A B BC
B
a definitive sheet
by chad valencia, ucla mathematics major
version 2.0.2000, rev 1
c
a
x
A
b
C
AB
personal notes
trig in a nutshell
calculus
sin x =a/c= opposite/hypotenuse
cos x = b/c = adjacent/hypotenuse
sec x = c/b = hypotenuse/adjacent
csc x = c/a = hypotenuse/opposite
tan x = a/b = sin x/cos x = opposite/adjacent
cot x = b/a = cos x/sin x = adjacent/opposite
y
Range:
( - ¥, k ]
-¥ < y £ k
Sample Function f(x)
absolute maximum
Odd/Even Identities
sin (- x) = - sin x
cos (- x) = cos x
tan (- x) = - tan x
cot (- x) = - cot x
sec (- x) = sec x
csc (- x) = - csc x
k
3 step test for continuity:
relative maximum
1. f(c) exists
f '(x)=0
2. lim exists
x->c
Double Angle Identities
sin 2x = 2 sin x cos x
cos 2x = cos ² x - sin ² x
cos ² x = 1+ cos 2x
2
sin ² x = 1- cos 2x
2
3. lim = f(c)
inflection point
f ''(x)=0
x->c
sin ² x + cos ² x = 1
1 + tan ² x = sec ² x
1 + cot ² x = csc ² x
relative minimum
f '(x)=0
a
c
b
zero
f (x)=0
x
e
d
-
+
e
d
b
f (x) = a
e
f ''(x)
b
if: a b = x
Area = òaf (x) dx
lim (1+ 1n
log a x = b
n
n ®¥
)= e
trig derivatives
derivatives
definition of the derivative:
f ' (x) = lim f (x + h) - f (x)
h
h -> 0
Addition Rule
f ' (u + v) = f ' u + f ' v
ln (xy) = ln x + ln y
ln (x/y) = lnx - lny
ln x = n ln x
ln e = e ln x = x
ln 1 = 0 ln e = 1
1
+
c
x
f (x) = log a x
1
-
f '(x)
log x = log10x
logex = ln x
logs in a nutshell
Domain:
( - ¥, e ]
-¥ < x £ e
+
cos (a+b) = cos a cos b - sin a sin b
sin (a+b) = sin a cos b + cos a sin b
cos (a - b) = cos a cos b + sin a sin b
sin (a - b) = sin a cos b - cos a sin b
Product Rule
f ' (u v) = udv + vdu
Power Rule
f ' ( x c ) = c x c -1
Quotient Rule
v du - u dv
u
f'( )=
v²
v
(Lo D Hi minus Hi D Lo
over Lo Lo)
Chain Rule
(f o g)' = f ' g '
Inverse Trig
Standard Trig
1 du
(d/dx)(csc u) = - csc u cot u
d
-1
sin
u = 1 - u² dx
(d/dx)(sec u) = sec u tan u
dx
(d/dx)(cot u) = - csc ² u
1
-1
d
tan u = 1 + u² du
(d/dx)(tan u) = sec ² u
dx
dx
(d/dx)(cos u) = - sin u
1
du
-1
d
(d/dx)(sin u) = cos u
dx sec u = |u| u² - 1 dx
volumes & areas
transcendental derivatives
l’Hôpital’s Rule
When
Mean Value Theorem
f (b) - f (a)
= f ' (c)
b-a
¥
0
= 0 OR ¥
f ' (x)
lim f (x) = xlim
® a g ' (x)
x®a
lim
x®a
f (x)
g(x)
d
dx
ln u
d u
a
dx
g(x)
integrals
Second Fundamental
Theorem of Calculus
(Leibniz's Rule)
b
òaf (x) dx = F(a) - F(b)
d
dx
where F is the
antiderivative of f
ò
v (x)
f (t) dt
u (x)
= f (v) dv - f (u) du
dx
dx
Trapezoidal Rule
T = b - a (y0 + 2y1 + 2y2+ ... + 2yn - 1+ yn )
2n
a & b = bounds
n = number of intervals
(Use h(b1+b2)/2 for
trapezoids of
