Flash Card Construction Instructions
*** THESE CARDS ARE FOR CALCULUS HONORS, AP CALCULUS AB AND AP
CALCULUS BC. AP CALCULUS BC WILL HAVE ADDITIONAL CARDS FOR THE
COURSE (IN A SEPARATE FILE).
The left column is the question and the right column is the answers. Cut out the
flash cards and paste the question to one side of a note card and the answer to the
other side. Be careful to paste the correct answer to its corresponding question!
COMMON FORMULAS/TRIGONOMETRY/GEOMETRY
Midpoint formula
⎛ x1 + x2 y1 + y2 ⎞
,
⎜
⎟
⎝ 2
2 ⎠
Distance formula (between 2 points)
d = (x2 − x1 )2 + (y2 − y1 )2
Quadratic Formula
−b ± b 2 − 4ac
2a
Pythagorean Theorem
a2 + b2 = c2
sin θ =
opp
y
1
and and
hyp
r
csc θ
cosθ =
adj
x
1
and and
hyp
r
sec θ
tan θ =
opp
y
1
and and
adj
x
cot θ
cot θ =
adj
x
1
and and
opp
y
tan θ
csc θ =
hyp
1
r
and and
y
opp
sin θ
sec θ =
hyp
1
r
and and
adj
x
cosθ
Quotient Identity
tan u
sin u
cosu
Quotient Identity
cot u
cosu
sin u
Pythagorean Identities
sin 2 u + cos 2 u = 1
1 + tan 2 u = sec 2 u
1 + cot 2 u = csc 2 u
Area of a Circle/
Circumference of a circle
A = πr2
C = 2π r
Area of a Parallelogram
A = bh
Area of a
Trapezoid
1
h(b1 + b2 )
2
Area of a Triangle
1
bh
2
30-60-90 triangle
1) Hypotenuse is 2 time short leg
2) Long leg is 3 times short leg
45-45-90 triangle
1) Hypotenuse is 2 times leg
2) Two legs are equal
sin 0
sin 0 = 0
sin 30
sin 30 =
1
2
sin 45
sin 45 =
2
2
sin 60 =
3
2
sin 60
sin 90
sin 90 = 1
cos 0
cos 30
cos 0 = 1
cos 30 =
3
2
cos 45
cos 45 =
2
2
cos 60
cos 60 =
1
2
cos 90
cos 90 = 0
tan 0
tan 0 = 0
tan 30
tan 30 =
3
3
tan 45
tan 45 = 1
tan 60
tan 60 = 3
tan 90
Undefined
sin(α + β ) =
sin(α + β ) = sin α cos β + cos α sin β
sin(α − β ) =
sin(α − β ) = sin α cos β − cos α sin β
cos(α + β ) =
cos(α + β ) = cos α cos β − sin α sin β
cos(α − β ) =
cos(α − β ) = cos α cos β + sin α sin β
sin 2θ =
2sin θ cosθ
cos2θ =
cos2 θ − sin 2 θ
2 cos2 −1
1 − 2sin 2 θ
What is a “solution point”.
(x,y) pair that makes an equations with
an x and y true
P.1
How to find x and y intercepts of an
equation.
x-intercept set y=0 and solve for x
y-intercept set x=0 and solve for y
P.1
What are the three types of symmetry?
P.1
What are the 3 tests for symmetry?
y-axis (replacing x with –x yielding
original equation)
x-axis (replacing y with –y yielding
original equation)
origin (replacing x with –x and y with –y
yielding original equations
y-axis
x-axis
origin
P.1
How to find the points of intersections of
two equations?
Simultaneously solving equations
(elimination, substitution or using
intersect feature of calculator
P.1
The formula for finding the slope
between two points?
y2 − y1
x2 − x1
P.2
What are the 4 types of slope?
