Name: ____________________
Date: ____________ Period: __
Memory Quiz #10:
Differentiation Rules:
Basic Derivatives:
Product Rule:
1)
!
!"
($%) =
4)
uE tvE x
5)
!
!"
$
( )=
%
6)
VU
2)
VITY
!
!"
or
UV
!
(()*")
!"
!
!"
Quotient Rule:
7)
u.VE orVu uI2
8)
!
!"
!
!"
9)
!
!"
3)
!"
0 $
=
Csc2X
(+,$) =
(-.,") =
(/$ ) =
(", ) =
Tou
COSX
e
4
U
n I
n
(-/(") =
secxtanxseczxfkuj.CI
Chain Rule:
!
=
10)
!"
11)
DX
12)
U
u
!
!"
or
f
!
!
!"
(*1,") =
(()-") =
Sinx
((-(") =
CSCxcotX
More Derivatives:
13)
15)
17)
19)
!
!"
()*23 " =
!
!"
$
*1,23 1
!
!"
(-(23 " =
!
!"
$
-.,23 1
14)
!
!"
()-23 " =
16)
!
!"
$
-/(23 1
l t X2
a
=
42
92
U
18)
I
7
1 1
=
20)
l
U
TE
Equation of a Tangent Line:
You need a __________
Slope and a
_________.
point
Equation:
21)
y y
_m
X
X
! $
1
!"
!
!"
=
I
VE
=
A
+)41 " =
a
U
ful
Ina
U
I
Xina
Alternate Definition of the Derivative:
22)
05 ( = lim
x
c
fCx
X
Fcc
C
First Derivative Test
Distance, Velocity, & Acceleration:
36)
f(x) is decreasing when:
23)
f
24)
Critical Values at x:
X
O
or
Relative Minimum:
of
f
or
D
x
DNE
28)
DN E
to
changes from
t
Ct
to C
Absolute Max or Min:
Must check
39)
The integral of acceleration (ft/sec2) is:
40)
The derivative of velocity (ft/sec) is:
41)
The integral of velocity (ft/sec) is:
f’’(x) < 0 then f(x) is:
concave
32)
33)
up
Point of Inflection:
f
o
34) Relative
F
O
X
0
Exponential Growth & Decay:
35)
y=
C
Kt
is
speed
46)
increasing
The particle is moving left when:
47)
The particle is moving right when:
velocity
48)
x
negative
is
velocity
49)
decreasing
If acceleration and velocity have the
same signs, then:
is
positive
Accumulation =
ft
o
t
dt
Average Velocity =
final position initial position
total time
Maximum:
X
velocity
If acceleration and velocity have
different signs, then:
the speed is
and sign changes
Relative Minimum:
f
Speed is:
45)
Down
f’’(x) > 0 then f(x) is:
ft Isac
Velocity
the
Second Derivative Test:
ft
The derivative of position (ft) is:
43)
endpoints
y
31)
ft 1 see
Position
44)
The maximum value is a ____ value.
concave
ft Isac
Velocity
42)
29)
30)
function
position
x(t) =
acceleration
X changes from
27)
Relative Maximum:
DNE
o or
of x
flex
function
acceleration
a(t) =
38)
o
x
F
26)
37)
f(x) is increasing when:
F
25)
LO
X
velocity position
v(t) =
50)
Displacement =
time
final
Initial time
51)
dt
V E
Total Distance =
time
final
initial time
V
t
dt
St
Riemann Sums
52)
The Fundamental Theorem of Calculus:
A Riemann Sum means:
70)
Approximation
Rectangular
DO NOT:
integral
the
evaluate
Instead:
add up the area of rectangles
Average Rate of Change (ARoC):
53)
msec =
fca
f b
b
54) Instantaneous
thigh
x
∫ -/(8 $ !$ =
57)
ta
8
56)
∫ (-( $ !$ =
,
∫ $ !$ =
U
Ntl
59)
60)
61)
∫ -., $ !$ =
∫
∫
1$ !$
!$
$
an
=
=
flxth
tC
fC
h
∫ /$ !$ =
65)
71)
U
Derivative of an Inverse Function:
72)
If f has an inverse function g then:
4 " =
I
t
C
f
n t
73)
C
74)
75)
SinutC
Inlcosul
1h
66)
∫ -/( $ *1, $ !$ =
67)
∫ !$ =
Sinn
∫ (-( $ !$ =
69)
∫ (-( $ ()* $ !$ =
C
tatan
!$
∫
$ $8 218
!$
∫
18 2$8
=
=
at
to
sea
sin
to
Itc
Trapezoidal Rule
76)
For uneven intervals,
to calculate area of
you may need at
a time and total
one trapezoid
for approximation
ATRAP=
secutC
77)
In ICSCutcotultC
CSC U
!$
∫ 18 9$8 =
to
Utc
68)
glx
More Integrals:
i
e Utc
∫ -/( $ !$ =
In Isecuttanultc
∫ ()* $ !$ =
f glu
5
COS Utc
∫ ()- $ !$ =
63)
x
Corollary to FTC:
InlultC
∫ *1, $ !$ =
64)
nu
Tna
62)
_f
x
a
CotutC
htt
58)
F
Where
F
F b
1
Rate of Change (IRoC):
Basic Integrals:
55)
: 0 " !" =
! 4($)
:
0 * !* =
!" 1
a
6*1, =
F
;
C
Eh
b
For even intervals:
=
∫< > ? @? ≈
tbd
bzna Yot2yit2yzt
2yn ityn
Area and Solids of Revolution:
Note: (a,b) are x-coordinates and
(c,d) are y-coordinates
Volume by Washer Method:
78) About x-axis:
V=
tfb
rCxD2d
Rc D2
Volume by Cross Sections:
General equations for known cross section
where base is the distance between two
curves and a and b are the limits of
integration.
