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AP Calculus AB definitions and theorems
Definitions and theorems that you should know
53
Mathematics
03/31/2011

Term
 [image]
Definition
 [image]
Term
 [image]
Definition
 [image]
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 Definition:A tangent line is...
Definition
 ...the line through a point on a curve with slope equal to the slope of the curve at that point.
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 Definition:A secant line is...
Definition
 ...the line connecting two points on a curve.
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 Definition:A normal line is...
Definition
 ...the line perpendicular to the the tangent line at the point of tangency.
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 Definition:[image] is continuous at [image] when...
Definition
 1. [image] exists;2. [image] exists; and3. [image].
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 Limit definition of the derivative of [image]:[image]
Definition
 [image]
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 Alternate definition of derivative of [image] at [image]:[image]=
Definition
 [image]
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 What [image] tells you about a function
Definition
 • slope of a curve at a point• slope of tangent line• instantaneous rate of change
Term
 Definition:Average rate of change is...
Definition
 [image]
Term
 Rolle's Theorem: If [image] is continuous on [image], differentiable on [image], and...
Definition
 ...[image], then there exists a value of [image] such that [image].
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 Mean Value Theorem for Derivatives:If [image] is continuous on [image] and differentiable on [image], then...
Definition
 ...there exists a value of [image] such that [image].
Term
 Extreme Value Theorem:If [image] is continuous on a closed interval, then...
Definition
 ...[image] must have both an absolute maximum and an absolute minimum on the interval.
Term
 Intermediate Value Theorem:If [image] is continuous on [image], then...
Definition
 ...[image] must take on every [image]-value between [image] and [image].
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 If a function is differentiable at a point, then...
Definition
 ...it must be continuous at that point. (Differentiability implies continuity.)
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 Four ways in which a function can fail to be differentiable at a point
Definition
 •Discontinuity•Corner•Cusp•Vertical tangent line
Term
 Definition:A critical number (a.k.a. critical point or critical value) of [image] is...
Definition
 ...a value of [image] in the domain of [image] at which either [image] or [image] does not exist.
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 If [image], then...
Definition
 ...[image] is increasing.
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 If [image], then...
Definition
 ...[image] is decreasing,
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 If [image], then...
Definition
 ...[image] has a horizontal tangent.
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 Definition:[image] is concave up when...
Definition
 ...[image] is increasing.
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 Definition:[image] is concave down when...
Definition
 ...[image] is decreasing.
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 [image] means that [image] is...
Definition
 concave up (like a cup).
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 [image] means that [image] is...
Definition
 concave down(like a frown).
Term
 Definition:A point of inflection is a point on the curve where...
Definition
 ...concavity changes.
Term
 To find a point of inflection,…
Definition
 … look for where [image] changes signs, or, equivalently, where [image] changes direction.
Term
 To find extreme values of a function, look for where…
Definition
 … [image] is zero or undefined (critical numbers).
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 At a maximum, the value of the derivative…
Definition
 … [image] changes from positive to negative.(First Derivative Test)
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 At a minimum, the value of the derivative…
Definition
 … [image] changes from negative to positive. (First Derivative Test)
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 The Second Derivative Test: If [image]…
Definition
 …and [image], then [image] has a maximum; if [image], then [image] has a minimum.
Term
 Position function[image]
Definition
 [image], the antiderivative of velocity
Term
 Velocity function[image]
Definition
 [image], the derivative of position, as well as [image], the antiderivative of acceleration
Term
 Acceleration function [image]
Definition
 [image], the derivative of velocity, as well as [image], the second derivative of position
Term
 A particle is moving to the left when…
Definition
 …[image].
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 A particle is moving to the right when…
Definition
 …[image].
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 A particle is not moving (at rest) when…
Definition
 …[image].
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 A particle changes direction when…
Definition
 …[image] changes signs.
Term
 To find displacement of a particle with velocity [image] from [image] to [image], calculate this:
Definition
 [image]
Term
 To find total distance traveled by a particle with velocity [image] from rom [image] to [image], calculate this:
Definition
 [image]
Term
 Area between curves
Definition
 [image]
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 Volume of a solid with cross-sections of a specified shape
Definition
 [image]
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 Volume using discs
Definition
 [image]"perpendiscular"
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 Volume using washers(discs with holes)
Definition
 [image]
Term
 Trapezoidal rule for approximating [image]
Definition
 [image]
Term
 Average value of [image] on [image]
Definition
 [image]
Term
 L'Hôpital's rule for indeterminate limitsIf [image] or [image],
Definition
 then [image], if the new limit exists.
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 Mean Value Theorem for Integration:If [image] is continuous on [image], then...
Definition
 ...there exists a value of [image] such that [image].
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 Fundamental Theorem of Calculus (part 1)[image]
Definition
 [image]
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 Fundamental Theorem of Calculus (part 2)[image] =
Definition
 [image], where [image] is an antiderivative of [image]
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 A differential equation is...
Definition
 …an equation containing one or more derivatives.
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 To solve a differential equation,...
Definition
 ...first separate the variables (if needed) by multiplying or dividing, then integrate both sides.
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 Exponential Growth and Decay:If [image], then...
Definition
 [image], where [image] is the quantity at [image], and [image] is the constant of proportionality.
Term
 [image]
Definition
 the amount which that quantity has changed from [image] to [image]
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