← Return to where you were reading
Glossary
Definitions of key terms as they are used in the HTML sections of Calculus: Newton, Leibniz, and Robinson Meet Technology. Each source link opens the textbook section that establishes or reviews the term.
No glossary entries match that search.
A
- Absolute approximation error
- The absolute value of the difference between an exact value and its approximation.
- Source: Section 10.1
- Acceleration at a moment
- The rate at which a rate-of-change function varies during a moment.
- Source: Section 7.2
- Accumulation function
- A function generated by accumulating variations in its value, where those variations occur at some rate over moments of its independent variable.
- Source: Section 5.1
- Antiderivative
- A function whose exact rate-of-change function (derivative) is a specified function. Any two antiderivatives of the same function differ by a constant.
- Source: Section 8.0
- Approximate rate of change function
- A function whose values approximate another function’s rate of change, commonly by using a difference quotient over a nonzero interval.
- Source: Section 6.1
- Argument
- The expression that represents the value at which a function is actually evaluated.
- Source: Section 3.14
C
- Change
- In this textbook, replacing one thing with another. This is distinguished from vary (change in progress) and variation (completed change).
- Source: Section 3.1
- Closed form definition
- A function definition that specifies how to compute its values in a finite number of operations on variables or familiar functions.
- Source: Section 8.0
- Closed interval
- An interval that contains both endpoints, conventionally written with brackets, such as [2, 3].
- Source: Section 7.1
- Completed variation
- Accumulation over all completed incremental intervals before the interval containing the independent variable’s current value.
- Source: Section 5.1
- Constant
- A number or symbol used to represent a quantity whose measure is the same within all situations under consideration.
- Source: Section 8.0
- Constant rate of change
- A relationship between two smoothly and continuously varying quantities in which all variations in one are proportional to corresponding variations in the other.
- Source: Section 4.1
- Continuous function
- A function that is not discontinuous at any value in the interval under consideration.
- Source: Section 3.17
- Convention
- One of several possible ways of doing or writing something that a community agrees to use in order to aid communication; a convention is not a rule.
- Source: Section 3.5
- Correspondence point
- A point whose coordinates are values that co-occur in related quantities; its location represents those values simultaneously.
- Source: Section 3.9
- Current variation
- Accumulation within the current incremental interval, from that interval’s left endpoint to the independent variable’s current value.
- Source: Section 5.1
D
- Decreasing function
- A function f is decreasing over an interval when f(a) is greater than f(b) whenever a is less than b for all a and b in that interval.
- Source: Section 3.17
- Dependent variable
- The variable whose values are determined by values of the independent variable.
- Source: Section 3.17
- Derivative
- The exact rate-of-change function derived from a known exact accumulation function.
- Source: Section 8.0
- Differential change
- A varying change in a variable’s value, typically varying within an interval of fixed length represented by an incremental change such as Δx.
- Source: Section 8.0
- Discontinuous function
- A function that, at some value in its domain, fails to keep all nearby output values within every chosen tolerance over any sufficiently small surrounding interval.
- Source: Section 3.17
- Domain of a function
- All values of a function’s independent variable for which the function has a value.
- Source: Section 3.17
E
- Essentially equal
- Two function values are essentially equal at a moment when, for any chosen tolerance, there is a sufficiently small surrounding interval throughout which their difference stays within that tolerance.
- Source: Section 4.9
- Even function
- A function f for which f(−x) = f(x) for every x in its domain; its Cartesian graph is symmetric about the y-axis.
- Source: Section 3.17
- Exact accumulation function
- A function whose values give exact accumulated amounts; it can be reconceptualized as a starting value plus exact net accumulation from an exact rate-of-change function.
- Source: Section 6.1
- Exact net accumulation
- The net amount accumulated from a starting value a to a current value x as accumulation proceeds at an exact rate over infinitesimal intervals.
- Source: Section 8.0
- Exact rate of change function
- A function whose value at each argument gives another function’s momentary exact rate of change at that argument.
- Source: Section 4.9
- Exponential function
- A function of the form f(x) = bˣ, in which the exponent varies and the base is constant.
- Source: Section 6.3
F
- Function
- A relationship between an independent and a dependent variable in which each value of the independent variable is related to exactly one value of the dependent variable.
- Source: Section 3.17
- Fundamental Theorem of Calculus
- For an exact accumulation function f with exact rate-of-change function r, the net accumulation of r from a to x equals f(x) − f(a).
