Glossary of Key Terms
Glossary of Key Terms
| Term | Definition |
| Average Cost (AC) | The total cost divided by the number of units produced. AC(q) = C(q) / q. |
| Average rate of change | The difference quotient, [f(x + Δx) – f(x)] / Δx, representing the change in a function y with respect to a change in its variable x. |
| Break-Even Analysis | A method used in business to determine the number of units that must be sold for total revenue to equal total cost. |
| Chain Rule | A rule for finding the derivative of a composite function. For functions of one variable, if y is a function of u and u is a function of x, then dy/dx = (dy/du) * (du/dx). For partial derivatives, if z is a function of x and y, which are both functions of t, then dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt). |
| Cobb-Douglas Production Function | A formula used to model production levels, where output Q is regarded as a function of capital investment K and labor force size L, often in the form Q(K, L) = AK^α L^(1-α). |
| Composite Function | A function formed from two functions, g(u) and h(x), by substituting h(x) for u in the formula for g(u), resulting in g[h(x)]. |
| Concave downward | The shape of a function’s graph on an interval where its derivative f’ is decreasing. This occurs where the second derivative f” is less than zero. |
| Concave upward | The shape of a function’s graph on an interval where its derivative f’ is increasing. This occurs where the second derivative f” is greater than zero. |
| Corner Point | In a feasibility region, a point where two of the boundary lines intersect. |
| Critical Point | A point on a function’s graph where the derivative is either zero or undefined. For a function of two variables, f(x, y), it is a point (a, b) where both fx(a, b) = 0 and fy(a, b) = 0. |
| Dependent variable | The output variable of a function (often y), whose value depends on the independent variable. |
| Derivative | The limit of the difference quotient as Δx approaches zero, representing the instantaneous rate of change of a function. Denoted as f'(x) or dy/dx. |
| Difference quotient | The ratio [f(x + Δx) – f(x)] / Δx, which represents the average rate of change of a function over a small interval. |
| Differential | An expression used to approximate the change in a function. For a function y = f(x), the differential dy is f'(x)Δx. For z = f(x, y), the total differential dz is (∂z/∂x)Δx + (∂z/∂y)Δy. |
| Domain of a Function | The set of all values of the independent variable(s) for which a function can be evaluated. |
| Elasticity of Demand (η) | A measure of the sensitivity of demand to price changes, given by the formula η = (p/q) * (dq/dp). It represents the percentage change in demand due to a 1 percent increase in price. |
| Feasibility Region | In linear programming, the solution set of the system of inequality constraints, representing all possible valid solutions. |
| Function (of one variable) | A rule that matches each number x in a set (the domain) with exactly one number y in another set (the range). |
| Function (of two variables) | A rule that assigns to each ordered pair of real numbers (x, y) in a set one and only one real number f(x, y). |
| Independent variable | The input variable of a function (often x), which can be chosen freely. |
| Inflection Point | A point on the graph of a function where the concavity changes (from upward to downward, or vice versa). |
| Instantaneous Rate of Change | The rate of change of a function at a specific point, given by its derivative. |
| Lagrange Multipliers | A technique used to find the relative maximum or minimum of a function of two variables subject to a constraint. The multiplier λ represents the rate of change of the optimal value M with respect to a change in the constraint constant K. |
| Linear Function | A function that changes at a constant rate with respect to its independent variable, with an equation of the form y = mx + b. Its graph is a straight line. |
| Linear Programming (LP) | A method for optimizing (maximizing or minimizing) a linear objective function subject to a system of linear constraint inequalities. |
| Marginal Analysis | In economics, the use of the derivative to approximate the change in a function resulting from a 1-unit change in its variable. |
| Marginal Cost | The derivative of the total cost function, C'(x), which approximates the cost of producing the (x + 1)st unit. |
| Marginal Revenue | The derivative of the total revenue function, R'(x), which approximates the revenue derived from the sale of the (x + 1)st unit. |
| Market Equilibrium | The point where the supply and demand curves intersect. The corresponding price is the equilibrium price, at which the quantity supplied equals the quantity demanded. |
| Objective Function | In linear programming, the linear function that is to be optimized (maximized or minimized). |
| Parabola | The U-shaped graph of a quadratic function y = ax^2 + bx + c (where a ≠ 0). |
| Partial Derivative | The derivative of a function of multiple variables with respect to one of those variables, treating the other variables as constants. |
| Percentage Rate of Change | The rate of change of a quantity expressed as a percentage of the size of that quantity; given by the formula 100 * [f'(x) / f(x)]. |
| Point-Slope Form | An equation of a line, y – y0 = m(x – x0), that passes through the point (x0, y0) and has slope m. |
| Range of a Function | The set of all possible output values (y or f(x)) of a function. |
| Relative Maximum/Minimum | A point on a graph that is higher (maximum) or lower (minimum) than any nearby points. Also called local extremum. |
| | Saddle Point | A critical point of a function of two variables that is neither a relative maximum nor a relative minimum. | | Second Derivative | The derivative of a function’s derivative, denoted by f”(x) or d²y/dx². It measures the rate of change of the rate of change. | | Second Partials Test | A test using second-order partial derivatives to classify a critical point of a function of two variables as a relative maximum, relative minimum, or saddle point. | | Slope | A measure of the steepness of a line; the amount by which the y-coordinate changes when the x-coordinate is increased by 1. Formula: (y2 – y1) / (x2 – x1). | | Slope-Intercept Form | An equation of a line, y = mx + b, whose slope is m and whose y-intercept is the point (0, b). | | Total differential | An expression that approximates the change in a function of two or more variables, given by dz = (∂z/∂x)Δx + (∂z/∂y)Δy. | | Vertex | The “peak” or “valley” of a parabola. |