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Calculus (Columbia) > Calculus (Columbia) > Flashcards

Calculus (Columbia) Flashcards

1
Q

Envision
how a
function is like a
machine.

A
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2
Q

A
_____ f is a
rule that assigns a
unique value
y = f(x) to each
input x ∈ D?

A

function

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3
Q

A
function f is a
_____ that assigns a
unique value
y = f(x) to each
input x ∈ D?

A

rule

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4
Q

A
function f is a
rule that assigns a
_____
y = f(x) to each
input x ∈ D?

A

unique value

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5
Q

A
function f is a
rule that assigns a
unique value
y = f(x) to each
_____ x ∈ D?

A

input

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6
Q

The

  • *_____** of function f is the
  • *set** of all
  • *x ∈ ℝ**
    s. t.
  • *f(x) is defined**.
A

domain

Picture:
A porn set where a domain-atrix with a red x over her nethers whipping an unsure British Red Guard gimp (who says “∈ℝ…”) while she screams “F(X) IS DEFINED!”

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7
Q

The

  • *domain** of function f is the
  • *_____** of all
  • *x ∈ ℝ**
    s. t.
  • *f(x) is defined**.
A

set

Picture:
A porn set where a domain-atrix with a red x over her nethers whipping an unsure British Red Guard gimp (who says “∈ℝ…”) while she screams “F(X) IS DEFINED!”

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8
Q

The

  • *domain** of function f is the
  • *set** of all
  • *_____**
    s. t.
  • *f(x) is defined**.
A

x ∈ ℝ

Picture:
A porn set where a domain-atrix with a red x over her nethers whipping an unsure British Red Guard gimp (who says “∈ℝ…”) while she screams “F(X) IS DEFINED!”

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9
Q

The

  • *domain** of function f is the
  • *set** of all
  • *x ∈ ℝ​**
    s. t.
  • *_____**.
A

f(x) is defined

Picture:
A porn set where a domain-atrix with a red x over her nethers whipping an unsure British Red Guard gimp (who says “∈ℝ…”) while she screams “F(X) IS DEFINED!”

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10
Q

In set notation, the
domain of function f is
D = {_____ | f(x) is defined}.

A

x ∈ ℝ

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11
Q

In set notation, the
domain of function f is
D = {x ∈ ℝ | _____}.

A

f(x) is defined

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12
Q

The
domain of
f(x) = x2 is
_____.

A

D = ℝ

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13
Q

The
domain of
any polynomial is
_____.

A

D = ℝ

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14
Q

In set notation,
the
domain of
1
x − 1
is _____.

A

D = {x ∈ ℝ | x ≠ 1}

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15
Q

In inverval notation,
the
domain of
1
x − 1
is _____.

A

D = (−∞, 1) ∪ (1, ∞)

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16
Q

(a, b)
is an
[open / closed / neither]
interval (or set).

A

()pen

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17
Q

[a, b]
is an
[open / closed / neither]
interval (or set).

A

[ losed

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18
Q

[a, b)
is an
[open / closed / neither]
interval (or set).

A

neither

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19
Q

What is
[a, b]
in
interval notation?

A

[a, b] = {x ∈ ℝ | a < x < b}

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20
Q

What is
(a, b)
in
interval notation?

A

(a, b) = {x ∈ ℝ | a < x < b}

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21
Q

Envision

A ∩ B.

A
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22
Q

Envision

A ∪ B.

A
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23
Q

The
_____ of function f is the
set of all
possible values of y = f(x)
for some
x ∈ D.

A

range

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24
Q

The
range of function f is the
_____ of all
possible values of y = f(x)
for some
x ∈ D.

A

set

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25
Q

The
range of function f is the
set of
_____ of y = f(x)
for some
x ∈ D.

A

all possible values

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26
Q

The
range of function f is the
set of all
possible values of y = f(x)
for some
_____.

A

x ∈ D

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27
Q

In set notation, the
range of function f is
R = {_____ | y = f(x) for some x ∈ D}.

A

y ∈ ℝ

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28
Q

In set notation, the
range of function f is
R = {y ∈ ℝ | _____ for some x ∈ D}.