- Source: Section 6.2
G
- Global maximum
- A maximum value of a function that is greater than every other value of the function on the interval under consideration.
- Source: Section 7.1
- Global minimum
- A minimum value of a function that is less than every other value of the function on the interval under consideration.
- Source: Section 7.1
- Graph of a mathematical statement
- The set of ordered pairs whose coordinate values make a mathematical statement involving two variables true.
- Source: Section 3.5
I
- Increasing function
- A function f is increasing over an interval when f(a) is less than f(b) whenever a is less than b for all a and b in that interval.
- Source: Section 3.17
- Incremental change
- A fixed-size change in a variable’s value, commonly represented by notation such as Δx, Δy, or Δt.
- Source: Section 8.0
- Independent variable
- The variable whose values are controlled or taken as determining the dependent variable’s values.
- Source: Section 3.17
- Integral
- The function representing exact net accumulation from a known exact rate-of-change function over an interval.
- Source: Section 8.0
- Integrating
- Determining exact net accumulation from a known exact rate-of-change function.
- Source: Section 8.0
- Interval
- A set of real numbers between two endpoint values; endpoints may be included or excluded.
- Source: Section 3.17
- Inverse function
- Functions h and k are inverses when each undoes the other: applying one and then the other returns the original input throughout the relevant domains.
- Source: Section 3.18
L
- Local maximum
- A maximum value relative to nearby values that is not the global maximum on the interval under consideration.
- Source: Section 7.1
- Local minimum
- A minimum value relative to nearby values that is not the global minimum on the interval under consideration.
- Source: Section 7.1
M
- Maximum value
- A function value at x = k where the exact rate of change is zero and changes from positive to negative around k.
- Source: Section 7.1
- Minimum value
- A function value at x = k where the exact rate of change is zero and changes from negative to positive around k.
- Source: Section 7.1
N
- Net accumulation function
- A function whose values give the net amount accumulated from a starting value to a current value as accumulation proceeds at a specified rate.
- Source: Section 9.1
O
- Odd function
- A function f for which f(−x) = −f(x) for every x in its domain; its Cartesian graph has 180-degree rotational symmetry about the origin.
- Source: Section 3.17
- One-to-one function
- A function in which no two different values of the independent variable are related to the same dependent-variable value.
- Source: Section 3.17
- Open form definition
- A function definition that gives a conceptual outline of the function’s meaning without finite instructions for computing each value.
- Source: Section 8.0
- Open interval
- An interval that excludes both endpoints, conventionally written with parentheses, such as (2, 3).
- Source: Section 7.1
P
- Parameter
- A notation representing a quantity that is constant in one situation but can vary from one situation to another.
- Source: Section 8.0
- Parameterization
- A representation that defines coordinates or related variables as functions of a common parameter.
- Source: Section 12.1
- Parametrically defined relationship
- A relationship between variables in which each variable is defined as a function of an additional common variable.
- Source: Section 12.1
- Point of inflection
- A value x = k where a function’s continuous second-order rate-of-change function is zero and changes sign around k.
- Source: Section 7.2
- Power function
- A function of the form f(x) = xᵇ, in which the exponent is constant.
- Source: Section 6.3
Q
R
- Range of a function
- All values that a function relates to values of its independent variable.
- Source: Section 3.17
- Rate of change at a moment
- The value that makes a function’s nearby values essentially equal to its current value plus that rate multiplied by a differential change through a sufficiently small interval.
- Source: Section 4.9
- Relative approximation error
- The absolute approximation error divided by the absolute value of the nonzero exact value.
- Source: Section 10.1
- Riemann sum
- The name used in many textbooks for approximating accumulation by treating the rate as constant over each incremental interval.
- Source: Section 10.1
S
- Simpson’s rule
- The name used in many textbooks for the quadratic method of approximating accumulation.
- Source: Section 10.1
T
- Taylor polynomial
- A polynomial approximation built so its value and successive rate-of-change values agree with those of a target function at a selected value.
- Source: Section 10.2
- Trapezoid rule
- The name used in many textbooks for approximating accumulation by assuming constant acceleration, so the rate is approximated linearly over each interval.
- Source: Section 10.1
U
- Unsimplified expression
- An expression left in a form that preserves its component operations and can retain information about the quantities and relationships used to construct it.
- Source: Section 8.0
V
- Variable
- A notation used to represent the value of a quantity whose measure varies within a situation.
- Source: Section 8.0