June 8 and Later

Question 1

𝚫
means
_____.

Answer 1

Question 2

Average rate of change
is associated with
[tangent / secant].

Answer 2

secant

Question 3

Instantaneous rate of change
is associated with
[tangent / secant].

Answer 3

tangent

Question 4

Touching at one point
is associated with
[tangent / secant].

Answer 4

tangent

Question 5

Derivative
is associated with
[tangent / secant].

Answer 5

tangent

Question 6

𝚫f = f(x₂) − f(x₁)
𝚫x x₂ − x₁

is associated with
[ instantaneous rate of change /
average rate of change ].

Answer 6

average rate of change

Question 7

𝚫f = f(x₂) − f(x₁)
𝚫x x₂ − x₁

is associated with
[tangent / secant].

Answer 7

secant

Question 8

lim_𝚫x→0 𝚫f
𝚫x

is associated with
[ instantaneous rate of change /
average rate of change ].

Answer 8

instantaneous rate of change

Question 9

df (x)
dx

is associated with
[ instantaneous rate of change /
average rate of change ].

Answer 9

instantaneous rate of change

Question 10

To talk about
instantaneous rate of change,
you must specify a
_____.

Answer 10

location

Question 11

To talk about
average rate of change,
you must specify an
_____.

Answer 11

interval

Question 12

Answer 12

Question 13

Answer 13

Question 14

Answer 14

Question 15

Answer 15

Question 16

Envision a

• *graph** of how the definition of
• *derivative** works?

Answer 16

Question 17

The
equation for
average rate of change is below.

How does it relate to the
equation for
instantaneous rate of change?

Answer 17

Question 18

This

lim_𝚫x→0 f(x + 𝚫x) − f(x)
𝚫x

may be
described verbally as
“the limit of the
_____ of f(x) over the interval [x, x + 𝚫x] as
𝚫x→0.

Answer 18

average rate of change

Question 19

This

lim_𝚫x→0 f(x + 𝚫x) − f(x)
𝚫x

may be
described verbally as
“the limit of the
average rate of change of f(x) over
the interval
_____ as 𝚫x→0.

Answer 19

[x, x + 𝚫x]

Question 20

To
evaluate a limit,

i.e.
lim_x→1 √(x² + 4) − 2
x²

• *first**
• *_____**.

Answer 20

plug in the limit point value

(here, x = 1)

Question 21

In
evaluating a limit,

i.e.
lim_x→1 √(x² + 4) − 2
x²

if you
plug in the
_____
and get a
well-defined result,
that’s the limit.

Answer 21

limit point value

Question 22

In
evaluating a limit,

i.e.
lim_x→1 √(x² + 4) − 2
x²

if you
plug in the
limit point value
and get a
_____,
that’s the limit.

Answer 22

well-defined result

Question 23

How would you write this

lim_𝚫x→0 f(x + 𝚫x) − f(x)
𝚫x

in
Leibniz notation?

Answer 23

df (x)
dx

Question 24

How would you write this

lim_𝚫x→0 f(x + 𝚫x) − f(x)
𝚫x

in
prime notation?

Answer 24

f’(x)

Question 25

This

f’(x)

is written in
[Leibniz / prime] notation.

Answer 25

prime

Question 26

This

df (x)
dx

is written in
[Leibniz / prime] notation.

Answer 26

Leibniz

Question 27

First Primary Interpretation of the Derivative
(analytical):

f’(x) is the

• *_____** of f(x) at the
• *value x** ( or at (x, f(x))

Answer 27

instantaneous rate of change

Question 28

Second Primary Interpretation of the Derivative
(geometric):

f’(x) is the

• *_____** of the line
• *tangent** to the graph of f(x) at the point
• *(x, f(x))**

Answer 28

slope

Question 29

f(x) is a function.

Its derivative,
df
dx

= lim_𝚫x→0 𝚫f
𝚫x

= f’(x)

• *=** lim_𝚫x→0 f(x + 𝚫x) − f(x)
• *𝚫x**

is
_____ function.

Answer 29

another

Question 30

What happens if you

• *immediately** use the
• *plug-in rule** for limits on a
• *derivative**?

Answer 30

You get
0/0.

Question 31

With derivatives,
_____ is called
indeterminate form,
meaning that there’s
not enough information
to determine whether
the limit exists.

Answer 31

0/0

Question 32

With derivatives,
0/0 is called
_____,
meaning that there’s
not enough information
to determine whether
the limit exists.