P.2
positive, negative, zero, undefined
What is the point slope form of the
equation of a line?
y − y1 = m(x − x1 )
P.2
What are the relationships of slopes
between parallel lines and perpendicular
lines.
parallel lines (same slope),
perpendicular lines (negative reciprocal
slopes)
P.2
How do you calculate an average rate of
change?
f (b) − f (a)
b−a
P.2
What is the slope-intercept equation of a
line?
y = mx + b
P.2
What is the relationship between a
relation and a function?
Function has each x pointing to only one
y value
What does “one-to-one” mean?
each y value is pointed to by only one xvalue
What does “onto” mean?
range consists of all of Y
P.3
P.3
P.3
How do you prove a graph is a function?
passes the Vertical Line Test
P.3
What are the 3 categories of elementary
functions?
P.3
What is the leading coefficient test for
polynomials?
a. algebraic (polynomial, radical,
rational)
b. trigonometric
c. exponential and logarithmic
a. even exponent of leading coefficient
i. leading coefficient > 0 up/up
ii. leading coefficient < 0 down/down
b. odd exponent of leading coefficient
iii. leading coefficient > 0 down left/up right
iv. leading coefficient < 0 up left/down right
P.3
What is an “odd” function?
(symmetric about origin)
What is an even function?
(y-axis symmetry)
What is the relationship of the domain
and range in inverse functions?
The domains and ranges are swapped
P.3
P.3
P.4
How can you determine if a function has
an inverse?
Original function will pass the
Horizontal Line Test
P.4
How can you visually determine of two
functions are inverses of each other?
P.4
The two functions will be reflected about
the line y = x
What are the domains and ranges of
arcsin?
Domain: −1 ≤ 𝑥 ≤ 1
!!
!
Range ! ≤ 𝑦 ≤ !
P.4
What are the domains and ranges of
arccos?
Domain: −1 ≤ 𝑥 ≤ 1
Range 0 ≤ 𝑦 ≤ 𝜋
P.4
What are the domains and ranges of
arctan?
Domain: −∞ < 𝑥 < ∞
!!
P.4
𝑎!
1
P.5
𝑎! 𝑎!
P.5
(𝑎 ! )!
𝑎 !!!
𝑎 !"
P.5
(𝑎𝑏)!
𝑎! 𝑏!
P.5
𝑎!
𝑎!
𝑎 !!!
𝑎
( )!
𝑏
𝑎!
𝑏!
𝑎!!
1
𝑎!
P.5
P.5
P.5
!
Range ! < 𝑦 < !
𝑙𝑛𝑒 !
x
𝑒 !"#
x
What are the domains and ranges of
𝑙𝑛𝑥?
Domain:(0, ∞)
P.5
P.5
P.5
What are the domains and ranges of
𝑒!?
P.5
Range (−∞, ∞)
Domain:(−∞, ∞)
Range 0, ∞)
What is the formula for finding a secant
line?
M sec =
f (x + Δx) − f (x)
Δx
1.1
What is the concept of a limit?
If f(x) becomes arbitrarily close to a
single number L as x approaches c from
either side the limit of f(x), as x
approaches c, is L
What is a generic definition of a tangent
line?
A line that touches curve at one point
1.1
1.1
What are the 3 conditions that need to
be met for a limit to exist?
a.
b.
c.
1.2
What are the 3 conditions where a limit
fails to exist?
lim f (x) exists
x→a +
lim f (x) exists
x→a −
lim f (x) = lim− f (x)
x→a +
x→a
a. unbounded behavior (vertical
asymptote)
b. limit from the left not equal to the
limit from the right
c. oscillating behavior
1.2
lim f (x) = f (c)
What is “well-behaved” function?
x→c
1.3
What are the 3 basic types of algebraic
functions?
a. polynomial
b. rational
c. radical
1.3
What are techniques for finding limits?
1.3
a.
b.
c.
d.
direct substitution (plug n chug)
dividing out (factoring)
rationalizing the numerator
make a table/graph
0 ∞
or
0 ∞
What are the indeterminate forms of a
function?
1.3
sin x
x→0
x
1
1 − cos x
x
0
lim
1.3
lim
x→0
1.3
!
e
lim (1 + 𝑥)!
