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AP Calculus First Semester Vocabulary

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Section 1

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Mean Value Theorem

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Calculus Math

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Cards (38)

Section 1

(38 cards)

Mean Value Theorem

Front

If y=f(x) is continuous at every point of the closed interval [a,b] and the differentiable at every point of its interior (a,b), then there is at least one point c in (a,b) at which f'(c)=(f(b)-f(a))/(b-a); the average rate of change equals the instantaneous rate of change

Back

Derivative calculus

Front

The branch of mathematics that deals with derivatives

Back

Monotonic function

Front

A function that is always increasing on an interval or always decreasing on an interval

Back

Extreme Value Theorem

Front

If f is continuous on a closed interval [a,b], then f has both a maximum value and a minimum value on the interval

Back

Tangent line

Front

To the graph of the function y=f(x) at a point x=a where f' exists, the line through (a, f(a)) with slope f'(a)

Back

Second derivative test

Front

If f'(c)=0 and f''(c)<0 then f has a local maximum at x=c; If f'(c)=0 and f''(c)>0 then f has a local minimum at x=c

Back

Definite integral

Front

Back

Increasing Function

Front

line goes up from left to right

Back

Absolute minimum

Front

The function f has an ___ value f(c) at a point c in its domain D if and only if f(x)>f(c) for all x in D the lowest point below the entire domain of a function or relation

Back

Differentiation

Front

If f'(x) exists, the function f is at x. A function that is able at every point of its domain is a ___ function; The process of finding the derivative

Back

Product rule

Front

onedtwo+twodone

Back

Absolute maximum

Front

The function f has an __ value f(c) at a point c in its domain D if and only if f(c)<f(c) for all x in D the highest point over the entire domain of a function or relation

Back

Point of inflection

Front

is a point on a curve at which the curve changes from being concave (concave downward) to convex (concave upward), or vice versa. AKA when f''(x)=0

Back

Quotient rule

Front

(lodhi-hidlo)/lolo

Back

Absolute value function

Front

a function written in the form y = /x/, and the graph is always in the shape of a v

Back

Optimization

Front

In an application, maximizing or minimizing some aspect of the system being modeled

Back

Decreasing function

Front

If f' < 0 at each point of (a,b), then f decreases on [a,b]

Back

Discontinuity

Front

If a function f is not continuous at a point c, then c is a point of __ of f

Back

Range

Front

The set of output values of a function.

Back

Integral Calculus

Front

The branch of mathematics that deals with integrals

Back

Concave down

Front

The graph of a differentiable function y=f(x) is _____ on an open interval I if y' is decreasing on I (y' is decreasing, y'' is negative)

Back

Average Rate of Change

Front

Amount of Change/Time F(b)-F(a)/(b-a)

Back

Critical point

Front

A point (value) in the interior of the domain of a function f at which f'=o or f' does not exist

Back

Domain

Front

The set of input values of a function.

Back

Chain Rule

Front

If y=f(u) is differentiable at the point u=g(x), and g is differentiable at x, then the composite function (fog)(x)=f(g(x)) is differentiable at x, and (fog)'(x)=f'(g(x))*g'(x)

Back

First derivative test

Front

For a continuous function f, (1) if f' changes sign from positive to negative at a critical point c, then f has a local maximum value at c; (2) if f' changes sign from negative to positive at a critical point c, then f has a local minimum value at c (3) if f' does not change sign at a critical point c, then f has no local extreme value at c; (4) if f'<0 (f'>0) for x>a where a is a left endpoint in the domain of f, then f has a local maximum (minimum) value at a;(5) if f'<0 (f'>0) for x<b where b is the right endpoint in the domain of f, then f has a local minimum (maximum) value at b

Back

Concave up

Front

The graph of a differentiable function y=f(x) is ___on an open interval I if y' is increasing on I (y' is increasing, y'' is positive)

Back

Implicit differentiation

Front

A process for finding dy/dx when y is implicitly defined as a function of x by an equation of the form f(x,y)=0

Back

Velocity

Front

The rate of change of position with respect to time

Back

Local minimum

Front

The function f has a value f(c) at a point c in the interior of its domain if and only if f(x)>= f(c) for all x in some open interval containing c. The function has a _ value at an endpoint c if the inequality holds for all x in some half-open domain interval containing c

Back

Local Maximum

Front

The function f has a _ value f(c) at a point c in the interior of its domain if and only if f(x)<=f(c) for all x in some open interval containing c. The function has a _ value at an endpoint if the inequality holds for all x in some half-open domain interval containing c

Back

Derivative

Front

The __ of y with respect to x at a is, geometrically, the slope of the tangent line to the graph of f at (a, f(a)).

Back

Continuity

Front

The limit of f(x) as x approaches a from either direction is equal to f(a), as long as a is in the domain of f(x).

