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✨AP Calculus AB/BC Unit 5 Vocabulary

58 essential vocabulary terms and definitions for Unit 5 – Analytical Applications of Differentiation

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✨Unit 5 – Analytical Applications of Differentiation
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✨Unit 5 – Analytical Applications of Differentiation

5.10 Introduction to Optimization Problems

TermDefinition
derivativeThe instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point.
maximum valueThe largest output value that a function attains on a given interval.
minimum valueThe smallest output value that a function attains on a given interval.
optimizationThe process of finding the minimum or maximum value of a function on a given interval.

5.1 Using the Mean Value Theorem

TermDefinition
average rate of changeThe change in the value of a function divided by the change in the input over an interval [a, b], calculated as (f(b) - f(a))/(b - a).
continuousA function that has no breaks, jumps, or holes in its graph over a given interval.
differentiableA property of a function that has a derivative at every point in an interval, meaning the function is smooth with no sharp corners or cusps.
instantaneous rate of changeThe rate at which a function is changing at a specific point, represented by the derivative at that point.
Mean Value TheoremA theorem stating that if a function is continuous on a closed interval and differentiable on the open interval, there exists at least one point where the instantaneous rate of change equals the average rate of change over that interval.

5.11 Solving Optimization Problems

TermDefinition
applied contextsReal-world situations or practical problems where mathematical functions are used to model and solve problems.
maximum valueThe largest output value that a function attains on a given interval.
minimum valueThe smallest output value that a function attains on a given interval.

5.12 Exploring Behaviors of Implicit Relations

TermDefinition
critical pointA point in the domain of a function where the derivative is zero or undefined, which are candidates for local and absolute extrema.
derivativeThe instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point.
first derivativeThe derivative of a function, denoted f', which describes the rate of change and indicates where a function is increasing or decreasing.
implicit differentiationA technique for finding the derivative of a function defined implicitly by differentiating both sides of an equation with respect to the independent variable.
implicit relationA relation defined by an equation in which the dependent variable is not explicitly solved in terms of the independent variable.
implicitly defined functionA function defined by an equation relating x and y, where y is not explicitly solved in terms of x.
second derivativeThe derivative of the first derivative, denoted f'', which describes the concavity of a function and indicates where it is concave up or concave down.

5.2 Extreme Value Theorem, Global vs Local Extrema, and Critical Points

TermDefinition
continuousA function that has no breaks, jumps, or holes in its graph over a given interval.
critical pointA point in the domain of a function where the derivative is zero or undefined, which are candidates for local and absolute extrema.
Extreme Value TheoremA theorem stating that if a function is continuous on a closed interval [a, b], then the function must attain both a minimum and maximum value on that interval.
first derivativeThe derivative of a function, denoted f', which describes the rate of change and indicates where a function is increasing or decreasing.
maximum valueThe largest output value that a function attains on a given interval.
minimum valueThe smallest output value that a function attains on a given interval.
relative extremaMaximum or minimum values of a function at a point relative to nearby points.

5.3 Determining Intervals on Which a Function is Increasing or Decreasing

TermDefinition
decreasingAn interval on which a function's output values are getting smaller as the input values increase, corresponding to where the first derivative is negative.
first derivativeThe derivative of a function, denoted f', which describes the rate of change and indicates where a function is increasing or decreasing.
increasingAn interval on which a function's output values are getting larger as the input values increase, corresponding to where the first derivative is positive.

5.4 Using the First Derivative Test to Determine Relative (Local) Extrema

TermDefinition
first derivativeThe derivative of a function, denoted f', which describes the rate of change and indicates where a function is increasing or decreasing.
relative extremaMaximum or minimum values of a function at a point relative to nearby points.

5.5 Using the Candidates Test to Determine Absolute (Global) Extrema

TermDefinition
absolute extremaThe maximum or minimum values of a function over its entire domain or a specified interval.
critical pointA point in the domain of a function where the derivative is zero or undefined, which are candidates for local and absolute extrema.
derivativeThe instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point.
endpointsThe boundary points of a closed interval where a function's absolute extrema may occur.

5.6 Determining Concavity

TermDefinition
concave downA property of a function where the graph curves downward, occurring when the function's derivative is decreasing on an interval.
concave upA property of a function where the graph curves upward, occurring when the function's derivative is increasing on an interval.
points of inflectionPoints on the graph of a function where the concavity changes from concave up to concave down or vice versa.
second derivativeThe derivative of the first derivative, denoted f'', which describes the concavity of a function and indicates where it is concave up or concave down.

5.7 Using the Second Derivative Test to Determine Extrema

TermDefinition
absolute maximumThe highest value of a function over its entire domain or a specified interval.
absolute minimumThe lowest value of a function over its entire domain or a specified interval.
continuousA function that has no breaks, jumps, or holes in its graph over a given interval.
critical pointA point in the domain of a function where the derivative is zero or undefined, which are candidates for local and absolute extrema.
global extremumThe absolute maximum or minimum value of a function over its entire domain or a specified interval.
relative maximumA point where a function reaches a highest value in a neighborhood around that point.
relative minimumA point where a function reaches a lowest value in a neighborhood around that point.
second derivativeThe derivative of the first derivative, denoted f'', which describes the concavity of a function and indicates where it is concave up or concave down.

5.8 Sketching Graphs of Functions and Their Derivatives

TermDefinition
analytical representationThe representation of a function or its derivatives using equations and algebraic expressions.
derivativeThe instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point.
first derivativeThe derivative of a function, denoted f', which describes the rate of change and indicates where a function is increasing or decreasing.
function behaviorThe characteristics of a function including its increasing/decreasing intervals, concavity, extrema, and end behavior.
graphical representationThe visual display of a function or its derivatives on a coordinate plane.
key featuresImportant characteristics of a function including extrema, inflection points, intervals of increase/decrease, and concavity.
numerical representationThe representation of a function or its derivatives using tables of values or numerical data.
second derivativeThe derivative of the first derivative, denoted f'', which describes the concavity of a function and indicates where it is concave up or concave down.

5.9 Connecting a Function, Its First Derivative, and its Second Derivative

TermDefinition
derivativeThe instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point.
function behaviorThe characteristics of a function including its increasing/decreasing intervals, concavity, extrema, and end behavior.
key featuresImportant characteristics of a function including extrema, inflection points, intervals of increase/decrease, and concavity.