!→!
1.3
What are the 3 conditions that need to
be met for continuity?
a.
b.
f(a) defined
lim f (x) exists
c.
f(a) = lim f (x)
x→a
x→a
1.4
What is the concept of a “continuous”
function?
when a graph can be drawn without
lifting the pencil
1.4
What is the concept of “everywhere
continuous”?
continuous over the entire number line
1.4
What are 3 types of discontinuity?
a.
b.
c.
hole
infinite (vertical asymptote)
jump
1.4
What is the concept of a “one-sided”
limit?
1.4
when only the limit from the left or the
limit from the right of x=c is defined.
What are 5 types of functions that are
continuous at every point in their
domain?
a.
b.
c.
d.
e.
polynomial functions
rational functions
radical functions
trigonometric functions
exponential and logarithmic
1.4
What does the Intermediate Value
Theorem state?
If f is continuous on the closed interval
[a,b] and k is any number between f(a)
and f(b), then there exists at least one
number c in [a,b] such that f(c) =k
1.4
What is a vertical asymptote?
1.5
How can you determine the difference
between when a hole exists and a
vertical asymptote exists?
Vertical line that is approached but
never touched (end behavior) and is a
result of the denominator of a rational
expression being undefined
If you can cancel a factor out of
denominator it is a hole
1.5
What is a horizontal asymptote?
Horizontal line that is approached but
never touched (end behavior) and is a
result of the denominator growing faster
than the numerator
1.6
𝑐
!→! 𝑥 !
0
𝑐
!→!! 𝑥 !
0
lim
1.6
lim
1.6
lim 𝑒 !
!→!!
0
1.6
lim 𝑒 !!
!→!
1.6
0
!!
!
𝑥>0
1.6
(sneaky technique)
!!
!!
𝑥<0
1.6
What are the 3 tests for determining
horizontal asymptotes?
1.6
1
1
(sneaky technique)
num exponent > den exponent, no
asymptote
num exponent < den exponent, y=0
num exponent = den exponent,
leadingcoefficient
y=
leadingcoefficient
f (x + Δx) − f (x)
Δx→0
Δx
What is the definition of the
derivative of a function using
limits?
f '(x) = lim
2.1
𝑓 𝑥 − 𝑓(𝑐)
!→!
𝑥−𝑐
What is an alternate form of the
derivative function using limits?
𝑓 ! 𝑐 = lim
2.1
f (x + Δx) − f (x)
Δx
What is the difference quotient?
2.1
What are the 3 cases where a derivative
fails to exist?
2.1
a.
b.
c.
any point of discontinuity
cusp
vertical tangent line
Differentiation Rules:
Constant Rule
d
[c] = 0
dx
Differentiation Rules:
Simple Power Rule
d n
[x ] = nx n −1
dx
Differentiation Rules:
Constant Multiple Rule
d
[cf (x)] = cf '(x)
dx
Differentiation Rules:
Sum and Difference Rules
d
[ f (x) ± g(x)] = f '(x) ± g '(x)
dx
2.2
2.2
2.2
2.2
d
[sin x]
dx
2.2
cos x
d
[cos x] =
dx
−sin x
𝑑 !
[𝑒 ]
𝑑𝑥
𝑒!
2.2
2.2
What is the standard
position function?
𝑠 𝑡 = −16𝑡 ! + 𝑉! 𝑡 + 𝑆!