Answer 32

indeterminate form

Question 33

With derivatives,
0/0 is called
indeterminate form,
meaning that there’s
_____
to determine whether
the limit exists.

Answer 33

not enough information

Question 34

With derivatives,
0/0 is called
indeterminate form,
meaning that there’s
not enough information
to determine whether
_____.

Answer 34

the limit exists

Question 35

_____ is
built into the
definition of the
derivative.

Answer 35

Indeterminate form

Question 36

To find the

• *slope** of a line
• *tangent** to points on this function,

f(x) = 2x³ − 6x² + x + 3,

you
_____.

Answer 36

differentiate it.

f’(x) = 6x² − 12x + 1.

Question 37

There is a line

• *tangent** to this function at the point
• *(−0.5, 0.75)**.

The slope of line at that point is
8.5.

What is an
equation for that tangent line?

Answer 37

y − 0.75 = 8.5(x + 0.5)

Question 38

What is the
point−slope form of a
line?

Answer 38

y − y₀ = m(x − x₀)

Question 39

A

• *function** f(x) is
• *_____** at x = a if
• *lim_x→​a f(x) = f(a).**

Answer 39

continuous

Question 40

A

• *function** f(x) is
• *continuous** at x = a if
• *_____ = f(a).**

Answer 40

lim_x→​a f(x)

Question 41

A

• *function** f(x) is
• *continuous** at x = a if
• *lim_x→​a f(x) _____ f(a).**

Answer 41

=

Question 42

A

• *function** f(x) is
• *continuous** at x = a if
• *lim_x→​a f(x) = _____.**

Answer 42

f(a)

Question 43

Is this function

• *continuous** at
• *x = a**?

If not,
why not?

Answer 43

• *No:**
• *lim_x→a f(x) ≠ f(a)**.

Question 44

Is this function

• *continuous** at
• *x = a**?

If not,
why not?

Answer 44

• *No:**
• *f(a) is undefined**.

Question 45

Is this function

• *continuous** at
• *x = a**?

If not,
why not?

Answer 45

• *No:**
• *lim_x→a f(x) DNE**.

Question 46

“Continuity is a
_____ concept.”

Answer 46

point-by-point

Question 47

A function f(x) is called

• *continuous** if it’s
• *continuous** at every
• *_____**.

Answer 47

x ∈ D

Question 48

f(x) = 1 , x ≠ 0
x

Is f(x)
**continuous** at
x = 0?

Answer 48

Yes:
x = 0 is not in the domain of this function, so it’s technically continuous.

Question 49

f(x) = 1
x

Is f(x)
**continuous**?

Answer 49

No:
f(x) is not continuous at x = 0.

Question 50

What is a

• *math term** for
• *plugging this hole**?

Answer 50

continuously extending

Question 51

How might you
continuously extend f(x)?

Answer 51

Redefine it:

Question 52

Continuously extending a
function
turns it into a
_____.

Answer 52

piecewise function

Question 53

f_c(x) is the
_____ of f(x).

Answer 53

continuous extension

Question 54

At a
_____, a function is
continuous if it is
continuous in the
one-sided sense.

Answer 54

boundary point

Question 55

At a
boundary point, a function is
continuous if it is
continuous in the
_____.

Answer 55

one-sided sense

Question 56

There are basically
two (related) definitions of
continuity: one for
_____ and one for
interior points.

Answer 56

boundary points

Question 57

There are basically

• *two (related) definitions** of
• *continuity**: one for
• *boundary points** and one for
• *_____**.

Answer 57

interior points

Question 58

f(x) = 1 , x ≠ 0
x

Is f(x)
**continuous**?

Answer 58

Yes:
f(0) is not in its domain.

Question 59

Graphically,
how can you tell whether a
function is
continuous?

Answer 59

You can

• *graph** it without lifting your
• *stylus**.

Question 60

A

• *rational function** is continuous for
• *_____**.

Answer 60

x ∈ ℝ,
so long as the
denominator ≠ 0.

Question 61

A

• *polynomial** is continuous for
• *_____**.

Answer 61

x ∈ ℝ

Question 62

Given that
f(x) and g(x) are
continuous at x = a,

f(x) ± g(x) is
_____ at x = a.

Answer 62

continuous

Question 63

Given that
f(x) and g(x) are
continuous at x = a,

f(x) g(x) is
_____ at x = a.