Back

Asymptote

Front

A line that a graph approaches but never crosses; limit as x-> + or - infinity gives horizontal (value is a number); limit as x->a from the left or right gives vertical _ (value is +/- infinity)

Back

Antiderivative

Front

A function F(x) is an ___ of a function f(x) if F'(x)=f(x) for all x in the domain of F

Back

Acceleration

Front

The derivative of a velocity function with respect to time

Back

Newton's method

Front

a method for finding successively better approximations to the roots (or zeroes) of a real-valued function

Back

Intermediate Value Theorem

Front

If a and b are any two points in an interval on which f is differentiable, then f' takes on every value between f'(a) and f'(b)

Back

AP Calculus First Semester Vocabulary

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Mean Value Theorem

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Cards (38)

Mean Value Theorem

If y=f(x) is continuous at every point of the closed interval [a,b] and the differentiable at every point of its interior (a,b), then there is at least one point c in (a,b) at which f'(c)=(f(b)-f(a))/(b-a); the average rate of change equals the instantaneous rate of change

Derivative calculus

The branch of mathematics that deals with derivatives

Monotonic function

A function that is always increasing on an interval or always decreasing on an interval

Extreme Value Theorem

If f is continuous on a closed interval [a,b], then f has both a maximum value and a minimum value on the interval

Tangent line

To the graph of the function y=f(x) at a point x=a where f' exists, the line through (a, f(a)) with slope f'(a)

Second derivative test

If f'(c)=0 and f''(c)<0 then f has a local maximum at x=c; If f'(c)=0 and f''(c)>0 then f has a local minimum at x=c

Definite integral

Increasing Function

line goes up from left to right

Absolute minimum

The function f has an ___ value f(c) at a point c in its domain D if and only if f(x)>f(c) for all x in D the lowest point below the entire domain of a function or relation

Differentiation

If f'(x) exists, the function f is __ at x. A function that is __able at every point of its domain is a ___ function; The process of finding the derivative

Product rule

onedtwo+twodone

Absolute maximum

The function f has an __ value f(c) at a point c in its domain D if and only if f(c)<f(c) for all x in D the highest point over the entire domain of a function or relation

Point of inflection

is a point on a curve at which the curve changes from being concave (concave downward) to convex (concave upward), or vice versa. AKA when f''(x)=0

Quotient rule

(lodhi-hidlo)/lolo

Absolute value function

a function written in the form y = /x/, and the graph is always in the shape of a v

Optimization

In an application, maximizing or minimizing some aspect of the system being modeled

Decreasing function

If f' < 0 at each point of (a,b), then f decreases on [a,b]

Discontinuity

If a function f is not continuous at a point c, then c is a point of __ of f

Range

The set of output values of a function.

Integral Calculus

The branch of mathematics that deals with integrals

Concave down

The graph of a differentiable function y=f(x) is _____ on an open interval I if y' is decreasing on I (y' is decreasing, y'' is negative)

Average Rate of Change

Amount of Change/Time F(b)-F(a)/(b-a)

Critical point

A point (value) in the interior of the domain of a function f at which f'=o or f' does not exist

Domain

The set of input values of a function.

Chain Rule

If y=f(u) is differentiable at the point u=g(x), and g is differentiable at x, then the composite function (fog)(x)=f(g(x)) is differentiable at x, and (fog)'(x)=f'(g(x))*g'(x)

First derivative test

Concave up

The graph of a differentiable function y=f(x) is ___on an open interval I if y' is increasing on I (y' is increasing, y'' is positive)

Implicit differentiation

A process for finding dy/dx when y is implicitly defined as a function of x by an equation of the form f(x,y)=0

Velocity

The rate of change of position with respect to time

Local minimum

The function f has a __ value f(c) at a point c in the interior of its domain if and only if f(x)>= f(c) for all x in some open interval containing c. The function has a ___ value at an endpoint c if the inequality holds for all x in some half-open domain interval containing c

Local Maximum

The function f has a ___ value f(c) at a point c in the interior of its domain if and only if f(x)<=f(c) for all x in some open interval containing c. The function has a ___ value at an endpoint if the inequality holds for all x in some half-open domain interval containing c

Derivative

The __ of y with respect to x at a is, geometrically, the slope of the tangent line to the graph of f at (a, f(a)).

Continuity

The limit of f(x) as x approaches a from either direction is equal to f(a), as long as a is in the domain of f(x).

Asymptote

A line that a graph approaches but never crosses; limit as x-> + or - infinity gives horizontal __ (value is a number); limit as x->a from the left or right gives vertical ___ (value is +/- infinity)

Antiderivative

A function F(x) is an ___ of a function f(x) if F'(x)=f(x) for all x in the domain of F

Acceleration

The derivative of a velocity function with respect to time

Newton's method

a method for finding successively better approximations to the roots (or zeroes) of a real-valued function

Intermediate Value Theorem

If a and b are any two points in an interval on which f is differentiable, then f' takes on every value between f'(a) and f'(b)

If f'(x) exists, the function f is at x. A function that is able at every point of its domain is a ___ function; The process of finding the derivative

The function f has a value f(c) at a point c in the interior of its domain if and only if f(x)>= f(c) for all x in some open interval containing c. The function has a _ value at an endpoint c if the inequality holds for all x in some half-open domain interval containing c

The function f has a _ value f(c) at a point c in the interior of its domain if and only if f(x)<=f(c) for all x in some open interval containing c. The function has a _ value at an endpoint if the inequality holds for all x in some half-open domain interval containing c

A line that a graph approaches but never crosses; limit as x-> + or - infinity gives horizontal (value is a number); limit as x->a from the left or right gives vertical _ (value is +/- infinity)