APPENDIX
B

Glossary

absolute extrema point The highest or lowest point on a graph.

acceleration The rate of change of velocity.

accumulation function A function defined by a definite integral; it has a variable in one or both of its limits of integration.

antiderivative The opposite of the derivative; if f(x) is an antiderivative of g(x), then , where C is a constant.

antidifferentiation The process of creating an antiderivative or integral.

asymptote A line representing an unattainable value that shapes a graph; because the graph cannot achieve the value, the graph bends toward that line but won’t intersect it.

average value of a function The value, f(c), guaranteed by the Mean Value Theorem for Integration found via the equation .

Chain Rule The derivative of the composite function h(x) = f(g(x)) is h′(x) = f′(g(x)) ∙ g′(x).

cofunction Trigonometric functions with the same name, apart from the prefix “co-,” like sine and cosine or tangent and cotangent.

concavity Describes how a curve bends; a curve that can hold water poured into it from the top of the graph is concave up, whereas one that cannot hold water is concave down.

conjugate A binomial whose middle sign is the opposite of another binomial with the same terms (e.g., are conjugates).

constant A polynomial of degree 0; a real number.

constant of integration The unknown constant that results from an indefinite integral, usually written as C in your solution; it is a required piece of all indefinite integral solutions.

continuous A function f(x) is continuous at x = c if .

coterminal angles Angles that have the same function value, because the space between them is a multiple of the function’s period.

critical number An x-value that causes a function to equal zero or become undefined.

cubic A polynomial of degree 3.

definite integral An integral that contains limits of integration; its solution is a real number.

degree The largest exponent in a polynomial.

derivative The derivative of a function f(x) at x = c is the slope of the tangent line to f at x = c, usually written f′ (c).

difference quotient One of two formulas that define a derivative:

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differentiable Possessing a derivative at the specific x-value; if a function does not have a derivative at the given x-value, it is said to be “nondifferentiable” there.

differential equation An equation containing a derivative.

displacement The total change in position counting only the beginning and ending position; if the object in question changes direction any time during that interval of time, it does not correctly reflect the total distance traveled.

domain The set of possible inputs for a function.

essential discontinuity See infinite discontinuity.

Euler’s Method A technique used to approximate solutions to a differential equation when you can’t apply separation of variables.

everywhere continuous A function that is continuous at every x in its domain.

exponential growth and decay A population grows or decays exponentially if its rate of change is proportional to the population itself—in other words, , where k is a constant and P is the size of the population.

extrema point A high or low point in the curve, a maximum or a minimum, respectively; it represents an extreme value of the graph, whether extremely high or extremely low, in relation to the points surrounding it.

Extreme Value Theorem If a function f(x) is continuous on the closed interval [a,b], then f(x) has an absolute maximum and an absolute minimum on [a,b].

factoring Reversing the process of multiplication. The results of the factoring process can be multiplied together to get the original quantity.

family of solutions Any mathematical solution containing “+ C”; it compactly represents an infinite number of possible solutions, each differing only by a constant.

function A relation such that every input has exactly one matching output.

greatest common factor The largest quantity by which all the terms of an expression can be divided evenly.

implicit differentiation Allows you to find the slope of a tangent line when the equation in question cannot be solved for y.

indefinite integral An integral that does not contain limits of integration; its solution is the antiderivative of the expression (and must contain a constant of integration).

indeterminate form An expression whose value is unclear; the most common indeterminate forms are .

infinite discontinuity Discontinuity caused by a vertical asymptote. Also called essential discontinuity.

inflection points Points on a graph where the concavity changes.

integer A number without a decimal or fractional part.

integral The opposite of the derivative; if f(x) is the integral of g(x), then , where C is a constant.

integration The process of creating an antiderivative or integral.

intercept Numeric value at which a graph hits either the x- or y-axis.

Intermediate Value Theorem If a function f(x) is continuous on the closed interval [a,b], then for every real number d between f(a) and f(b), there exists a c between a and b so that f(c) = d.

irrational root An x-intercept that cannot be written as a fraction.

jump discontinuity Occurs when no general limit exists at the given x-value because the left- and right-hand limits are not equal.

left sum A Riemann approximation in which the heights of the rectangles are defined by the values of the function at the left-hand side of each interval.

left-hand limit The height a function intends to reach as you approach the given x-value from the left.