-4.9 can be substituted if calculating in
meters instead of feet
2.2
d
[tan x] =
dx
sec 2 x
d
[csc x] =
dx
−csc x cot x
d
[sec x] =
dx
sec x tan x
d
[cot x] =
dx
−csc 2 x
2.3
2.3
2.3
2.3
Differentiation Rules:
Product Rule
f (x)g'(x) + g(x) f '(x)
first d second + second d first
2.3
Differentiation Rules:
Quotient Rule
g(x) f '(x) − f (x)g'(x)
g(x)2
bottom d top – top d bottom over
bottom squared
2.3
Differentiation Rules:
Chain Rule
f '(g(x))g'(x)
d outer d inner (don’t touch the stuff)
Differentiation Rules:
General Power Rule
nu n−1u'
d
[sin u] =
dx
(cosu)u'
d
[cosu] =
dx
(−sin u)u'
d
[tan u] =
dx
(sec 2 u)u'
d
[cot u] =
dx
−(csc 2 u)u'
2.4
2.4
2.4
2.4
2.4
2.4
d
[secu] =
dx
2.4
(secu tan u)u'
d
[cscu] =
dx
−(cscucot u)u'
𝑑
[ln 𝑥]
𝑑𝑥
1
,𝑥 > 0
𝑥
𝑑
[ln| 𝑢|]
𝑑𝑥
𝑢!
𝑢
log ! 𝑥
1
𝑙𝑛𝑥
𝑙𝑛𝑥 𝑜𝑟
𝑙𝑛𝑎
𝑙𝑛𝑎
𝑑 !
[𝑎 ]
𝑑𝑥
𝑙𝑛𝑎 𝑎 !
2.4
2.4
2.4
2.4
2.4
𝑑 !
[𝑎 ]
𝑑𝑥
𝑙𝑛𝑎 𝑎!
𝑑𝑢
𝑑𝑥
𝑑
[log ! 𝑥]
𝑑𝑥
1
𝑙𝑛𝑎 𝑥
2.4
2.4
𝑑
[log ! 𝑢]
𝑑𝑥
1 𝑑𝑢
𝑢!
𝑜𝑟
𝑙𝑛𝑎 𝑢 𝑑𝑥
𝑙𝑛𝑎 𝑢
𝑑 !
[𝑒 ]
𝑑𝑥
𝑒 ! 𝑢!
What is the explicit form of an
equation?
when an equation is solved for one
variable
Inverse functions have what types
of slopes at inverse pairs of points?
reciprocal slopes
2.4
2.4
2.5
2.6
d
[arcsin u] =
dx
u'
1− u 2
2.6
d
[arccosu] =
dx
1− u 2
d
[arctan u] =
dx
u'
1+ u 2
−u'
2.6
2.6
d
[arc cot u] =
dx
2.6
d
[arcsecu] =
dx
2.6
−u'
1+ u 2
u'
u u 2 −1
d
[arc cscu] =
dx
−u'
u u 2 −1
2.6
What is a related rate derivative
usually taken with respect to?
time
2.7
What is the formula for the volume
of a cone?
𝑉=
𝜋 !
𝑟 ℎ
3
2.7
What is the formula for the volume
of a sphere?
2.7
4
𝑉 = 𝜋𝑟 !
3
What is a another name for a
tangent line of approximation
called?
linear approximation
What method uses a tangent
line to approximate the y-values
of a function?
Newton’s method
2.8
2.8
What is a “maximum”?
f(c) > all f(x) on an interval
What is a “minimum”?
f(c) < all f(x) on an interval
What is the difference between critical
numbers and critical points?
critical numbers are x-values and critical
points are (x,y). Critical numbers are
found when f '(c) = 0 or where f '(c)
does not exist.
3.1
3.1
3.1
What theorem state if f is continuous
on a closed interval [a,b], then
f has both a minimum and a maximum
on the interval
3.1
Extreme Value
Theorem
Where does the derivative fail to
identify possible extrema?
endpoints
What does Rolle’s Theorem state?
if f(a) = f(b) then there exists at least one
number c in (a,b) such that f '(c) = 0
3.1
3.2
What does the Mean Value Theorem
state?
f '(c) =
f (b) − f (a)
b−a
3.2
What are two major similarities between
Rolle’s Theorem and the Mean Value
Theorem?
Function must be 1) continuous and
2) differentiable
3.2
What is meant by “increasing” in terms
of a derivative?
3.3
f '(x) > 0 for all x in (a,b)
f '(x) < 0 for all x in (a,b)
What is meant by “decreasing” in terms
of a derivative?