different height)
(use b x h for approximation)
RRAM
MRAM
Power Rule:
a
òx dx = xa+1 +C
a+1
x¹-1
Average Value
b
Avg. (f (x)) = 1 . f (x) dx
b-a
ò
a
du
dx
dx
ln a
Standard Trig
ò
ò
òu
Alternating Signs
Tabular Integration
ò [(algebraic)(trigonometric/e)]dx
ex: ò 2x³cosx dx
+ cosx
2x³
- sinx
6x²
12x
+ -cosx
12
- -sinx
cosx
0
2x³sinx + 6x²cosx - 12x sin x - 12 x cos x +C
d
du
u² - a²
=
1
a
-1
sec |
u
|+
a
transcendental integrals
du
òu =
ln |u| + C
u
u
ò ln x dx = x ln x - x + C
ò e du = e + C
u
u
ò a du = lnaa+ C
ò
px + q = A
+
(x+a)(x+b) (x+a)
B
(x+b)
px + q = A +
B
(x+a)2
(x+a)
(x+a)2
px2 - qx + r
=
A + Bx + C
(x+a)(x2+bx+c)
(x+a) (x2+bx+c)
3
3
SA(sphere) = 4 p r ²
A
=s²4 3
velocity & motion
ò
ò
a(t)
s(t) or x(t)
position
acceleration
v(t)
d
velocity
d
the velocity equation
s(t) = ½ g t ² + vO t + sO
g = - 32 ft / s ², - 9.8 m / s ²
disc & shell methods
Volume
Disc
(no hole)
C
partial fractions
integration by parts
òudv = uv - òvdu
Logarithmic
Inverse Trig
Algebraic
Trigonometric
Euler's Constant (e)
=a
e
ò sin x dx = - cos x + C
ò tan x dx = - ln |cos x| + C
ò cos x dx = sinx + C
ò cot x dx = ln |sin x| + C
ò sec² x dx = tan x + C
ò sin² x dx = x - sin 2x + C
2
4
ò csc² x dx = - cot x + C
ò cos² x dx = x + sin 2x + C
ò sec x tan x dx = sec x + C
2
4
ò csc x cot x dx = - csc x + C
Inverse Trig
du
-1
du
-1 u
=1
tan au + C
a
a² - u² = sin a + C
a² + u²
Rectangular Approximation Methods (RAM)
Priority:
u
u du
trigonometric integrals
First Fundamental
Theorem of Calculus
LRAM
d u
e =
dx
1 du
= u dx
V(cone) = 1 p r ² h
V(sphere) = 4 p r ³
Disc
w/ Hole
Shell
d
b
b
2
2
ò r dx pò(R - r )dx 2pòr h dy
Y-Axis pò r dy pò(R - r )dy 2pòr h dx
X-Axis
2
p
a
a
d
b
2
2
c
c
d
2
a
c
r = radius R = Outside radius
r = inside radius
r = radius
h = height
trig substitutions
Use:
If you see:
2
2
a +x
x = a tan q
2
2
a -x
x = a sin q
2
2
x -a
x = a sec q
solved through trig substitution:
òsec u du = ln |sec u + tan u| + C
personal notes
A B BC
improper integrals
¥
1
calculus
Comparison Test
P Series Test:
1
xp
a definitive sheet
by chad valencia, ucla mathematics major
version 2.0.2000, rev 1
if f(x) < convergent function,
f(x) is convergent
Converges if p > 1
Diverges if 0 < p < 1
if f(x) > divergent function,
f(x) is divergent
Limit Comparison Test:
Let f(x) be a known convergent or divergent function:
lim f(x)= L 0 < L < ¥
x ® ¥ g(x)
f(x) & g(x) both converge or both diverge
¥
Greek letter
sigma
(sum of)
sequences
Limits of Common Sequences:
lim ln n =
Convergence/Divergence:
Let L be a finite number
n
lim
lim an = L
n ®¥
lim
x
=
n!
