Math Flash cards
Algebra/Trig Review Flash Cards
Formula (Precalculus)
quadratic formula
Formula(Precalculus)
Formula for the slope of a line
Formula (Precalculus)
equation of a circle
Definition (Trigonometry)
sec, csc, tan, cot
Math Flash cards
Changes
Formula (Precalculus)
equation of a line in various forms
Definition (Precalculus)
definition of a circle
Definition (Trigonometry)
sin, cos, tan
Formula (Trigonometry)
Fundamental Trig Identities
Mark Stankus has modifed these flash cards to be specific to the course Math 142 at the California Polytechnic State University of San Luis Obispo.
c
Copyright 2009
Djun. M. Kim. This work is licensed under a Creative
Commons Attribution-Noncommercial-Share Alike 2.5 Canada License.
http://viu.ca/math
http://teaching.puregin.org/math-flash-cards
These cards are part of a project to create free,
high quality, printable flash cards for mathematics
study. You are free to share, modify, and distribute
these for non-commercial use, as long as you mention the source
URL. Suggestion for additional maR
terial and
corrections are welcome. Parts of this
set based on LATEX source from Jason Underdown
(http://www.physics.utah.edu/˜jasonu).
c
Copyright 2009
Djun. M. Kim. This work is licensed under a Creative
Commons Attribution-Noncommercial-Share Alike 2.5 Canada License.
http://viu.ca/math
Form
Equation
point–slope
y − y1 = m(x − x1 )
slope–intercept
y = mx + b
standard
Ax + By + C = 0
http://teaching.puregin.org/math-flash-cards
The solutions or roots of the
quadratic equation
ax2 + bx + c = 0 are given by
√
−b ± b2 − 4ac
x=
2a
A circle with center (h, k) and radius r is
the set of points whose distance from the
point (h, k) is exactly r.
m=
y2 − y1
x2 − x1
Often the idea of circle and the circle together with its
inside are used interchangably.
hyp
θ
adj
opp
opp
sin θ =
hyp
adj
cos θ =
hyp
opp
tan θ =
adj
sin2 x + cos2 x = 1
sec2 x − tan2 x = 1
csc2 x − cot2 x = 1
The equation of a circle of radius r
centered at the point (a, b) is:
(x − a)2 + (y − b)2 = r2
sec θ =
1
cos θ
tan θ =
sin θ
cos θ
cot θ =
cos θ
sin θ
csc θ =
1
sin θ
Formula (Trigonometry)
Double Angle Identities
Formula (Trigonometry)
Formula with sin(t) and cos(t)
Formula (Trigonometry)
Formula with csc(t) and cot(t)
Definition (Precalculus,Calculus I)
Definition of odd function
Math Flash cards
Changes
Formula (Trigonometry)
Half Angle Identities
Formula (Trigonometry)
Formula with sec(t) and tan(t)
Definition (Precalculus,Calculus I)
Definition of even function
Math Flash cards
Integral Calculus Flash Cards
Antiderivative (Calculus I)
Z
xn dx
(n 6= −1)
Integral Calculus
1
sin2 x = (1 − cos 2x)
2
1
cos2 x = (1 + cos 2x)
2
sec2 (t) = 1 + tan2 (t)
tan2 (t) = sec2 (t) − 1
f (−x) = f (x)
for all x
sin 2x = 2 sin x cos x
cos 2x = cos2 − sin2 x
cos 2x = 2 cos2 x − 1
cos 2x = 1 − 2 sin2 x
sin2 (t) + cos2 (t) = 1
sin2 (t) = 1 − cos2 (t)
cos2 (t) = 1 − sin2 (t)
csc2 (t) = 1 + cot2 (t)
cot2 (t) = csc2 (t) − 1
Examples: x2 , cos(x)
These cards are part of a project to create free, high
quality, printable flash cards for mathematics study.