3.3
f '(x) = 0 for all x in (a,b)
What is meant by “constant” in terms of
a derivative?
3.3
What does “strictly monotonic” mean”?
When a function is either increasing or
decreasing on entire interval
3.3
a.
if f '(x) changes from increasing to
decreasing at x =c then f '(c) is a relative
What does the first derivative test state?
maximum
b.
if f '(x) changes from decreasing to
increasing at x =c then f '(c) is a relative minimum
c.
if f '(x) does not change signs at x =c then
f '(c) is a neither a relative maximum or relative
3.3
minimum
How do you use the second derivative to
determine concavity?
a.
if f ''(x) > 0 , for all x in an interval
f is concave upward
b.
if f ''(x) < 0 , for all x in an interval
f is concave downward
3.4
What are “points of inflection”?
3.4
where f ''(c) = 0 or f ''(c) is undefined
(where a graph goes from concave
upward to concave downward or vice
versa
How do you use the second derivative to
determine relative extrema using critical
numbers?
3.4
a.
if f ''(c) > 0 ,then f(c) is a relative
minimum
b.
if f ''(c) < 0 ,then f(c) is a relative
maximum
c.
if f ''(c) = 0 then use must use the
First Derivative Test
In optimization problems what
is the equation that is to
be optimized called?
primary equation
What is a differential equation?
an equation that contains a derivative
What is the equation for
a tangent line of approximation
(linear approximation)?
𝑦 = 𝑓 𝑐 + 𝑓′(𝑐)(𝑥 − 𝑐)
3.6
3.7
3.7
0𝑑𝑥
C
4.1
∫ du =
u+C
∫ kf (x)dx
k ∫ f (x)dx
∫ [ f (x) ± g(x)]dx
∫ f (x)dx ± ∫ g(x)dx
∫ x dx =
x n+1
+C
n +1
∫ cos x dx =
sin x + C
∫ sin x dx =
− cos x + C
4.1
4.1
4.1
n
4.1
4.1
4.1
∫ (sec x)dx
2
tan x + C
4.1
∫ sec x tan x dx
4.1
sec x + C
∫ (csc x)dx =
− cot x + C
∫ csc x cot x dx
− csc x + C
2
4.1
4.1
𝑒! + 𝐶
𝑒 ! 𝑑𝑥
4.1
𝑎 ! 𝑑𝑥
(
1 !
)𝑎 + 𝐶
𝑙𝑛𝑎
4.1
1
𝑑𝑥
𝑥
ln |𝑥| + 𝐶
To change a general solution
into a particular solution what is
needed?
an initial condition
!
!!! 𝑎! is what type of notation?
sigma notation
4.1
4.1
4.2
!
𝑐
4.2
!→!
!
𝑖
4.2
𝑐𝑛
!→!
𝑛(𝑛 + 1)
2
!
𝑖
!
𝑛(𝑛 + 1)(2𝑛 + 1)
6
𝑖
!
𝑛! (𝑛 + 1)
4
!→!
4.2
!
!→!
4.2
!
Left Rectangle Rule
𝑏−𝑎
(𝑓(𝑥!) + ⋯ 𝑓 𝑥!!! )
𝑛
Right Rectangle Rule
𝑏−𝑎
(𝑓(𝑥!) + ⋯ 𝑓 𝑥! )
𝑛
4.2
4.2
!
The definite integral as the
area of a region
𝑓 𝑥 𝑑𝑥
!
4.3
!
𝑓 𝑥 𝑑𝑥
0
!
4.3
!
!
𝑓 𝑥 𝑑𝑥
−
!
𝑓 𝑥 𝑑𝑥
!
4.3
!
!
𝑓 𝑥 𝑑𝑥
!
with point c between a and b
4.3
!
𝑓 𝑥 𝑑𝑥 +
!
𝑓 𝑥 𝑑𝑥
!
!
!
𝑘𝑓(𝑥)𝑑𝑥)
𝑘
!