n=0
0
n
n ®¥
1
n
lim (1+ 1n
n
n ®¥
lim (1+
n ®¥
x
n
Geometric Series
)= e
n
)= e
b
a
dy
) dx
1+(dx
2
Arc Length (y-axis):
d
dx 2
+ dy dy
c
1( )
Surface Area:
b
2
dx
2p ay 1+ (dy
dx)
)
Sn= a(1-r
(1-r)
Work
Work = Force x Distance
= Density x Volume x Distance
= Density x Sum of Areas x Distance
Hooke’s Law for Springs
Force = f(x) = kx
(k is a constant, x is the distance)
Work = ò f(x)dx, from position A to position B
T - T S= (T0 - TS )e
T = temperature of object at a given time
Ts = temperature of surroundings
To = temperature at time zero
t = time
Polar
Basic Shapes
(pink = cosine, blue = sine)
Circles
Lemniscates
Spiral of
Archimedes
r = a cos q
r² = a cos q
r = aq
r = a sin q
r² = a sin q
¥
2 x ( ) ( )
lim San= r
Speed
dy
(dx
dt) + (dt)
2
2
r = nlim
®¥
Velocity Equation for Vectors:
(-½ g t² + S0)j + V0 t
where v0 = #( cos t i + sin t j)
# = initial velocity/muzzle speed
Rectangular
x
n
an
3.
Power Series
1. Use ratio test on the
absolute value of the series.
Converges if r < 1
No conclusion if r = 1 2. Set r < 1 to find the interval
3. To check bounds, plug
Divergent if r > 1
into original equation
4. Take ½ of the interval
to find radius of convergence
Error (Alternating Series/Taylor)
Error = Actual - Approximate
Error < First Unused Term
taylor/maclaurin series
Polar Conversion
(x,y) <-> (r,q)
x² + y² = r²
y
x = r cos q tan q =x
y = r sin q
(x,y)
Root Test
an > an+1 for all n.
lim an = 0
n ®¥
2.
Sbn
an+1
r = nlim
®¥
an
n-1
(-1) an
n=1
Converges if:
1. All terms are positive
0<r<¥
San & Sbn
both converge or both diverge
Ratio Test
0
Alternating Series Test (AST)
Let Sbn be a known
convergent or divergent series:
Vector Valued Functions
r(t) = x(t)i + y(t)j
r'(t) = x'(t)i + y'(t)j
òr(t)dt = (òx(t)dt + c)i + (òy(t)dt + c)j
y
k
Limit Comparison Test
lim an ¹
n ®¥
series is divergent
Comparison Test
if San < convergent series,
San is convergent
if San > divergent series,
San is divergent
If integral converges, series converges
If integral diverges, series diverges
n ®¥
Unit Vector:
v
||v||
an
n=k
ò ax dx
n=k
T(t) = r ' (t)
If T = u1i + u2j
||r ' (t)|| N = -u2i + u1j
- kt
n=1
¥
an
( ) ( )
Unit Tangent and
Unit Normal Vectors
||v|| = a² + b²
Newton’s Law of Cooling
¥
vectors
Vector Length
(magnitude/norm):
n-1
Integral Test
( )
Notation
v = ai + bj
<a,b>
¥
a
a r = 1-r
|r| <1
if |r| > 1, series diverges
First Derivative: Second Derivative:
Arc Length:
b
dy
dy
dx 2 dy 2
d
dt
dx
dt + dt
dy = dt
d²y =
a
dx
dx²
dx
dx
dt
dt
Surface Area (x-axis):
Surface Area (y-axis):
b
b
dx 2 dy 2
dx 2 dy 2
dt
dt
+
dt + dt
dt
dt
a
a
2 y ( ) ( )
¥
n
x
Nth Term Test for Divergence
Given:
If:
Infinite Series:
Finite Series:
extraneous bc concepts parametric
Arc Length (x-axis):
first number
n ®¥
n ®¥
lim an¹ L
n
series
n ®¥
n ®¥
Divergent
sequence
0
n
|x|<1
x = 0(fraction)
lim n n= lim n = 1
n ®¥
Convergent
Sa
BC
last number
Taylor Polynomial
P(x) = f (a) + f ' (a)(x - a) + f ''(a)(x - a)2 + f ''' (a)(x - a)3 + … + f n (a)(x - a)n
2!