You are free to share, modify, and distribute these
for non-commercial use, as long as you mention the
source URL. Suggestion for additional material and
corrections are welcome. If you found these useful,
consider contributing back to this project with your
time.
f (−x) = −f (x)
for all x
Examples: x, sin(x)
c
Copyright 2009
Djun. M. Kim. This work is licensed under a Creative
Commons Attribution-Noncommercial-Share Alike 2.5 Canada License.
http://viu.ca/math
http://teaching.puregin.org/math-flash-cards
xn+1
+C
n+1
Mark Stankus has modifed these flash cards to be specific to the course Math 142 at the California Polytechnic State University of San Luis Obispo.
c
Copyright 2009
Djun. M. Kim. This work is licensed under a Creative
Commons Attribution-Noncommercial-Share Alike 2.5 Canada License.
http://viu.ca/math
This only works if n is a constant. We know this from
Calculus I if n is a rational number. We will learn this
in Calculus II if n is an irrational number, like n = π.
http://teaching.puregin.org/math-flash-cards
Antiderivative (7.3)
Z
Antiderivative (7.3)
Z
x
ax dx
e dx
Integral Calculus
Antiderivative (Calculus I)
Integral Calculus
Antiderivative (Calculus I)
Z
Z
sin x dx
cos x dx
Integral Calculus
Antiderivative (7.2)
Z
Integral Calculus
Antiderivative (Calculus I)
Z
1
dx
x
sec2 x dx
Integral Calculus
Antiderivative (Calculus I)
Z
Integral Calculus
Antiderivative (Calculus I)
Z
2
csc x dx
sec x tan x dx
Integral Calculus
Antiderivative (Calculus I)
Z
Integral Calculus
Antiderivative (7.2)
Z
csc x cot x dx
Integral Calculus
tan x dx
Integral Calculus
ax
+C
ln a
ex + C
This only works for constants a
with a 6= 1.
sin x + C
− cos x + C
tan x + C
ln |x| + C
sec x + C
− cot x + C
ln | sec x| + C
− csc x + C
Antiderivative (7.2)
Antiderivative (7.2)
Z
Z
cot x dx
sec x dx
Integral Calculus
Antiderivative (7.2)
Integral Calculus
Antiderivative (7.6)
Z
Z
csc x dx
√
dx
− x2
a2
Integral Calculus
Antiderivative (7.6)
Z
Integral Calculus
Antiderivative (7.6)
dx
2
a + x2
Z
dx
x x 2 − a2
√
Integral Calculus
First Step (Calculus I)
Z
Integral Calculus
Antiderivative (Calculus I)
Z
2
sin (x) dx
sin2 (10x) dx
Integral Calculus
Antiderivative (Calculus I)
Integral Calculus
Vocabulary (Calculus I)
Upper limit of integration in
Z
2
cos (75x) dx
Z2
x2 dx
7
Integral Calculus
Integral Calculus
ln | sec x + tan x| + C
sin−1
x
+C
a
ln | sin x| + C
ln | csc x − cot x| + C
or
− ln | csc x + cot x| + C
or
ln | − csc x + cot x| + C
This only works for a constant a.
1
x
sec−1 | | + C
a
a
1
x
tan−1 + C
a
a
This only works for constants
nonzero a.
Use sin2 (x) = 12 (1 − cos(2x)).
Use sin2 (10x) = 12 (1 − cos(20x)).
2
Use cos2 (75x) = 12 (1 + cos(150x)).
Vocabulary(Calculus I)
Vocabulary(Calculus I)
Lower limit of integration in
Z2
Integrand of
Z2
2
x dx
7
x2 dx
7
Integral Calculus
Vocabulary(Calculus I)
Integral Calculus
Property (Calculus I)
Variable of integration of
Z2
Za
f (x) dx
x2 dx
a
7
Integral Calculus
Integral Calculus
Property (Calculus I)
Property (Calculus I)
Za
Za
f (x) dx if f is an even function
−a
f (x) dx if f is an odd function
−a
Integral Calculus
True or False (Calculus I)
Zb
?
Match It! (7.6)
Zb
f (x) dx=
a
Integral Calculus
f (u) du
Domain of arcsin(x)
a
Integral Calculus
Match It! (7.6)
Integral Calculus
Match It! (7.6)
Range of arcsin(x)
Integral Calculus
Domain of arccos(x)
Integral Calculus
x2
7
0
x
0
Za
f (x) dx
2
0
Graph an example to understand
this!
[−1, 1]
[−1, 1]
This is a great property because plugging in zero is
often easier than plugging in −a.
True. Think in terms of graphs.
