Given that f(x) is continuous on an open interval that contains x = c, if f'(c) = 0 and f''(c) < 0, then...

f(x) has a local maximum at x = c

Which method is most efficient in determining if a function is increasing or decreasing over a given interval?

Check to see if f'(x) > 0 or f'(x) < 0

How can we create a slope field for the image below?

Plug appropriate coordinates into the given equation, y'

Given that f(x) is continuous on an open interval that contains x = c, if f'(c) = 0 and f''(c) > 0, then...

f(x) has a local minimum at x = c

A critical point (or value) of f(x) is when...

f'(c) = 0 or if it is undefined

Which method can be used to determine the relative maximum or minimum of the continuous function, f(x)?

Find the crit. pts of f(x), use the first derivative test

Which are two scenarios that tells us a function can fail to be differentiable at a point?

cusp and oscillating discontinuity

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AP Calculus AB Quiz Review: Definitions, Theorems, and other facts

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1
The ∫ csc²(x) dx is...
tan(x) + C
sin(x) + C
-cot(x) + C
-cos(x) + C
2
A particle changes direction when...
v(t) changes sign.
a(t) changes sign.
The particle never changes direction.
s(t) changes sign.
3
Which one is a Left-Riemann Sum?
B
None of them
A
C
4
Which point is an absolute or global maximum?
(4,-1)
(2,1)
(-3,2)
(-1,-3)
5
Given that f(x) is continuous on an open interval that contains x = c, if f'(c) = 0 and f''(c) < 0, then...
f(x) has a local minimum at x = c
the test fails and there is neither a local maximum or minimum
f(x) has a local maximum at x = c
the limit of the function approaches infinity as x -> c
6
Which method is most efficient in determining if a function is increasing or decreasing over a given interval?
Check to see if f'(x) > 0 or f'(x) < 0
Using your finger and tracing along the f"(x) curve
Plugging in all of the x-values on a given interval into f(x)
Using indirect substitution
7
The derivative of e^x is...
ln|x|
0
1
ex
8
A particle is moving along the x-axis. Which direction is used to describe the motion of the particle?
in and out
left and right
60°
up and down
9
Determine the integral of the following.
arccot(x) + C
arctan(x) + C
arcsin(x) + C
arccos(x) + C
10
Which choice represents the area of a rectangle?
(1/2)(base)(height)
(base)(height)(width)
(base)(height)
πr2
11
If f'(x) = 0, then...
f(x) = 0
f(x) has a horizontal tangent
f(x) is decreasing
f(x) is increasing
12
If f(x) is concave down in the interval between x and a, then it's linear approximation will be...
an exact measure
an overestimation
undefined
an underestimation
13
Which is the correct logarithmic expansion for ln(a/b)?
ln(a) + ln(b)
ln(a) * ln(b)
ln(a) - ln(b)
ln(a) / ln(b)
14
How can we create a slope field for the image below?
Plug appropriate coordinates into the given equation, y'
Plug appropriate coordinates into the equation, y''
Draw random slopes and hope for the best
Use the Mean Value Theorem
15
Geometrically, the average rate of change is represented by a...
polynomial
tangent line
piecewise function
secant line
16
Based on the intermediate value theorem, the function f(x) must be continuous over...
a partially closed interval
a closed interval
an open interval
a partially open interval
17
The derivative of csc(x) is...
csc(x)cot(x)
sec(x)tan(x)
-csc(x)cot(x)
sin(x)cos(x)
18
Let f be a function that satisfies the following three hypotheses (shown). Which theorem is it?
Rolle's Theorem
