Concepts

A payment of the same amount at the end of each of n periods. Its present value is a(1 − (1 + r)⁻ⁿ)/r. See Fact 8.9.

A function F with F′(x) = f(x) on an interval. Also called an indefinite integral of f, written ∫ f(x) dx = F(x) + C. See Definition 7.12.

A(t) = the area under the graph of f between a and t. Its derivative is f(t): it is an antiderivative of f (Fact 7.17).

The m × (n + 1) array (A | b) of the system Ax = b: the coefficient matrix with the column b of constants written to its right. See Definition 13.15.

Stopping Gaussian elimination at the staircase and solving from the last equation upwards, inserting each value found into the equation above.

The number a in aᵗ: the number that is raised to a power. For an exponential function it is the factor by which the value is multiplied in each unit step. See Definition 2.3.

A set in the plane is bounded if it is contained in some disc of finite radius. See Definition 11.8.

The line $px + qy = m$: the bundles that cost exactly the whole income.

The set of all bundles $(x, y)$ with $x \ge 0,$ $y \ge 0$ that a consumer with income $m$ can afford at the prices $p$ and $q$: $px + qy \le m.$

“Other things being equal”: the economist's name for holding the other variables fixed, which is what a partial derivative does. See Watch out 8.18.

The derivative of a composition $F(x)=g(f(x))$ is the outer derivative, evaluated at the inner value, times the derivative of the inside. In symbols $F'(x)=g'(f(x))\cdot f'(x).$ With $z=g(y)$ and $y=f(x)$ this reads $dz/dx=dz/dy\cdot dy/dx.$ See Fact 3.16.

A set in the plane is closed if it contains all its boundary points. A set cut out by weak inequalities (≤, ≥, =) between continuous functions is closed. See Definition 11.8.

F(x, y) = A·xᵃ·yᵇ with A, a, b > 0: the workhorse production function of the course. See the text after Definition 8.13.

One of the constants $a_0, a_1, \ldots, a_n$ of a polynomial; $a_0$ is the constant term.

The m × n matrix A = (aᵢⱼ) of the coefficients of a linear system: one row per equation, one column per unknown.

Computing a determinant of order 3 as entry times the order-2 determinant left when the entry's row and column are deleted, summed along a row or a column, with the sign (−1) to the power i + j for the entry in row i, column j. See Method 14.7.

A matrix with only one column, written as a bold lower-case letter. Its entries are its components.

Closed and bounded. On a nonempty compact set a continuous function attains a maximum and a minimum (Fact 11.9). See Definition 11.8.

An entry of a vector.

A function is concave on an interval if every chord between two points of its graph lies on or below the graph. For a twice differentiable f this means f″ ≤ 0 there. See Definition 6.19.

For an expression √A − B, the conjugate is √A + B. Their product is A − B², with no root left, which is why multiplying and dividing by the conjugate helps with a difference that involves a root.

A system is consistent if it has at least one solution. See Definition 12.5.

The arbitrary constant C in ∫ f(x) dx = F(x) + C. Two antiderivatives on an interval differ by a constant and nothing more (Fact 7.13); in a cost problem C is the fixed cost.

The right-hand sides b₁, …, bₘ of a linear system. Together they form the column vector b.

f is continuous at a if the limit of f(x) as x → a equals f(a): f(a) exists, the limit exists, and the two agree. Continuous on an interval means continuous at each point of it; a function that is not continuous at a is discontinuous there. See Definitions 5.13 and 5.14.

Several level curves of f at different heights, drawn together and labelled with their heights. See Definition 9.1.

An improper integral or an infinite series converges if the limit that defines it exists and is finite. See Definition 8.5.

A function is convex on an interval if every chord between two points of its graph lies on or above the graph. For a twice differentiable f this means f″ ≥ 0 there. See Definition 6.19.

A set S in the plane that contains the whole straight segment between any two of its points. A disc, a rectangle, a half-plane and the whole plane are convex; a ring is not. See Definition 10.9.