π π
[− , ]
2 2
Match It! (7.6)
Match It! (7.6)
Range of arccos(x)
Domain of arctan(x)
Integral Calculus
Match It! (7.6)
Integral Calculus
Match It! (7.6)
Range of arctan(x)
Domain of arcsec(x)
Integral Calculus
Match It! (7.6)
Integral Calculus
Formula (7.6)
d
arcsin(x)
dx
Range of arcsec(x)
Integral Calculus
Integral Calculus
Formula (7.6)
Formula (7.6)
d
arcsin(u)
dx
d
arccos(x)
dx
Integral Calculus
Formula (7.6)
Integral Calculus
Formula (7.6)
d
arccos(u)
dx
Integral Calculus
d
arctan(x)
dx
Integral Calculus
(−∞, ∞)
[0, π]
|x| > 1
π π
(− , )
2 2
d
1
arcsin(x) = √
dx
1 − x2
d
−1
arccos(x) = √
dx
1 − x2
d
1
arctan(x) =
dx
1 + x2
from 0 to π except π/2
d
1
du
arcsin(u) = √
dx
1 − u2 dx
d
−1 du
arccos(u) = √
dx
1 − u2 dx
Formula (7.6)
Formula (7.6)
d
arctan(u)
dx
d
arcsec(x)
dx
Integral Calculus
Formula (7.6)
Integral Calculus
Match It!(7.6)
d
arcsec(u)
dx
Graph of y = arcsin(x)
Integral Calculus
Match It!(7.6)
Integral Calculus
Match It!(7.6)
Graph of y = arccos(x)
Graph of y = arctan(x)
Integral Calculus
Match It!(7.6)
Integral Calculus
Match It!(7.6)
Graph of y = arcsec(x)
Graph of y = arcsin(x − 2)
Integral Calculus
Match It!(7.6)
Integral Calculus
Match It!(7.6)
Graph of y = arccos(x − 2)
Integral Calculus
Graph of y = arctan(x − 2)
Integral Calculus
d
1 du
arctan(u) =
dx
1 + u2 dx
d
1
arcsec(x) = √
dx
|x| x2 − 1
FILL IN
d
1
du
arcsec(u) = √
dx
|u| u2 − 1 dx
FILL IN
FILL IN
FILL IN
FILL IN
FILL IN
FILL IN
Match It!(7.6)
Match It!(7.6)
Graph of y = arcsec(x − 2)
Graph of y = 3 arcsin(x)
Integral Calculus
Match It!(7.6)
Integral Calculus
Match It!(7.6)
Graph of y = 3 arccos(x)
Graph of y = 3 arctan(x)
Integral Calculus
Match It!(7.6)
Integral Calculus
Match It!(7.6)
Graph of y = 3arcsec(x)
Graph of y = 3 + arcsin(x)
Integral Calculus
Match It!(7.6)
Integral Calculus
Match It!(7.6)
Graph of y = 3 + arccos(x)
Graph of y = 3 + arctan(x)
Integral Calculus
Match It!(7.6)
Integral Calculus
Match It!(7.6)
Graph of y = 3 + arcsec(x)
Integral Calculus
Graph of y = −3 + arcsin(x)
Integral Calculus
FILL IN
FILL IN
FILL IN
FILL IN
FILL IN
FILL IN
FILL IN
FILL IN
FILL IN
FILL IN
Match It!(7.6)
Match It!(7.6)
Graph of y = −3 + arccos(x)
Graph of y = −3 + arctan(x)
Integral Calculus
Match It!(7.6)
Integral Calculus
Match It!(7.6)
Graph of y = −3 + arcsec(x)
Compute lim + arcsin(x)
x→−1
Integral Calculus
Match It!(7.6)
Integral Calculus
Match It!(7.6)
Compute lim − arcsin(x)
x→−1
Compute lim arcsin(x)
x→0
Integral Calculus
Match It!(7.6)
Integral Calculus
Match It!(7.6)
Compute lim + arccos(x)
x→−1
Compute lim − arccos(x)
x→−1
Integral Calculus
Match It!(7.6)
Integral Calculus
Match It!(7.6)
Compute lim arccos(x)
x→0
Integral Calculus
Compute lim arctan(x)
x→−∞
Integral Calculus
FILL IN
FILL IN
FILL IN
Draw the graph of y = arcsin(x).
Draw the graph of y = arcsin(x).
Does not make sense!
graph of y = arcsin(x).
Draw the
Draw the graph of y = arccos(x).
Draw the graph of y = arccos(x).
π
2
Draw the graph of y = arctan(x).
Draw the graph of y = arccos(x).