Intermediate Value Theorem
Squeeze Theorem
Mean Value Theorem
19
If a function is continuous, then...
it is always differentiable
it is never differentiable
it is sometimes differentiable
20
The following identity represents what?
The Pythagorean Theorem
The derivative of a function
The Average Value
The average rate of change
21
Which choice represents the area of a circle?
πr2
(base)(height)(width)
(1/2)(base)(height)
(base)(height)
22
Which choice represents the area of a triangle?
(base)(height)(width)
πr2
(base)(height)
(1/2)(base)(height)
23
Jhoel wants to fill an object with sand (image shown) @ a rate of 2.9 cm³ / sec. Which formula should he be considering?
Vcylinder = (⫪r2)(height)
Vpyramid = (1/3)(base)(height)
Vsphere = (4/3)(⫪r3)
Vcone = (1/3)(⫪r2)(height)
24
A particle is moving to the left on the x-axis when...
v(t) < 0
s(t) < 0
There is no situation where the particle is moving.
a(t) < 0
25
Given that f(x) is continuous on an open interval that contains x = c, if f'(c) = 0 and f''(c) > 0, then...
the limit of the function approaches infinity as x -> c
f(x) has a local maximum at x = c
f(x) has a local minimum at x = c
the test fails and there is neither a local maximum or minimum
26
The following formula is used when you want to find the...
average rate of change of a function
critical point of a function
average value of a function
slope of a tangent line
27
If f'(x) > 0, then...
f(x) is constant
f(x) is decreasing
f(x) is increasing
f(x) = 0
28
The ∫ sin(x) dx is...
sin(x) + C
-cot(x) + C
tan(x) + C
-cos(x) + C
29
At which x-value is there a jump discontinuity?
x=0
x=-2
x=5.5
x=1
30
The derivative of ln(x) is...
1/x
1/x + C
1/x2
1/c
31
Which of the three Riemann sums shows an underestimation?
None of them
C
A
B
32
A particle is not moving on the x-axis when...
a(t) = 0
There is no situation where the particle is not moving.
s(t) = 0
v(t) = 0
33
A critical point (or value) of f(x) is when...
f'(c) = a finite value
f'(c)=0 only
the limit of f(x) as x->c exists
f'(c) = 0 or if it is undefined
34
Which of the following is a method that can be used in determining the limit of a function?
Slope Formula
Pythagorean Theorem
Midpoint Formula
Conjugates
35
Which is the correct logarithmic expansion for ln(a^b)?
b + ln(a)
ln(ab)
b * ln(a)
b - ln(a)
36
The integral of 1/x is...
ln(x) + C
ln|x|
ln(x)
ln|x| + C
37
The derivative of sin(x) is...
cos(x)
-sin(x)
sin(x)
-cos(x)
38
Shannel wants to fill up a cup w/ tea (image shown) at a rate of 0.3 in³ / sec. Which formula should she be considering?
Vpyramid = (1/3)(base)(height)
Vcone = (1/3)(⫪r2)(height)
Vsphere = (4/3)(⫪r3)
Vcylinder = (⫪r2)(height)
39
Which type of number does not belong to the set of real numbers?
decimals
fractions
whole numbers
imaginary numbers
40
If k is a constant, then d/dx of k³ is...
1
3k
0
k/3
41
Csc(x) can be rewritten as...
1/sin(x)
1/tan(x)
1/cos(x)
1/sec(x)
42
The quotient rule of derivatives is expressed as...
(f / g)' = (f'g - fg') - g2
(f / g)' = (f'g - fg') * g2
(f / g)' = (f'g - fg') / g2
(f / g)' = (f'g - fg') + g2
43
Which theorem is described in the image shown?
Mean Value Theorem
Rolle's Theorem
Intermediate Value Theorem
Squeeze Theorem
44
The following exponent rule, (x^a)^b, is equivalent to...
xa*b
xa+b
xa-b
xa/b
45
The image shown is of what type of discontinuity?
Jump Discontinuity
Infinite discontinuity
Removable Discontinuity
Oscillating Discontinuity
46
Which method can be used to determine the relative maximum or minimum of the continuous function, f(x)?
Substitute random values to narrow down the highest or lowest possible y