A

y = f(x)

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29
Q

In set notation, the
range of function f is
R = {y ∈ ℝ | y = f(x) for some _____}.

A

x ∈ D

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30
Q

The
domain of a function is the
[input / output] of that function
and represents the set of
all possible
[independent / dependent] variables
for that function.

A

input

independent

  • INput*
  • INdependent*
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31
Q

The
range of a function is the
[input / output] of that function
and represents the set of
all possible
[independent / dependent] variables
for that function.

A

output

dependent

“If you’re PUT OUT in the cold, you’re DEPENDENT on the kindness of others.”

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32
Q

In interval notation,
what is the
domain of this function?

A

D = [a, b]Next

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33
Q

In interval notation,
what is the
range of this function?

A

R = [c, d]

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34
Q

Analytically,
what is
Step 1 to determining whether a value is in the
range of a function?

A

Substitute
the value for the dependent variable
(i.e. y = f(x))

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35
Q

Analytically,
what is
Step 2 to determining whether a value is in the
range of a function?

A

Solve
for the independent variable.
(i.e. x)

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36
Q

After you’ve done all the work,
how do you know whether a value is in the
range of a function?

A

If you can
solve for the independent variable,
then the value is
in the range.

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37
Q

Given

f(x) = 1
x

is

  • *0** in the
  • *range** of f(x)?
A

No,
because you can’t substitute 0 for f(x) and then solve.

There is no such x-value.

38
Q

Graphically,
how can you determine whether a graph is a
function?

A

The
vertical line test.
(not rigorous)

39
Q

How do you know whether a

  • *graph**
  • *passes** the vertical line test?
A

If you
can’t draw a vertical line that
passes through more than one point
on the graph, then it
passes and is a
function.

40
Q

This
fails the vertical line test, so it’s
not a function.

What do we have here?

A

An
implicit relation
between x and y.
(both are independent variables)

41
Q

A

  • *_____** function has
  • *different functional forms** over
  • *different domains**.
A

piecewise

42
Q

A

  • *piecewise** function has
  • *_____** over
  • *different domains**.
A

different functional forms

43
Q

A

  • *piecewise** function has
  • *different functional forms** over
  • *_____**.
A

different domains

44
Q

How might this
function be defined?

A
45
Q

Analytically,
polynomials are expressed as

p(x) = c0 + c1x + c2x2 + … + cjxn,

  • *cj ___** and
  • *n > 0 is an integer**
A

cj ∈ ℝ

46
Q

Analytically,
polynomials are expressed as

p(x) = c0 + c1x + c1x2 + … + cjxn,

  • *cj ∈ ℝ** and
  • *_____**
A

n > 0 is an integer

(n ∈ ℤ, n > 0)

47
Q

How many
variables can be in a polynomial?

A

One

48
Q

In a
polynomial, the
coefficients can be
_____.

A

any real number

49
Q

In a
polynomial, the
exponents of the variables can be
_____.

A

any nonnegative integer

50
Q

A

  • *_____** is a
  • *ratio** of
  • *polynomials**.
A

rational function

51
Q

A

  • *rational function** is a
  • *_____** of
  • *polynomials**.
A

ratio

52
Q

A

  • *rational function** is a
  • *ratio** of
  • *_____**.
A

polynomials

53
Q

Analytically,
in an
odd function,
f(−x) = _____

A

−f(x)

54
Q

Graphically,
an
odd function will
reflect about _____.

A

the y- and x-axes

or the

origin

O<span>dd</span>

O<span>rigin</span>

55
Q

Analytically,
in an
even function,
f(−x) = _____

A

f(x)

56
Q

Graphically,
an
even function will
reflect about _____.

A

the y-axis

57
Q

Another way to
describe an
odd function is
_____.

A

odd symmetric

58
Q

Another way to
describe an
even function is
_____.

