APPENDIX
C

Glossary

absolute convergence Describes when the series converges. If converges, then so does .

acceleration The rate of change of the velocity.

alternating series A series whose consecutive terms alternate between positive and negative values.

Alternating Series Test Let a_n = (-1)ⁿb_n with b_n > 0 for all n. If b_n is decreasing for all n, and then is convergent.

antiderivative The expression from which the derivative was found.

average value of a function The average value of a function f(x) on the interval [a,b] is .

Chain Rule The rule used for differentiating functions that are composed of other functions. If f(x) = g(k(x)), then f'(x) = g'(k(x))k'(x).

Comparison Test If two positive series, and , with the property that a_k > b_k for k ≥ m, then we can conclude that if the series converges, then so does the series . Also, if the series diverges, then so does the series .

Comparison Test for Improper Integrals If f(x) and g(x) are continuous functions for x > a with f(x) > g(x) > 0, then (1) If converges, then so does If diverges, then so does .

conditionally convergent A series that is convergent but not absolutely convergent.

constant of integration The constant added to the end of an indefinite integral to indicate there is a family of functions that satisfy the given integral.

continuous at a point A function is continuous at a point x = c if and only if .

continuous function A function that is continuous at all points in its domain.

converge A sequence or series that is bounded by some finite value.

decreasing sequence A sequence in which a_{n + 1} ≤ a_n for all n.

definite integral An integral that contains a lower and upper bound and whose answer is a real number.

differential equation An equation that contains a derivative.

disk method The method used to calculate the volume of a solid of rotation when the interior of the solid contains no holes.

displacement The change in position of an object from the start to an end of its travel. This is not necessarily the same as the distance traveled.

Divergence Test The infinite series is divergent if .

divergent A sequence or series that does not have a limiting value.

dot product Given two vectors a = (a₁,a₂) and b = (b₁,b₂), the dot product is .

Euler’s Method A technique used to approximate a value of a function using a known value of the function and the derivative of the function.

exponential growth A process in which the rate of growth of a function is proportional to the current value of the function.

First Order Linear Differential Equation A differential equation of the form .

Fundamental Theorem of Calculus Given the function F(x) whose derivative is f(x), the Fundamental Theorem of Calculus says that .

geometric sequence A sequence in which the ratio of consecutive terms is a constant.

geometric series The sum of the terms of a geometric sequence.

implicit differentiation When equations cannot be written in the form y = f(x), the notation indicates that it is implied that y is a function of x so that whenever a term in y is differentiated, the Chain Rule needs to be applied by attaching the term to it.

improper integral An integral that has at least one of its bounds going to infinity or an integral that contains an infinite discontinuity within the bounds of integration.

increasing sequence A sequence in which a_{n +1} ≥ a_n for all n.

indefinite integral An integral that yields a family of functions each of whose derivative is the integrand of the integral.

infinite discontinuity Discontinuity caused by a vertical asymptote.

integrand The expression contained within the integral whose antiderivative is being sought. f(x) is the integrand in the .

integration by parts A consequence of the product rule for differentiation: d(uv) = udv + vdu becomes udv = d(uv) – vdu so that .

Integration Test If f(x) is a continuous, positive, decreasing function on [1,∞] with f(n) = a_n for all positive integers n, then the series converges if and only if converges.

interval of convergence The interval on which a power series converges and goes from c – r to c + r where c is the point about which the power series was constructed and r is the radius of convergence.

Lagrange Error Estimate (or Lagrange Remainder) Given a Taylor Series T(x) for f(x), the Lagrange Remainder, R(x), for the difference f(x) – T(x) is where f^{(n + 1)}(x*) is the nth derivative evaluated at the point x*.

L’Hopital’s Rule Given , with both f(x) and g(x) differentiable at x = c. If or , then .

Limit Comparison Test Given two positive series and , let . If L is positive and finite, then both and converge or both diverge.

logistic growth A phenomenon which looks like exponential growth but eventually slows and reaches a plateau.

MacLaurin Series A series of the form that approximates a function f(x) about the point x = 0.

Mean Value Theorem If a function f(x) is defined and continuous on [a,b] and differentiable of (a,b), there is a value of c in the interval (a,b) so that the instantaneous rate of change at c is equal to the average rate of change over [a,b]. That is, there exists a value of c in (a,b) so that .

p-series A series of the form . The series converges when p > 1 and diverges when p ≤ 1.

parametric equations The coordinates x and y are each written in terms of a third parameter, usually t.

partial fraction decomposition A technique used to separate a fraction into the sum and difference of smaller fractions.

partition A partition is a subset of the interval [a,b] on which an integral is being computed.

piece-wise (split domain) function A function that is defined by different rules for specific subsets of the domain is called a piece-wise function.

polar coordinates A coordinate system in which the location of a point is determined by the radius of a circle drawn from a fixed point and the measure of an angle drawn from a fixed ray.

positive series A series that only contains positive terms.

power series A series centered at x = c that has the form .

Product Rule If f(x) = g(x) × k(x), then f(x) = g(x)k'(x) + g'(x)k(x).

Quotient Rule If , then .

radius of convergence The distance from a central value about which a power series will converge. See also interval of convergence.

Ratio Test Given a series , let . If r < 1, then converges. If r > 1, then diverges. If r = 1, then no conclusion about can be drawn from this test.

Riemann Sum A Riemann Sum takes the interval [a,b], divides it into a number of partitions, and computes the area under the curve for each partition for the purpose of estimating the integral .

Root Test Given a series , let . If r < 1, then is absolutely convergent. If r > 1, then divergent. If r = 1, then no conclusion about can be drawn from this test.

scalar A quantity that has magnitude only.

Second Fundamental Theorem of Calculus If , then Q'(x) = f(k(x))k'(x) – f(g(x))g'(x).

separation of variables A technique used to solve simple differentiable equations. All terms of a given variable are moved to one side of the equation.

sequence A listing of numbers generated by a mathematical rule.

series The sum of the terms in a sequence.

shell method A procedure used to compute the volume of a solid of rotation by accumulating the surface area of a shell and multiplying it by the thickness.

Simpson’s Rule Simpson’s Rule uses the area under a parabolic arc determined by three data points to approximate the area under a curve.

slope field A visual representation of a differential equation. It is traditional to use lattice points (points whose coordinates are integers), compute the slope of the tangent line to the function using the differential equation, and draw a small line segment with that slope at that point.

Squeeze Theorem If a_n ≤ b_n ≤ c_n for n ≥ k, where k is some constant, with and , then .

strictly decreasing sequence A sequence in which a_{n + 1} < a_n for all n.

strictly increasing sequence A sequence in which a_{n + 1} > a_n for all n.

Taylor Series A series of the form that approximates a function f(x) about a point x = c.

transcendental function A function that cannot be written as a polynomial or ratio of polynomials.

trapezoidal rule The trapezoidal approximation of a Riemann Sum has the height of each trapezoid as the width of the interval, while the bases of the trapezoid are the functional values at each endpoint.

u-substitution A technique used when both a function and its derivative appear in the integrand.

vector A quantity that has both magnitude and direction.

washer method The method used to calculate the volume of a solid of rotation when the interior of the solid contains holes.