If |A| ≠ 0, the square system Ax = b has exactly one solution, with xⱼ = |Aⱼ| / |A|, where Aⱼ is A with its j-th column replaced by b. See Facts 14.4 and 14.11.

f is decreasing on an interval if f(x₂) ≤ f(x₁) whenever x₂ is larger than x₁ in the interval. See Definition 2.22.

For f continuous on [a, b] and F any antiderivative: the number F(b) − F(a), written ∫ from a to b of f(x) dx. It is the signed area between the graph and the x-axis. See Definition 7.16.

The highest power $n$ whose coefficient $a_n$ is not zero.

The number of freely chosen unknowns in the solution of a consistent linear system: one for each column of the finished staircase without a leading entry (Method 13.16, Fact 13.20).

In z = f(x, y), the variable z whose value the function determines; in economics, the endogenous one. See Definition 8.13.

The derivative of f at a, written f'(a), is the slope of the tangent to the graph at (a, f(a)) (Definition 2.15). It is computed as the limit of the Newton quotient as h tends to 0 (Definition 2.17).

A number computed from a square matrix. For order 2 it is a₁₁a₂₂ − a₂₁a₁₂, written |A| or det(A); it may be negative. It is not zero exactly when the matrix has an inverse. See Definitions 14.1 and 14.6.

f is differentiable at a if the limit of the Newton quotient as h tends to 0 exists, that is, if f'(a) exists. See Definition 2.17.

Leibniz's way of writing the derivative: dy/dx, or d/dx in front of f(x), read “differentiate with respect to x”. It means the same as f'(x).

k = 1/(1 + r), the factor by which a payment loses value today for each period it is delayed, at interest rate r. See Fact 8.9.

The number $\Delta = b^2 - 4ac$ of a quadratic $ax^2 + bx + c$: two roots if it is positive, one if it is zero, none if it is negative.

The distance between x = (x₁, …, xₙ) and a = (a₁, …, aₙ) is the square root of (x₁ − a₁)² + ⋯ + (xₙ − aₙ)²: Pythagoras in n dimensions. See Definition 9.9.

An improper integral or an infinite series diverges if the limit that defines it does not exist or is not finite. See Definition 8.5.

The set $D$ of all $x$ for which a function $f$ is defined.

The time in which an exponential function A·aᵗ with base a above 1 doubles: ln 2 / ln a. It depends neither on A nor on the starting moment. See Fact 2.13.

The percentage change in a quantity per one per cent change in a variable. The elasticity of output F with respect to capital K is (K/F)·F′_K; for a Cobb–Douglas function it is the exponent of K. See the text after Example 8.17 (Lecture 9 defines partial elasticities in general).

The three operations on a system of linear equations: interchange two equations; multiply an equation by a non-zero scalar; add a multiple of one equation to a different equation. They do not change the set of solutions. See Definition 13.13.

One of the numbers of a matrix. The entry aᵢⱼ sits in row i and column j — row first, then column. See Definition 12.1.

Two equation systems are equivalent if they have exactly the same solutions. Elementary row operations turn a system into an equivalent one (Fact 13.14).

The number r in xʳ, or t in aᵗ: the power to which the base is raised. See Definition 2.1.

A function of the form f(t) = A·aᵗ, where the base a is a positive constant: the base is fixed and the exponent varies. Each unit step in t multiplies the value by a. See Definition 2.3.

A maximum point or a minimum point of f over a set S. See Definition 6.8.

If f is continuous on a closed and bounded interval [a, b], then f attains both a maximum and a minimum on [a, b]. See Fact 6.9.

$Q$ is a factor of the polynomial $P$ if the remainder on dividing $P$ by $Q$ is zero, so that $P(x) = q(x)\,Q(x).$

If f′ ≥ 0 to the left of c and f′ ≤ 0 to the right of c, throughout an interval I, then c is a maximum point of f over the whole of I; with the inequalities reversed, a minimum point. See Fact 6.18.

At an interior point c of an interval where f is differentiable, if c is a maximum or a minimum point then f′(c) = 0. Necessary, not sufficient. See Fact 6.13.

An unknown whose column of the finished staircase has no leading entry. It may be chosen freely, for instance as a parameter t, and the other unknowns are then determined (Method 13.16).

