just need to classify a single critical point, rather than build a whole sign chart.

$[a, b]$

a closed, bounded interval

you need to guarantee that an absolute maximum or minimum actually exists before searching for it, such as when output is capped by capacity.

output or another decision variable is restricted to a closed interval, such as a factory's minimum and maximum capacity.

a cost or area depends on two design variables linked by a fixed relationship, such as a fixed volume or a fixed budget.