89)
Squares:
V=
f
About y-axis:
V=
79)
ftp.pcyDZ lrCYDZDY
80) About
V=
f
x
base ohd
Where h is the height of the rectangles.
Equilateral:
V=
81)
21T
Triangles:
Rectangles:
V=
y RCyDdy
About y-axis:
V=
2dX
90)
Volume by Shell Method:
x-axis:
21T
base
91)
RCxDdx
base 2dx
Isosceles Right:
V=
92)
Area Between Two Curves:
Slices ^ to x-axis:
A=
82)
I
83)
fix
d
gc
Semi Circles:
V=
radius 2dx
El
Where radius is ½ distance between the two
fly glyDdy
Volume by Disk Method:
About x-axis:
V = IT
2dx
curves.
84)
Horizontal Asymptotes:
If the largest exponent in the numerator
is > the largest exponent in the denominator
then
94)
Rex
About y-axis:
V=
rly 2dy
IT
lim >(?) =
E→±H
Curve Sketching Sign Chart
86)
f
U
n
i
87)
f’
t
f’’
DNE
If the largest exponent in the numerator
is = the largest exponent in the denominator
then
95)
lim >(?) =
E→±H
e
b
If the largest exponent in the numerator
is < the largest exponent in the denominator
then
96)
r
is
t
lim >(?) =
E→±H
88)
base 2dx
93)
Slices ^ to y-axis:
A=
85)
It
O
Rolle’s Theorem
Extreme Value Theorem:
If the function f(x) is
97)
If the function f(x) is
98)
aib
Differentiable on Aib
f a f b
Continuous on
Continuous on
Then
Then
the function is guaranteed to have
an absolute maximum and absolute
minimum on the interval
there exists at least one
number X C in Ca b such
that f c 0
Intermediate Value Theorem:
Mean Value Theorem
If the function f(x) is
99)
Continuous
y
is between
f
a
Continuous
and fcb
y
b
Pythagorean Identities:
Definition:
Ln 1 = 0
Ln e =
102)
106)
107)
108)
Ln (M/N) =
103)
Ln M
104)
109)
Ln (MN) =
LnMtLnN
Sin2ftcos20
Cotto I
Ittanzo
=1
111)
= csc q
2
112)
= sec2q
110)
113)
COS2X
I
Sin 2x
2sin2X
Ill
cos x =
tz
2
cos 2x
It cos 2x
cotq =
114)
tanq =
Sino
cost
Sino
Sin(2x) =
Cos(2x) =
sin2x =
Quotient Identities:
2sinxcosx
Ln MP
105)
a
Power-Reducing Formulas:
Double Angle Formulas:
Ln N
p • Ln M =
Ca b
Trigonometric Identities:
Logarithms:
101)
on
Then
There exists at least one
number X C in Ca b that
f c fcb fca
there exists at least one number
X c in the open interval a b
Ln N = p Û ep = N
a b
on
Differentiable
Then
such that fcc
If the function f(x) is
100)
b
a
on
aib
cost
Reciprocal Identities:
115)
sec x =
I
cos X
116)
csc x =
Sinx
Unit Circle:
Radian Measure
tan N
cosN
117)
π
O
8
O
P
O
I
O
Q
118)
I
120)
124)
126)
E
Iz
129)
132)
0
127)
I
125)
E
128)
B
130)
133)
122)
ordne
x
123)
119)
0
121)
O
sin N
131)
I
O
I
I
E
2
134)
O
O
Parent Functions:
135)
0 " = "I
136)
0 " =
3
"8
0 " = "
137)
138)
0 " = /"
f
0 " = |"|
140)
0 " ="
141)
n
r
142)
i
0 " =
3
"
w
T
144)
0 " = LM "
145)
f
0 " = "8
V
0 " = ()-"
f
L
143)
18 − " 8
n
n
139)
0 " =
146)
0 " = -.,"
I