A

even symmetric

59
Q

Any

  • *_____** with
  • *all even powers** is
  • *even symmetric**.
A

polynomial

60
Q

Any

  • *polynomial** with
  • *all odd powers** is
  • *_____**.
A

odd symmetric

61
Q

Any

  • *polynomial** with
  • *_____** is
  • *odd symmetric**.
A

all odd powers

62
Q

Analytically,
f(x) is
non-decreasing over interval I = (a, b), if, for all
a < x1 < x2 < b,
_____.

A

f(x2) > f(x1)

63
Q

Analytically,
f(x) is
non-decreasing over interval I = (a, b), if, for all
a < x1 < x2 < b,
_____.

A

f(x2) > f(x1)

64
Q

Analytically,
f(x) is
_____ over interval I = (a, b), if, for all
a < x1 < x2 < b,
f(x2) > f(x1).

A

non-decreasing

65
Q

Analytically,
f(x) is
_____ over interval I = (a, b), if, for all
a < x1 < x2 < b,
f(x2) < f(x1).

A

non-increasing

66
Q

Analytically,
f(x) is
_____ over interval I = (a, b), if, for all
a < x1 < x2 < b,
f(x2) > f(x1).

A

increasing

67
Q

Analytically,
f(x) is
increasing over interval I = (a, b), if, for all
_____,
f(x2) > f(x1).

A

a < x1 < x2 < b

68
Q

Analytically,
f(x) is
increasing over interval I = (a, b), if, for all
a < x1 < x2 < b,
_____.

A

f(x2) > f(x1)

69
Q

Analytically,
f(x) is
_____ over interval I = (a, b), if, for all
a < x1 < x2 < b,
f(x2) < f(x1).

A

decreasing

70
Q

Analytically,
f(x) is
decreasing over interval I = (a, b), if, for all
a < x1 < x2 < b,
_____.

A

f(x2) < f(x1)

71
Q

Over
x ∈ [0, 1], this function is
[increasing / decreasing / neither increasing nor decreasing].

A

increasing

72
Q

Over
x ∈ [1, 2], this function is
[increasing / decreasing / neither increasing nor decreasing].

A

decreasing

73
Q

Over
x ∈ [0, 2], this function is
[increasing / decreasing / neither increasing nor decreasing].

A

neither increasing nor decreasing

74
Q

Pitfall:
When describing whether a
function is
increasing or decreasing, the
pitfall is
_____.

A

the interval

75
Q

y = mx + b
is a
_____ function.

A

linear

76
Q

y − y0 = m(x − x0)
is a
_____ function.

A

linear

77
Q

f(x) = _____
is a
power function.

A

{ xp | p ∈ ℝ }

78
Q

f(x) = { xp | p ∈ ℝ }
is a
_____ function.

A

power

79
Q

Which are
power functions?

f(x) = **x<sup>−1/2</sup>**
g(x) = **x<sup>π</sup>**
h(x) = **x<sup>√(−1/2)</sup>**
A

Yes

Yes

No

The exponent
must be a real number,
although it can be
rational or irrational.

80
Q

The
domain of a
power function is
_____.

A

D = ℝ

81
Q

The
domain of
rational function
R(x) = n(x)
d(x)
is _____.

A

D = { x ∈ ℝ | d(x) ≠ 0}

82
Q

Plainly,
how do you determine the
domain of a
rational function?

A

Set the denominator equal to zero.

The domain is everything but that.

83
Q

What is this
asking about?

A
84
Q

Analytically,
a function f(x) has a
_____ at level y = L if
limx→∞ = L
or
limx→−∞​ = L

A

horizontal asymptote

85
Q

Analytically,
a function f(x) has a
horizontal asymptote
at level y = L if
_____
or
_____.

A

limx→∞ = L
or
limx→−∞​ = L

86
Q

Plainly,
what is
Step 1 to finding a
horizontal asymptote?

A

Look to

  • *highest-degree term** in
  • *numerator and denominator**
87
Q

Plainly,
what is
Step 2 to finding a
horizontal asymptote?

A

See if the highest-degree terms in
numerator and denominator
approach a finite number.

88
Q
A

“When you’re taking a limit and the numerator and denominator are of the same degree, you’re typically going to get a finite number.”

89
Q
A
90
Q
A

Calculus (Columbia) (2 decks)

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