Find the crit. pts of f(x), use the first derivative test
The Squeeze Theorem
Using the average rate of change
47
Which are two scenarios that tells us a function can fail to be differentiable at a point?
Indirect substitution and a jump discontinuity
Reverberating discontinuity and a cusp
cusp and oscillating discontinuity
There are not two. It only happens when there is a cusp.
48
The notation, f'(x), is normally associated with...
acceleration
position
magnitude
velocity
49
Sec(x) can be rewritten as...
1/tan(x)
1/cos(x)
1/sec(x)
1/sin(x)
50
Based on the mean value theorem, the function f(x) must be differentiable over...
a partially closed interval
a closed interval
a partially open interval
an open interval
51
If a function is differentiable, then...
it is always continuous
it is never continuous
it is sometimes continuous
52
There is a removable discontinuity at x=4.
True
False
53
The ∫ sec(x)tan(x) dx is...
sec(x) + C
tan(x) + C
-cot(x) + C
-csc(x) + C
54
If k is a constant, then d/dx of kf(x) is...
kf'(x)
0
k'f(x) ⋅ f'(x)
1
55
Which theorem is described in the image shown?
Rolle's Theorem
Intermediate Value Theorem
Mean Value Theorem
Squeeze Theorem
56
A particle is moving to the right on the x-axis when...
v(t) > 0
There is no situation where the particle is moving.
s(t) > 0
a(t) > 0
57
The following is an example of a...
Slope Field
The Squeeze Theorem
Differentiable Function
Continuous funciton
58
The constant multiple rule of integration, ∫ kf(x) dx, is also expressed as...
k ⋅ f(x) + C
f(x) + C
k + C
k ∫ f(x) dx
59
The formula used to determine a linear approximation equation is...
y = mx + b
Midpoint Formula
a2 + b2 = c2
Quadratic Equation
60
The ∫ csc(x)cot(x) dx is...
tan(x) + C
sec(x) + C
-cot(x) + C
-csc(x) + C
61
Given that f(x) is continuous on an open interval that contains x = c, if f'(c) = 0 and f''(c) = 0, then...
f(x) has a local maximum at x = c
the limit of the function approaches infinity as x -> c
the test fails and there is neither a local maximum or minimum
f(x) has a local minimum at x = c
62
The derivative of cos(x) is...
sin(x)
cos(x)
-sin(x)
-cos(x)
63
If f'(x) < 0, then...
f(x) is decreasing
f(x) = 0
f(x) is constant
f(x) is increasing
64
ⁿ√(x) can be rewritten as...
x-n
x-1/n
xn
x1/n
65
Determine the integral of the following.
arcsin(x) + C
arccot(x) + C
arctan(x) + C
arccos(x) + C
66
The ∫ sec²(x) dx is...
sin(x) + C
-cot(x) + C
-cos(x) + C
tan(x) + C
67
Which point is an absolute or global minimum?
(4,-1)
(2,1)
(-1,-3)
(-3,2)
68
The derivative of f(x) + g(x) is...
f(x) * g(x)
f'(x)g(x) + f(x)g'(x)
f'(x) - g'(x)
f'(x) + g'(x)
69
When is there a point of inflection on a graph?
When the 1st derivative shows a change in slope
When the 2nd derivative shows a change in concavity
When the slope of the tangent is undefined only
When the slope of the tangent is = 0 only
70
In order to apply the Intermediate Value Theorem, your function must be...
Defined only
Continuous and/or differentiable
A step function
A piecewise function with a removable discontinuity
71
The function f is continuous at x=c if...
If the function is defined over the interval [a,b]
If there is only one type of discontinuity
f(c) exists, the lim of f(x) as x->c = L exists, and f(c) = L
If there is a removable discontinuity
72
If s(t) is a position function, then acceleration is represented as...
the integral of s(t)
s''(t)
s'(t)
0
73
Based on the mean value theorem, the function f(x) must be continuous over...
a partially closed interval
a closed interval