Answer 63

continuous

Question 64

Given that
f(x) and g(x) are
continuous at x = a and
k ∈ ℝ,

k f(x) is
_____ at x = a.

Answer 64

continuous

Question 65

Given that
f(x) and g(x) are
continuous at x = a,

f(x)
g(x)

is
_____ at x = a.

Answer 65

continuous (provided g(x) ≠ 0)

Question 66

The
difference between the
two defintions of continuity is
whether you can use a
_____.

Answer 66

one-sided limit

(boundary points only)

Question 67

Given:
c(x) = o(i(x)),

according to the
_____,
c(x) is
continuous at x = a if:

• lim_x→​a i(x) = L exists and
• o(x) is continuous at
• *lim_x→​a i(x) = L**.

Answer 67

continuity of function compositions theorem

Question 68

Given:
c(x) = o(i(x)),

according to the
continuity of function compositions theorem,
c(x) is
_____ if:

• lim_x→​a i(x) = L exists and
• o(x) is continuous at
• *lim_x→​a i(x) = L**.

Answer 68

continuous at x = a

Question 69

Given:
c(x) = o(i(x)),

according to the
continuity of function compositions theorem,
c(x) is
continuous at x = a​ if:

• _____ and
• o(x) is continuous at
• *lim_x→​a i(x) = L**.

Answer 69

lim_x→​a i(x) = L exists

Question 70

Given:
c(x) = o(i(x)),

according to the
continuity of function compositions theorem,
c(x) is
continuous at x = a​ if:

• lim_x→​a i(x) = L exists and
• _____ at
• *lim_x→​a i(x) = L**.

Answer 70

o(x) is continuous

Question 71

Given:
c(x) = o(i(x)),

according to the
continuity of function compositions theorem,
c(x) is
continuous at x = a​ if:

• lim_x→​a i(x) = L exists and
• o(x) is continuous at
• *_____**.

Answer 71

lim_x→​a i(x) = L​

Question 72

Given:
c(x) = o(i(x)),

for the
continuity of function compositions theorem
to apply at x = a,
i(x) needs to
_____ at x = a, but not to
be continuous at x = a.

Answer 72

have a limit

Question 73

Given:
c(x) = o(i(x)),

for the
continuity of function compositions theorem
to apply at x = a,
i(x) needs to
have a limit at x = a, but not to
_____ at x = a.

Answer 73

be continuous

Question 74

Best Practice:
When dealing with
inverse trig functions,
always
_____.

Answer 74

make a chart

Question 75

According to the
extreme value theorem,
a function f(x) over
x ∈ [b, d] has a
_____
f(x_min) = m if there exists
x_min ∈ [b, d] s.t.
f(x_min) = m < f(x) for all
x ∈ [b, d].

Answer 75

global minimum value

Question 76

According to the
extreme value theorem,
a function f(x) over
x ∈ [b, d] has a
_____
f(x_max) = M if there exists
x_max ∈ [b, d] s.t.
f(x_max) = M > f(x) for all
x ∈ [b, d].

Answer 76

global maximum value

Question 77

According to the
extreme value theorem,
a function f(x) over
x ∈ [b, d] has a
global minimum value
_____ if there exists
x_min ∈ [b, d] s.t.
f(x_min) = m < f(x) for all
x ∈ [b, d].

Answer 77

f(x_min) = m

Question 78

According to the
extreme value theorem,
a function f(x) over
x ∈ [b, d] has a
global maximum value
_____ if there exists
x_max ∈ [b, d] s.t.
f(x_max) = M > f(x) for all
x ∈ [b, d].

Answer 78

f(x_max) = M

Question 79

According to the
extreme value theorem,
a function f(x) over
x ∈ [b, d] has a
global minimum value
f(x_min) = m if there exists
_____ s.t.
f(x_min) = m < f(x) for all
x ∈ [b, d].

Answer 79

x_min ∈ [b, d]

Question 80

According to the
extreme value theorem,
a function f(x) over
x ∈ [b, d] has a
global maximum value
f(x_max) = M if there exists
_____ s.t.
f(x_max) = M > f(x) for all
x ∈ [b, d].

Answer 80

x_max ∈ [b, d]

Question 81

According to the
extreme value theorem,
a function f(x) over
x ∈ [b, d] has a
global minimum value
f(x_min) = M if there exists
x_min ∈ [b, d] s.t.
_____ for all
x ∈ [b, d].