3!
n!
¥
n
n
- a)
S f ( a)(x
n!
n =0
Polar
Limacons
Limacons
Cardioids
w/ Dimple
w/ Inner Loop
r = a + b cos q
r = a + b cos q r = a + b cos q r = a + b sin q
r = a + b sin q r = a + b sin q
|a|=|b|
|a|>|b|
|a|<|b|
Common MacLaurin Series
(r,q)
¥
2
4
6
( - 1)n x2n
cos x = 1- x + x - x + …=
(2n)!
n =0
2! 4! 6!
r
q
Polar Slope
r ' sin q + r cos q
r ' cos q - r sin q
S
sin x = x - x + x - x + … = ( - 1) x
S (2n+1)!
3! 5! 7!
e = 1 + x + x² + x³+... =
S xn!
2! 3!
¥
3
5
n
n =0
¥
x
Polar Area:
1 b2
dq
2 ar
r = a cos bq
r = a sin bq
Roses
If b is odd, b = number of petals
If b is even, 2b = number of petals
Polar Surface Area (x-axis):
b
2
dr 2
2p r sin q r + dq dq
a
Polar Arc Length:
b
2
dr 2
r+ dq dq
a
( )
( )
Polar Surface Area (y-axis):
b
2
dr 2
dq
2p a r cos q r + (dq)
hyperbolic trig functions
Every function splits
into Even and Odd parts:
f(x) = f(x) + f(-x) + f(x) - f(-x)
2
2
even
odd
cosh x = e x + e x
2
sinh x = e x - e x
2
tanhx = e x - e x
ex + e -x
sech x =
2
e x + ex
csch x =
2
e x - ex
cothx = e x + e x
ex - e -x
2n+1
7
sinh 2x = 2 sinh x cosh x
cosh 2x = cosh ² x - sinh ² x
cosh ² x = cosh 2x+1
2
sinh ² x = cosh 2x - 1
2
cosh² x - sinh² x = 1
tanh² x + sech ² x = 1
coth² x - csch² x = 1
Standard Hyperbolic Trig Derivatives
(d/dx)(csch u) = - csch u coth u (du/dx)
(d/dx)(sech u) = - sech u tanh u (du/dx)
(d/dx)(coth u) = - csch ² u (du/dx)
(d/dx)(tanh u) = sech ² u (du/dx)
(d/dx)(cosh u) = sinh u (du/dx)
(d/dx)(sinh u) = cosh u (du/dx)
n
n =0
¥
S
1 = 1 + x + x² + x³+... =
xn
1- x
n =0
¥
S
1 = 1 - x + x² - x³+... =
(-1)n xn
1+x
n =0
Inverse Hyperbolic Trig Derivatives
1 du
d
-1
dx cosh u = u² - 1 dx , u > 1
1
du
d
-1
dx sinh u =u² + 1 dx
1
-1
d
tanh u = 1 - u² du , |u|<1
dx
dx
1 du
-1
d
coth
u
=
, |u|>1
dx
1 - u² dx
1
du
-1
d
dx sech u =u u² - 1 dx , 0<u<1
1
du
-1
d
dx csch u =|u| u² - 1 dx , u ¹0
Hyperbolic Trig Integrals
ò sinh u du = cosh u + C
ò cosh u du = sinh u + C
ò sech² u du = tanh u + C
ò csch² u du = - coth u + C
ò sech x tanh u du =- sech u + C
ò csch x coth u du = - csch u + C
© 2000 Chad A. Valencia. All Rights Reserved.