Formula sheet

$x$

input (independent variable)

$y$

f(x) = output (dependent variable)

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

$f(x+h)$

f with x replaced by (x + h)

$h$

a small change in x

a question asks you to simplify the difference quotient, or the average rate of change from x to x + h.

$h$

horizontal shift (right if h > 0)

$k$

vertical shift (up if k > 0)

$A$

vertical stretch, flip across the x-axis if A < 0

a graph is a basic function moved, flipped or stretched, such as y = -(x - 2)^2 + 3 or y = \sqrt{x + 1} - 2.

$C(x)$

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 depends on a second, which depends on a third, such as cost in TL from cost in dollars from output.

$f^{-1}$

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

a demand or supply function is solved for one variable and you 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 a cost function, m is the variable cost per unit and b is the fixed cost.

$(x_1, y_1), (x_2, y_2)$

two points on the line

$m$

slope

you are given two points or one point and a slope.

$p$

price per unit

$v$

variable cost per unit

$F$

fixed cost

$x_{BE}$

break-even quantity

revenue and cost are both linear in x.

$a$

shape: a > 0 opens upward (minimum), a < 0 opens downward (maximum)

$c$

y-intercept, f(0)

you are given a quadratic in expanded form and need its graph.

$(h, k)$

the vertex

$k$

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

you need the optimum of a quadratic, or the centre of its graph, without derivatives.

$a, b, c$

the coefficients of ax^2 + bx + c

$b^2 - 4ac$

the discriminant

a quadratic does not factor easily and you need 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 (decimal)

$n$

compounding periods per year

$t$

years

interest compounds a fixed number of times a year: yearly n = 1, quarterly n = 4, monthly n = 12.

$P$

principal

$r$

annual rate

$t$

years

$e$

Euler's number, \approx 2.71828

the question says compounded continuously.

$b$

base, b > 0, b \neq 1

$x$

the argument, x > 0

$y$

the exponent

you convert between log form and exponential form: the first step in almost every log question.

$M, N$

positive numbers

$p$

any real exponent

you expand or combine logs, or bring an exponent down to solve for it.

$b$

the base

$x$

a positive number

you simplify expressions like ln(e^{3x}) or need log_3 20 on a calculator with only ln.

$r$

continuous annual growth rate (decimal)

$t_double$

doubling time in years

you need the doubling time, or the rule of 70 for a quick estimate.

$a$

the input value approached

$L$

the number f(x) approaches

$x \to a$

x close to a, from either side, never equal to a

you need what a function approaches near a point, especially where it misbehaves.

$x \to a^-$

from the left (below a)

$x \to a^+$

from the right (above a)

a graph or a piecewise function may jump at a, and you must decide whether the limit exists.

$f, g$

functions whose limits at a exist

a limit is built from pieces whose limits you know.

$n, m$

degrees of numerator and denominator

$a_n, b_m$

their leading coefficients

you need the long-run behaviour or the horizontal asymptote of a rational function.

$f(a)$

the function's value at a

$\lim_{x \to a} f(x)$

what f approaches near a

you must show (dis)continuity, especially at the boundary of a piecewise function.

$p, q$

polynomials

$u(x)$

the expression inside the root or the log

you must give the intervals on which a function is continuous.

$a, b$

endpoints of a closed interval

$c$

a point strictly between a and b

you must show a root (break-even point, equilibrium) exists without finding it.

$a$

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.