A rule $f\colon D\to T$ that assigns to each element $x$ of $D$ exactly one element $f(x)$ of $T.$

A rule that assigns to each pair (x, y) in its domain a single number z = f(x, y). See Definition 8.13.

Undoing a derivative and measuring area are the same problem: the area function is an antiderivative, so the area from a to b is F(b) − F(a). See Fact 7.17.

A series a + ak + ak² + ⋯ in which each term is k times the one before. For |k| < 1 its sum is a/(1 − k); for |k| ≥ 1 (and a ≠ 0) it diverges. See Fact 8.7.

The set of all points $(x, y)$ in the plane with $x$ in the domain and $y = f(x).$

The series 1 + 1/2 + 1/3 + ⋯ . Its terms tend to zero and it still diverges. See Watch out 8.12.

A linear system Ax = 0, with all constants zero. The homogeneous system corresponding to Ax = b has the same coefficient matrix and 0 in place of b. See Definition 13.22.

The n × n matrix Iₙ with 1 in every position on the main diagonal and 0 everywhere else. AI = IA = A, so it plays the part of the number 1. See Definition 13.3.

Differentiate both sides of the equation with respect to x, treating y as a function of x, so that every y leaves a y′ behind; then solve the result for y′. See Method 4.9.

An equation in x and y defines y implicitly as a function of x if, near the point of interest, it is satisfied by exactly one value of y for each x. The equation need not be solvable for y. See Definition 4.8.

An integral over an unbounded interval, defined as a limit: the integral of f from a to infinity is the limit, as b → ∞, of the integral from a to b. See Definition 8.5.

A rational function $P(x)/Q(x)$ in which the degree of $P$ is at least the degree of $Q;$ it can be divided.

A system is inconsistent if it has no solution. See Definition 12.5.

f is increasing on an interval if f(x₂) ≥ f(x₁) whenever x₂ is larger than x₁ in the interval. Flat stretches are allowed. See Definition 2.22.

In z = f(x, y), the variables x and y that are chosen freely; in economics, the exogenous ones. See Definition 8.13.

A quotient f(x)/g(x) is of indeterminate form 0/0 at a if numerator and denominator both tend to 0 as x → a, and of form ∞/∞ if both tend to ±∞. The limits of the parts do not determine the limit of the quotient. See Definition 5.18.

A level curve of a utility function: the bundles (x, y) a consumer likes equally, U(x, y) = u₀. Higher curves are better.

A sum a₁ + a₂ + a₃ + ⋯ with infinitely many terms. Its sum is the limit of its partial sums, if that limit exists and is finite. See Definition 8.5.

An interior point c where f″ changes sign: the curve bends one way on one side of c and the other way on the other. f″(c) = 0 there, but f″(c) = 0 alone is not enough. See Definition 7.6.

The number a · b = a₁b₁ + a₂b₂ + … + aₙbₙ for two n-vectors of the same dimension: multiply in pairs and add. See Definition 12.13.

The product rule read backwards: the integral of u·v′ equals u·v minus the integral of u′·v. It trades one integral for another. See Fact 7.21.

The chain rule read backwards: with u = g(x) and du = g′(x) dx, the integral of f(g(x))·g′(x) dx becomes the integral of f(u) du. The limits of a definite integral move with the variable. See Fact 8.1 and Method 8.2.

The number $b = f(0)$ in $f(x) = ax + b$: where the line crosses the $y$-axis.

If f is continuous on a closed interval [a, b], then f takes every value between f(a) and f(b) somewhere on [a, b]. In particular, if f(a) and f(b) have opposite signs, f has a root in (a, b). See Fact 6.1.

X is an inverse of the square matrix A if AX = XA = I. There is at most one, written A⁻¹. See Definition 14.15.

A square matrix that has an inverse. A square matrix is invertible exactly when its determinant is not zero. See Definition 14.15 and Fact 14.18.

A discontinuity at a where the limit of f(x) as x → a does not exist, so that no choice of the value f(a) can repair it. See Definition 5.15.

A level curve of a production function: the input combinations (K, L) that give one and the same output, F(K, L) = Y₀.