𝑓 𝑥 𝑑𝑥
!
4.3
!
!
𝑓 𝑥 ± 𝑔 𝑥 𝑑𝑥
!
!
𝑓 𝑥 𝑑𝑥 ±
!
𝑔 𝑥 𝑑𝑥
!
4.3
Trapezoidal Rule
4.3
Fundamental Theorem of
Calculus
𝑏−𝑎
[𝑓 𝑥! + 2𝑓 𝑥! + ⋯ 2𝑓(𝑥!!!)
2𝑛
+ 𝑓(𝑥!) ]
!
𝑓 𝑥 𝑑𝑥 = 𝐹 𝑏 − 𝐹(𝑎)
!
4.4
Mean Value Theorem
For Integrals
!
𝑓 𝑥 𝑑𝑥 = 𝑓(𝑐)(𝑏 − 𝑎)
!
4.4
Average value of a function
!
1
𝑓 𝑥 𝑑𝑥
𝑏−𝑎 !
Second Fundamental
Theorem of Calculus
!
𝑑
[ 𝑓 𝑡 𝑑𝑡] = 𝑓 𝑥
𝑑𝑥 !
Net Change Theorem
!
4.4
4.4
!
4.4
𝐹 ! 𝑥 = 𝐹 𝑏 − 𝐹(𝑎)
∫ u du =
u n+1
+C
n +1
n
4.5
𝑘𝑓 𝑥 𝑑𝑥
𝑘
!
𝑓 𝑥 𝑑𝑥 (even function)
!!
2
𝑓 𝑥 𝑑𝑥
4.5
!
𝑓 𝑥 𝑑𝑥
!
4.5
!
𝑓 𝑥 𝑑𝑥 (odd function)
!!
0
4.5
du
∫u =
ln u + C
∫ a du =
⎛ 1 ⎞ u
⎜
⎟a + C
⎝ ln a ⎠
4.6
u
4.6
∫ sin u du =
−cosu + C
4.6
∫ cosu du =
sin u + C
∫ tan u du =
− ln cosu + C
4.6
4.6
∫ cot u du =
ln sin u + C
∫ sec u du =
ln secu + tan u + C
∫ cscu =
− ln cscu + cot u + C
∫ sec u du =
tan u + C
4.6
4.6
4.6
2
4.6
∫ csc u du =
2
−cot u + C
4.6
∫ secu tan u du =
secu + C
∫ cscu cot u du =
−cscu + C
du
1
u
arctan + C
a
a
4.6
4.6
∫a +u =
2
4.7
2
du
∫ a −u =
u
arcsin + C
a
du
u
1
arcsec + C
a
a
2
2
4.7
∫u u −a =
2
4.7
2
What is a differential equation?
an equation that includes a derivative
What is Euler’s Method?
a numerical approach to approximating
the particular solution to a differential
equation
5.1
5.1
What is the solution to a exponential
growth or decay problem?
𝑦 = 𝐶𝑒 !"
5.2
What is k in a half-life problem?
1
ln (2)
𝑡
=𝑘
5.2
What is the process of collecting all
terms with x’s and y’s on opposite sides
of the equal sign called?
5.2
separation of variables
How do you find the area between two
curves?
b
∫ [ f (x) − g(x)]dx
a
6.1
Disk Method
Horizontal Axis of Revolution
b
π ∫ [R(x)]2 dx
a
6.2
Disk Method
Vertical Axis of Revolution
d
π ∫ [R(y)]2 dy
c
6.2
Washer Method
Horizontal Axis of Revolution
b
π ∫ ([R(x)]2 − [r(x)]2 )dx
a
6.2
Washer Method
Vertical Axis of Revolution
d
π ∫ ([R(y)]2 − [r(y)]2 )dy
c
6.2
Volume of solid with known cross
section perpendicular to x-axis
b
∫ A(x)dx
a
6.2
Volume of solid with known cross
section perpendicular to y-axis
d
∫ A(y)dy
c
6.2