L’Hôpital’s Rule If a limit results in an indeterminate form after substitution, you can take the derivatives of the numerator and denominator of the fraction separately without changing the limit’s value .

limit The height a function intends to reach at a given x-value, whether or not it actually reaches it.

limits of integration Small numbers next to the integral sign, indicating the boundaries when calculating area under the curve; in the expression , the limits of integration are 1 and 3.

linear approximation The equation of a tangent line to a function used to help approximate the function’s values lying close to the point of tangency.

linear expression A polynomial of degree 1.

logistic growth Begins quickly (it initially looks like exponential growth) but eventually slows and levels off to some limiting value; most natural phenomena, including population and sales graphs, follow this pattern rather than exponential growth.

Mean Value Theorem If a function f(x) is continuous and differentiable on a closed interval [a,b], then there exists a point c, a ≤ c ≤ b, so that .

Mean Value Theorem for Integration If a function f(x) is continuous on the interval [a,b], then there exists a c, a ≤ c ≤ b, such that .

midpoint sum A Riemann approximation in which the heights of the rectangles are defined by the values of the function at the midpoint of each interval.

nondifferentiable Not possessing a derivative.

nonremovable discontinuity A point of discontinuity for which no limit exists (e.g., infinite or jump discontinuity).

normal line The line perpendicular to a function’s tangent line at the point of tangency.

optimizing Finding the maximum or minimum value of a function given a set of circumstances.

parameter A variable into which you plug numeric values to find coordinates on a parametric equation graph.

parametric equations Pairs of equations, usually in the form of “x =” and “y =,” that define points of a graph in terms of yet another variable, usually t or θ.

period The amount of horizontal space it takes a periodic function to repeat itself.

periodic function A function whose values repeat over and over after a fixed interval.

point discontinuity Occurs when a general limit exists but the function value is not defined.

point-slope form A line containing the point (x₁,y₁) with slope m has equation y – y₁ = m(x – x₁).

position equation A mathematical model that outputs an object’s position at a given time, t.

Power Rule for Differentiation The derivative of the expression axⁿ with respect to x, where a and n are real numbers, is (a ∙ n)x^n–1.

Power Rule for Integration The integral of a single variable to a real-number power is found by adding 1 to the existing exponent and dividing the entire expression by the new exponent , assuming n ≠ –1.

Product Rule The derivative of f(x)g(x), with respect to x, is f(x) ∙ g ′(x) + f ′(x) ∙ g(x).

quadratic A polynomial of degree 2.

Quotient Rule If .

range The set of possible outputs for a function.

reciprocal The fraction with its numerator and denominator reversed (e.g., the reciprocal of is ).

related rates A problem that uses a known rate of change to compute the rate of change for another variable in the problem.

relation A collection of related numbers, usually described by an equation.

relative extrema point Occurs when that point is higher or lower than all of the points in the immediate surrounding area; visually, a relative maximum is the peak of a hill in the graph, and a relative minimum is the lowest point of a dip in the graph.

removable discontinuity A point of discontinuity for which a limit exists (i.e., point discontinuity).

Riemann sum An approximation for the area beneath a curve calculated by adding the areas of rectangles.

right sum A Riemann approximation in which the heights of the rectangles are defined by the values of the function at the right-hand side of each interval.

right-hand limit A function’s intended height as you approach the given x-value from the right.

Rolle’s Theorem If a function f(x) is continuous and differentiable on a closed interval [a,b] and f(a) = f(b), then there exists a c between a and b such that f′(c) = 0.

secant line A line that cuts through a graph, usually intersecting it in multiple locations.

separation of variables A technique used to solve basic differential equations; in it, you move the different variables of the equation to different sides of the equal sign in order to integrate each side of the equation separately.

sign graph See wiggle graph.

Simpson’s Rule The approximate area under the curve f(x) on the closed interval [a,b] using an even number of subintervals, n, is: .

slope Numeric value that describes the “slantiness” of a line.

slope field A tool to visualize the solution of a differential equation; a collection of line segments centered at points whose slopes are the values of the differential equation evaluated at those points.

slope-intercept form A line with slope m and y-intercept b has equation y = mx + b.

speed The absolute value of velocity.

symmetric function A function that looks like a mirror image of itself, typically across the x-axis, y-axis, or about the origin. Symmetry across the x-axis is possible as well but results in a graph that is not a function.

tangent line A line that skims across a curve, hitting it only once at the indicated location.

Trapezoidal Rule The approximate area beneath a curve f(x) on the interval [a,b] using n trapezoids is: .

u-substitution Integration technique that is useful when a function and its derivative appear in an integral.

velocity The rate of change of position; it includes a component of direction, and therefore may be negative.

vertical line test Tests whether or not a graph is a function; if any vertical line can be drawn through the graph that intersects the graph more than once, then the graph in question cannot be a function.

wiggle graph A segmented number line that describes the direction of a function and the signs of its derivative.