For an indeterminate form 0/0 or ∞/∞, the limit of f/g equals the limit of f′/g′ whenever that limit exists: numerator and denominator are differentiated separately. If f(a) = g(a) = 0 and g′(a) is not 0, the limit is f′(a)/g′(a). See Fact 5.19.

The constant λ in the Lagrangian ℒ = f − λ[g − c]. At the optimum it is the rate at which the optimal value changes with the constraint level c (Fact 11.21). See Definition 11.13.

For max (min) f(x, y) subject to g(x, y) = c, the function ℒ(x, y) = f(x, y) − λ[g(x, y) − c]. It agrees with f on the constraint set. See Definition 11.13.

The first non-zero entry of a row. In a staircase each row's leading entry stands strictly further right than the one above, and Gaussian elimination scales it to 1 (Method 13.16).

The term of highest degree of a polynomial, such as 3x² in 3x² − 5x + 1. As x → ∞ it decides the size of the polynomial.

The set of points (x, y) where f(x, y) = c, for one fixed height c. See Definition 9.1.

The number that $f(x)$ approaches as $x$ approaches $a.$ Exactly: $f(x)$ can be made as close to that number as we please by taking $x$ sufficiently close to $a,$ but not equal to $a.$ The value $f(a)$ plays no part. See Definition 3.1; Lecture 5 makes it precise.

Near a point a, f(x) ≈ f(a) + f′(a)(x − a): the function replaced by its tangent line at a, the only line with the same value and the same slope as f there. See Fact 4.15.

For n-vectors a and b and scalars t and s, the n-vector ta + sb. See Definition 12.12.

A function that can be written $f(x) = ax + b$ for constants $a$ and $b;$ its graph is a straight line.

A local maximum point or a local minimum point. See Definition 7.1.

A point c such that f(x) ≤ f(c) for every x in some interval around c, however small. It beats its neighbours, not necessarily the whole domain. See Definition 7.1.

A point c such that f(x) ≥ f(c) for every x in some interval around c, however small. See Definition 7.1.

To differentiate y = f(x): take logarithms of both sides and simplify, differentiate implicitly (the left-hand side gives y′/y), then multiply by y. See Method 4.12.

The entries a₁₁, a₂₂, …, aₙₙ of a square matrix of order n.

The derivative C'(Q) of the total cost C(Q): the extra cost of producing roughly one more unit. In economics “marginal” always means a derivative; R'(Q) is marginal revenue.

On an indifference curve, the rate at which a consumer is willing to give up y for x and stay equally happy: the slope −U′₁/U′₂ of the curve (Fact 9.15), often quoted without its minus sign.

A rectangular array of numbers with m rows and n columns, written A = (aᵢⱼ) with m × n. See Definition 12.1.

For A of order m × n and B of order n × p, the m × p matrix AB whose entry in row i, column k is the inner product of row i of A and column k of B. See Definition 12.17.

A point c in the set S with f(x) ≤ f(c) for every x in S. The number f(c) is then the maximum value of f over S. See Definition 6.8.

The value f(c) at a maximum point c: the largest value f takes on S. See Definition 6.8.

A point c in the set S with f(x) ≥ f(c) for every x in S. The number f(c) is then the minimum value of f over S. See Definition 6.8.

The value f(c) at a minimum point c: the smallest value f takes on S. See Definition 6.8.

What is obtained by differentiating $f$ again and again, $n$ times in all. The third derivative is written $f'''.$ See Definition 3.21.

The exponential function f(x) = eˣ, whose base is the number e = 2.718…; also written exp(x). See Definition 2.7.

For a positive number a, ln a is the power of e needed to get a. So ln 1 = 0 and ln e = 1. See Definition 2.8.

The quotient (f(a + h) − f(a)) / h: the slope of the secant through (a, f(a)) and (a + h, f(a + h)), and the average rate of change of f between a and a + h.

A method for solving f(x) = 0. From a guess x₀ put x₁ = x₀ − f(x₀)/f′(x₀), the point where the tangent at x₀ meets the x-axis, and repeat from x₁. See Method 6.5.

A square matrix whose determinant is not 0; the same as invertible. See Fact 14.18.