Answer 81

f(x_min) = M < f(x)

Question 82

According to the
extreme value theorem,
a function f(x) over
x ∈ [b, d] has a
global maximum value
f(x_max) = M if there exists
x_max ∈ [b, d] s.t.
f(x_max) = M > f(x) for all
_____.

Answer 82

x ∈ [b, d]

Question 83

Any maximum or minimum
(local or global) of f(x) is called an
_____.

Answer 83

extreme value

Question 84

_____
(local or global) of f(x) is called an
extreme value.

Answer 84

Any maximum or minimum

Question 85

Given
f(x) = x, x ∈ [0, 1),

where does f(x) have a
global maximum?

Answer 85

n/a

Question 86

Given
f(x) = x, x ∈ [0, 1),

where does f(x) have a
global minimum?

Answer 86

x = 0

Question 87

This function has
global max [x_max = 1/2 / y_max = 1/4]
at
[x_max = 1/2 / y_max = 1/4].

Answer 87

global max y_max = 1/4

at x_max = 1/2

Question 88

This function has
global min y_min = 0 at
_____.

Answer 88

x_min = {0, 1}

Question 89

[b, d] is a
_____, bounded
domain.

Answer 89

closed

(b, d are included)

Question 90

[b, d] is a
closed, _____
domain.

Answer 90

bounded

(b, d are finite)

Question 91

f(x) is

• *continuous** over
• *[b, d].**

It [must / might / cannot] have
global extremes?

Answer 91

must

f(x) is continuous over a closed and bounded domain, so it must have global extrema in that interval.

Question 92

f(x) is

• *not continuous** over
• *[b, d]**.

It [must / might / cannot] have
global extremes?

Answer 92

might

Continuity over a closed, bounded interval is sufficient (per the EVT), but it is
not necessary.

Question 93

f(x) is

• *continuous** over
• *[b, d)**.

It [must / might / cannot] have
global extremes?

Answer 93

might

f(x) is not known to be continuous over a closed and bounded domain, so the EVT doesn’t apply.

Question 94

According to the
intermediate value theorem,
suppose

• f(x) is
• *_____** over x ∈ [b, d],
• m = y_min is the global minimum over [b, d], and
• M = y_max is the global maximum over [b, d],

then, for any

• *y_int ∈ [m, M]**, there exists at least one value
• c ∈ [b, d*] s.t.
• *f(c) = y_int**.

Answer 94

continuous

Question 95

According to the
intermediate value theorem,
suppose

• f(x) is
• *continuous** over x ∈ [b, d],
• m = y_min is the global minimum over [b, d], and
• M = y_max is the global maximum over [b, d],

then, for any

• *_____**, there exists at least one value
• *c ∈ [b, d]** s.t.
• *f(c) = y_int**.

Answer 95

yint ∈ [m, M]

Question 96

According to the
intermediate value theorem,
suppose

• f(x) is
• *continuous** over x ∈ [b, d],
• m = y_min is the global minimum over [b, d], and
• M = y_max is the global maximum over [b, d],

then, for any

• *y_int ∈ [m, M]**, there exists at least one value
• *_____** s.t.
• *f(c) = y_int**.

Answer 96

c ∈ [b, d]

Question 97

According to the
intermediate value theorem,
suppose

• f(x) is
• *continuous** over x ∈ [b, d],
• m = y_min is the global minimum over [b, d], and
• M = y_max is the global maximum over [b, d],

then, for any

• *y_int ∈ [m, M]**, there exists at least one value
• *c ∈ [a, b]** s.t.
• *_____**.

Answer 97

f(c) = y_int

Question 98

Visualize the
intermediate value theorem:

Answer 98

Question 99

How does the

• *intermediate value theorem** relate to the
• *extreme value theorem**?

Answer 99

The IVT
depends on the
EVT.

The EVT tells us that if a function is continuous over a closed, bounded domain, there will be a global maximum and a global minimum.

The IVT adds that, in that domain, the function will output every y-value between the global extrema at least once.

Question 100

Trick:
“With
square roots, multiply the
numerator and denominator by the
_____.”

Answer 100

algebraic conjugate

Question 101

• *0** ≠ _____
• *0**

Answer 101

1

Question 102

• *0** = _____
• *0**

Answer 102

indeterminate form

Question 103

Does this
limit exist
at x = 1?

Answer 103

Yes

Question 104

Does this
limit exist
at x = 1?

Answer 104

Yes

Question 105

Does this
limit exist
at x = 1?

Answer 105

Yes