−
Match It!(7.6)
Match It!(7.6)
Compute lim arctan(x)
x→∞
Compute lim arctan(x)
x→0
Integral Calculus
Math Flash cards
Integral Calculus
Formula (5.2)
From Calculus I
Definition of Definite Integral of
continuous function
Integral Calculus
Integral Calculus
Math Flash cards
Math Flash cards
Differential Calculus Flash Cards
Changes
Integral Calculus
Definition (Calculus I)
f 0 (x)
Differential Calculus
Constant Multiple Rule (Calculus I)
d
cf (x)
dx
Differential Calculus
Integral Calculus
Constant Rule (Calculus I)
d
c
dx
Differential Calculus
Sum Rule (Calculus I)
d
[f (x) + g(x)]
dx
Differential Calculus
0
Draw the graph of y = arctan(x).
Zb
f (x) dx = lim
n→∞
a
n
X
f (ck )∆xk
i=1
Mark Stankus has modifed these flash cards to be specific to the course Math 142 at the California Polytechnic State University of San Luis Obispo.
c
Copyright 2009
Djun. M. Kim. This work is licensed under a Creative
Commons Attribution-Noncommercial-Share Alike 2.5 Canada License.
http://viu.ca/math
http://teaching.puregin.org/math-flash-cards
π
2
Draw the graph of y = arctan(x).
I hope that these flashcards help.
These cards are part of a project to create free, high
quality, printable flash cards for mathematics study.
You are free to share, modify, and distribute these
for non-commercial use, as long as you mention the
source URL. Suggestion for additional material and
corrections are welcome.
c
Copyright 2009
Djun. M. Kim. This work is licensed under a Creative
Commons Attribution-Noncommercial-Share Alike 2.5 Canada License.
http://viu.ca/math
0
http://teaching.puregin.org/math-flash-cards
f (x + h) − f (x)
h→0
h
lim
This only works for constants c.
cf 0 (x)
f 0 (x) + g 0 (x)
This only works for constants c.
Difference Rule (Calculus I)
d
[f (x) − g(x)]
dx
Differential Calculus
Quotient Rule (Calculus I)
d f (x)
dx g(x)
Differential Calculus
Derivative (Calculus I)
d n
[x ]
dx
Differential Calculus
Derivative (Calculus I)
d
cos x
dx
Differential Calculus
Derivative (Calculus I)
d
csc x
dx
Differential Calculus
Product Rule (Calculus I)
d
[f (x)g(x)]
dx
Differential Calculus
Chain Rule (Calculus I)
d
[f (g(x))]
dx
Differential Calculus
Derivative (Calculus I)
d
sin x
dx
Differential Calculus
Derivative (Calculus I)
d
sec x
dx
Differential Calculus
Derivative (Calculus I)
d
tan x
dx
Differential Calculus
f 0 (x)g(x) + f (x)g 0 (x)
f 0 (x) − g 0 (x)
f 0 (g(x))g 0 (x)
f 0 (x)g(x) − f (x)g 0 (x)
[g(x)]2
cos x
nxn−1
This only works for constants n. You learned this in
Calculus I for rational numbers n. We will learn this
in Calculus II for irrational numbers n, like π.
sec x tan x
− sin x
sec2 x
− csc x cot x
Derivative (Calculus I)
d
cot x
dx
Derivative (7.3)
d x
e
dx
Differential Calculus
Derivative (7.3)
(base e)
Differential Calculus
Derivative (7.2)
d x
a
dx
d
ln x
dx
(base a)
Differential Calculus
Derivative (7.3)
Differential Calculus
First Step (7.3)
d
loga x
dx
d
(cos(x))x
dx
Differential Calculus
Definition (7.2)
Differential Calculus
Formula (7.2)
d
ln(x)
dx
ln(x)
Differential Calculus
Formula (7.2)
Differential Calculus
Formula (7.2)
d
ln(u)
dx
Differential Calculus
d
ln(−x)
dx
Differential Calculus
ex
− csc2 x
1
x
ax ln a
This only works when a is a constant.
Use
• cos(x) = eln(cos(x))
• laws of exponents.
d
1
ln(x) =
dx
x
1
x ln a
This only works constants a and
a 6= 1.