The left-hand limit of f at a is the number f(x) approaches as x approaches a from below (x < a); the right-hand limit, from above (x > a). The ordinary limit exists exactly when both exist and are equal. See Definition 5.7 and Fact 5.8.

What must be given up of one good to get one more unit of another; on the budget line, $p/q$ units of good 2 per unit of good 1.

The order of a matrix is m × n: its number of rows, then its number of columns. See Definition 12.1.

The graph of a quadratic function: $\cup$-shaped if $a > 0$ and $\cap$-shaped if $a < 0.$

The derivative of f(x, y) with respect to one variable while the other is held fixed: ∂f/∂x = f′₁(x, y) holds y fixed, ∂f/∂y = f′₂(x, y) holds x fixed. See Definition 8.15.

The partial elasticity of z = f(x, y) with respect to x is (x/z)·∂z/∂x: the percentage change in z per one per cent change in x, with y held fixed. See Definition 9.13.

The n-th partial sum of a series is the sum of its first n terms, sₙ = a₁ + ⋯ + aₙ. See Definition 8.5.

One solution s of Ax = b, often found by giving the free unknowns the value 0. All solutions are s plus the solutions of Ax = 0 (Fact 13.23).

A payment of the same amount at the end of every period, for ever. At interest rate r a perpetuity paying P is worth P/r today. See Example 8.8.

A function of the form $P(x) = a_nx^n + \cdots + a_1x + a_0$ with constants $a_0, \ldots, a_n.$

A function of the form f(x) = A·xʳ for positive x, where A and r are constants: the base varies and the exponent is fixed. See Definition 2.1.

What a payment arriving later is worth today: at interest rate r per period, a payment P arriving in n periods has present value P/(1 + r)ⁿ, the sum which, invested today, grows to P by then. See Example 8.8.

The derivative of a product: $(fg)'=f'g+fg'.$ Each factor is differentiated in turn while the other is kept. See Fact 3.8.

A rational function $P(x)/Q(x)$ in which the degree of $P$ is less than the degree of $Q.$

A function of the form $f(x) = ax^2 + bx + c$ with $a \ne 0.$

The polynomial $q$ in $P(x) = q(x)\,Q(x) + r(x)$: the result of dividing $P$ by $Q,$ apart from the remainder.

The derivative of a quotient $f/g$ is $f'g-fg'$ divided by the square of $g,$ where $g(x)\neq 0.$ See Fact 3.11.

A quotient $R(x) = P(x)/Q(x)$ of two polynomials, defined for all $x$ with $Q(x) \ne 0.$

The relative rate of change of $f$ at $x$ is the quotient $f'(x)/f(x),$ the change in $f$ per unit of $f$ itself. Multiplied by $100$ it is the percentage rate of change. See Definition 3.14.

The polynomial $r$ in $P(x) = q(x)\,Q(x) + r(x);$ its degree is less than the degree of $Q.$

The error Rₙ₊₁(x) = f(x) − pₙ(x) of a Taylor approximation. By Taylor's formula it equals f⁽ⁿ⁺¹⁾(z)(x − a)ⁿ⁺¹/(n+1)! for some z between a and x, and it is at most M|x − a|ⁿ⁺¹/(n+1)! in size when |f⁽ⁿ⁺¹⁾| ≤ M there. See Facts 5.1 and 5.4.

A discontinuity at a where the limit of f(x) as x → a exists: redefining f(a) to be that limit makes f continuous at a. See Definition 5.15.

What scaling all inputs by t does to output. For A·xᵃ·yᵇ output is multiplied by t to the power a + b: increasing returns if a + b > 1, constant if a + b = 1, decreasing if a + b < 1. See Example 8.14.

A number $x_0$ with $P(x_0) = 0;$ also called a zero of $P.$

A matrix with only one row, written as a bold lower-case letter. Its entries are its components.

A critical point every disc around which contains a point where f is larger and a point where f is smaller. It is neither a local maximum nor a local minimum. See Definition 11.2.

A shortcut for order 3 only: copy the first two columns to the right, add the three products falling to the right and subtract the three rising. It does not generalise. See Method 14.9.