a partially open interval
an open interval
74
Anais wants to find the volume of a partially inflated basketball. Which shape would best represent this situation?
circle
cone
sphere
cylinder
75
Which theorem is described in the image shown?
The Fundamental Theorem of Calculus
Intermediate Value Theorem
Squeeze Theorem
Mean Value Theorem
76
A point of inflection is a point on the curve where...
there is a vertical tangent
the value is zero
there is a horizontal tangent
concavity changes
77
The graph f'(x) is shown below. At which x-value is there a maximum on f(x) from the choices given?
x = 0.5
x = 0
x = 0.28
x = -0.28
78
Based on the intermediate value theorem, the function f(x) must be differentiable over...
a partially open interval
an open interval
This theorem does not rely on differentiability
a closed interval
79
Which interval in the graph shown below represents negative area?
[c,b]
[a,c]
[b,c]
[a,b]
80
Which one is a Midpoint-Riemann Sum?
None of them
C
A
B
81
Using the Riemann sums helps to approximate the...
slope of a tangent line
hypotenuse of a right triangle
critical point of a function
area under the curve
82
A particle is moving along the y-axis. Which direction is used to describe the motion of the particle?
in and out
60°
up and down
left and right
83
The following is a graph of...
y = sec(x)
y = cos(x)
y = sin(x)
y = tan(x)
84
The derivative of -(cos(x)) is...
-sec2(x)
sin(x)
tan(x)
-sin(x)
85
Tan(x) can be rewritten as...
sin(x) / cos (x)
-sin(x) / cos(x)
cos(x) / sin(x)
-cos(x) / sin(x)
86
Cot(x) can be rewritten as...
-sin(x) / cos(x)
cos(x) / sin(x)
sin(x) / cos (x)
-cos(x) / sin(x)
87
The derivative of sec(x) is...
-tan(x)
sec(x)tan(x)
cos(x)
tan(x)
88
The graph f'(x) is shown below. At which x-value is there a minimum on f(x)?
x = 0.5
x = 0.28
x = 0
x = -0.28
89
If n is a constant, then d/dx of axⁿ is...
n*axn+1 / (n + 1)
n*axn-1
n*axn-1 / (n - 1)
n*axn+1
90
Which one is a Right-Riemann Sum?
A
None of them
B
C
91
Which theorem is shown?
The Fundamental Theorem of Calculus
The Mean Value Theorem
The Squeeze Theorem
The Intermediate Value Theorem
92
Which is the correct logarithmic expansion for ln(ab)?
ln(a) / ln(b)
ln(a) - ln(b)
ln(a) * ln(b)
ln(a) + ln(b)
93
The constant function rule of integration, ∫ k dx, is also expressed as...
kx
0
1
kx + C
94
The product rule of derivatives is expressed as...
(f ⋅ g)' = f'g + fg'
(f ⋅ g)' = f'g * fg'
(f ⋅ g)' = f'g - fg'
(f ⋅ g)' = f'g / fg'
95
Which interval in the graph shown below represents positive area?
[b,c]
[a,c]
[c,b]
[a,b]
96
To solve for a particular solution of a differential equation, which is a possible method you could use?
First separate the variables, if needed, and integrate both sides.
The Distance Formula
L'Hospital's Rule
Mean Value Theorem
97
A differential equation is...
the formula used to determine the Midpoint Riemann Sum
a point on f'(x)
the midpoint formula
an equation containing one or more derivatives
98
If f(x) is concave up in the interval between x and a, then it's linear approximation will be...
an overestimation
undefined
an exact measure
an underestimation
99
The derivative of cot(x) is...
tan(x)
cos(x)
sec2(x)
-csc2(x)
100
When would you need to apply the Squeeze Theorem?
To show that the AROC = IROC
To show that there exists a max and min value in an interval
To find the limit of one function using the help of 2 other functions
To show that there exists a specific y-value along a continuous function

AP Calculus AB Quiz Review: Definitions, Theorems, and other facts Flashcards | StudBud