Zx
ln(x) =
1
dt
t
1
d
1
ln(−x) =
dx
x
d
1 du
ln(u) =
dx
u dx
Formula (7.2)
Match It!(7.2)
d
ln(|x|)
dx
Domain of ln(x)
Differential Calculus
Match It!(7.2)
Differential Calculus
Match It!(7.2)
Domain of ln(u)
Domain of ln(−x)
Differential Calculus
Match It!(7.2)
Differential Calculus
Match It!(7.2)
Domain of ln(|x|)
Domain of ln(|u|)
Differential Calculus
Match It!(7.2)
Differential Calculus
Match It!(7.2)
Graph of y = ln(x)
Graph of y = ln(−x)
Differential Calculus
Match It!(7.2)
Differential Calculus
Match It!(7.2)
Graph of y = ln(|x|)
Differential Calculus
Compute lim ln(x)
x→−∞
Differential Calculus
x>0
d
1
ln(|x|) =
dx
x
x<0
u>0
u 6= 0
x 6= 0
FILL IN
FILL IN
FILL IN
Does not make sense!
Draw the graph of y = ln(x) to
memorize this.
Match It!(7.2)
Match It!(7.2)
Compute lim ln(x)
x→∞
Compute lim+ ln(x)
x→0
Differential Calculus
Match It!(7.2)
Differential Calculus
Match It!(7.2)
Compute lim− ln(x)
Compute lim ln(−x)
x→−∞
x→0
Differential Calculus
Match It!(7.2)
Differential Calculus
Match It!(7.2)
Compute lim ln(−x)
x→∞
Compute lim+ ln(−x)
x→0
Differential Calculus
Match It!(7.2)
Differential Calculus
Match It!(7.2)
Compute lim− ln(−x)
Compute lim ln(|x|)
x→−∞
x→0
Differential Calculus
Match It!(7.2)
Differential Calculus
Match It!(7.2)
Compute lim ln(|x|)
x→∞
Differential Calculus
Compute lim+ ln(|x|)
x→0
Differential Calculus
−∞
∞
Draw the graph of y = ln(x) to
memorize this.
Draw the graph of y = ln(x) to
memorize this.
∞
Does not make sense!
Draw the graph of y = ln(−x) to
memorize this.
Draw the graph of y = ln(x) to
memorize this.
Does not make sense!
Does not make sense!
Draw the graph of y = ln(−x) to
memorize this.
Draw the graph of y = ln(−x) to
memorize this.
∞
−∞
Draw the graph of y = ln(|x|) to
memorize this.
Draw the graph of y = ln(−x) to
memorize this.
−∞
∞
Draw the graph of y = ln(|x|) to
memorize this.
Draw the graph of y = ln(|x|) to
memorize this.
Match It!(7.2)
Definition (7.2)
e
Compute lim− ln(|x|)
x→0
Differential Calculus
Differential Calculus
Definition (7.3)
Formula (7.3)
d x
e
dx
ex
Differential Calculus
Differential Calculus
Formula (7.3)
Match It!(7.3)
d u
e
dx
Domain of ex
Differential Calculus
Match It!(7.3)
Differential Calculus
Match It!(7.3)
Range of ex
Graph of y = ex
Differential Calculus
Match It!(7.3)
Differential Calculus
Match It!(7.3)
Graph of y = e2x
Differential Calculus
Graph of y = ex+4
Differential Calculus
−∞
The number e is defined such that
ln(e) = 1.
d x
e = ex
dx
Draw the graph of y = ln(|x|) to
memorize this.
The function which is inverse to
y = ln(x).
d u
du
e = eu
dx
dx
(−∞, ∞)
FILL IN
(0, ∞)
FILL IN
FILL IN
Match It!(7.3)
Match It!(7.3)
Graph of y = ex−4
Graph of y = 2 + ex
Differential Calculus
Match It!(7.3)
Differential Calculus
Match It!(7.3)
Graph of e−x
Compute lim ex
x→−∞
Differential Calculus
Match It!(7.3)
Differential Calculus
Match It!(7.3)
Compute lim ex
x→∞
Differential Calculus
Match It!(7.3)
Compute lim− ex
x→0
Differential Calculus
Compute lim+ ex
x→0
Differential Calculus
FILL IN
FILL IN
FILL IN
0
Draw the graph of y = ex to memorize it.
1
∞
Draw the graph of y = ex to memorize it.
Draw the graph of y = ex to memorize it.
1
Draw the graph of y = ex to memorize it.