A real number, as opposed to a matrix or a vector. A matrix is multiplied by a scalar entry by entry. See Definition 12.8.

The straight line through two points P and Q of a graph. Its slope is the Newton quotient.

The derivative of the derivative of $f,$ written $f''(x)$ or $y''.$ It is the rate at which the slope changes. See Definition 3.21.

At an interior point c with f′(c) = 0: f″(c) < 0 gives a local maximum, f″(c) > 0 a local minimum, and f″(c) = 0 gives no information. See Fact 7.3.

A partial derivative of a partial derivative. There are four: f″₁₁, f″₁₂ (first argument, then second), f″₂₁ and f″₂₂. The two mixed ones, f″₁₂ and f″₂₁, agree for essentially every function in economics (Fact 8.21). See Definition 8.19.

The multiplier λ read as what one more unit of a constrained resource is worth to the decision maker: the change in the optimal value per unit of the constraint level. Its units are those of f per unit of g.

A square matrix with determinant 0. A singular matrix has no inverse. See Fact 14.18.

The number $a$ in $f(x) = ax + b$: the change in $f(x)$ when $x$ increases by one unit.

An ordered list (s₁, …, sₙ) that satisfies all the equations of a system simultaneously. See Definition 12.5.

A matrix with as many rows as columns (m = n): square of order n. Only a square matrix has a main diagonal.

A point c at which f′(c) = 0. The tangent there is horizontal. A critical point is the same thing. See Definition 6.12.

A local maximum point (x₀, y₀) where f(x, y) < f(x₀, y₀) for every other point near it, not just ≤. A strict local minimum point reverses the inequality. See Definition 11.1.

Concave with the chord strictly below the graph between its two ends; strictly convex likewise with the chord strictly above. See Definition 6.19.

f is strictly decreasing on an interval if f(x₂) is smaller than f(x₁) whenever x₂ is larger than x₁ in the interval. See Definition 2.22.

f is strictly increasing on an interval if f(x₂) is larger than f(x₁) whenever x₂ is larger than x₁ in the interval. See Definition 2.22.

A square matrix A with Aᵀ = A, that is, with the entry in row i, column j equal to the entry in row j, column i for all i, j. See Definition 13.11.

m equations aᵢ₁x₁ + … + aᵢₙxₙ = bᵢ in n unknowns x₁, …, xₙ, to hold at the same time; aᵢⱼ is the coefficient of xⱼ in equation i. See Definition 12.5.

A straight line that just touches a curve at a point P, running in the same direction as the curve does there. At the point (a, f(a)) it is the line y = f(a) + f'(a)(x − a). See Definition 2.15.

The Taylor polynomial of degree n of f about a is the polynomial whose value and first n derivatives at a are those of f: the sum of f⁽ᵏ⁾(a)(x − a)ᵏ/k! for k = 0, …, n. It approximates f near a. See Definition 4.19.

f(x) = pₙ(x) + Rₙ₊₁(x): the Taylor polynomial of degree n plus a remainder that has the form of the next term, with the derivative f⁽ⁿ⁺¹⁾ taken at some unknown point z between a and x. See Fact 5.1.

The base a > 0 for which the limit of (aʰ − 1)/h as h tends to 0 is exactly 1. It is irrational, e = 2.71828…, and it is the one base whose exponential is its own derivative. See Definition 4.1.

The transpose Aᵀ of an m × n matrix A is the n × m matrix whose r-th row is the r-th column of A: the array reflected in its main diagonal. See Definition 13.7.

The solution x = 0 of a homogeneous system Ax = 0. Every homogeneous system has it, so it is never inconsistent.

f*(c): the best attainable value of the objective when the constraint level is c. Its derivative is the multiplier: df*/dc = λ(c) (Fact 11.21). See Definition 11.20.

A matrix with only one row or only one column. A vector with n components is an n-vector.

The highest or lowest point of a parabola; it lies at $x^{*} = -b/2a.$

A set of points in the plane is the graph of a function if and only if every vertical line meets it at most once.

The m × n matrix 0 all of whose entries are 0. Adding it changes nothing: A + 0 = A (Fact 12.10).